Assessing the progress of trapped-ion processors towards fault-tolerant quantum computation

TL;DR

This paper introduces a fair criterion for quantum error correction, benchmarks trapped-ion processors using topological color codes, and guides future hardware improvements towards achieving fault-tolerance in quantum computing.

Contribution

It presents a new criterion for assessing quantum error correction progress and evaluates trapped-ion processors, offering detailed hardware models and realistic error analysis.

Findings

Two-species trapped-ion crystals are promising for fault-tolerant quantum computation.

The proposed criterion helps benchmark and guide hardware development.

Realistic error models improve the assessment of QEC performance.

Abstract

A quantitative assessment of the progress of small prototype quantum processors towards fault-tolerant quantum computation is a problem of current interest in experimental and theoretical quantum information science. We introduce a necessary and fair criterion for quantum error correction (QEC), which must be achieved in the development of these quantum processors before their sizes are sufficiently big to consider the well-known QEC threshold. We apply this criterion to benchmark the ongoing effort in implementing QEC with topological color codes using trapped-ion quantum processors and, more importantly, to guide the future hardware developments that shall be required in order to demonstrate beneficial QEC with small topological quantum codes. In doing so, we present a thorough description of a realistic trapped-ion toolbox for QEC, and a physically-motivated error model that goes beyond standard simplifications in the QEC literature. Our large-scale numerical analysis shows that two-species trapped-ion crystals in high-optical aperture segmented traps, with the improvements hereby described, are a very promising candidate for fault-tolerant quantum computation.

Tables

Table 1. Table 1: Protocol for assessing the beneficial role of QEC.

Step	Action
1	We select a qubit state at random.
2	We require Alice to encode it into the $n$ physical
	qubits of the code. She does so perfectly.
3	The $n$ physical qubits are subjected to
	environmental noise for a time $\tau /2$.
4	Optionally, Igor is asked to apply a full round of
	imperfect error correction.
5	The $n$ physical qubits are subjected to
	environmental noise for a further time $\tau /2$.
6	Bob takes the final state of the $n$ qubits, and performs
	an analysis so as to make his best guess of the state.
	He does so perfectly.

Table 2. Table 2: Current and anticipated gate-operation infidelities and durations . Single-qubit operations are a 90 degree rotation on the Bloch’s sphere, whole entangling operations correspond to fully entangling Mølmer-Sørensen operations (see Sec.  III ). The reported dual-species operations have been performed in a 9 Be + – 40 Ca + crystal. For the parameter marked by the ⋆ ⋆ \star symbol, i.e. the anticipated value of the qubit reset fidelity, numerical simulations were performed both for the value 5 ⋅ 10 − 3 ⋅ 5 superscript 10 3 5\cdot 10^{-3} and the value 1 ⋅ 10 − 4 ⋅ 1 superscript 10 4 1\cdot 10^{-4} .

Operation	Current	Current	Anticipated	Anticipated
	duration	infidelity	duration	Infidelity
Single-qubit gates	5$\mu$s	$5\cdot {10}^{-5}$	1$\mu$s	$1\cdot {10}^{-5}$
Entangling (2 qubits)	40$\mu$s	$1\cdot {10}^{-2}$	15$\mu$s	$2\cdot {10}^{-4}$
Entangling (5 qubits)	60$\mu$s	$5\cdot {10}^{-2}$	15$\mu$s	$1\cdot {10}^{-3}$
Dual species	60 $\mu$s	$3\cdot {10}^{-2}$	15 $\mu$s	$4\cdot {10}^{-4}$
entangling (2 qubits)
Dual species	80 $\mu$s	$5\cdot {10}^{-2}$	15 $\mu$s	$6\cdot {10}^{-4}$
entangling (3 qubits)
Dual species	-	-	15 $\mu$s	$2\cdot {10}^{-3}$
entangling (5 qubits)
Measurement	400$\mu$s	$1\cdot {10}^{-3}$	30$\mu$s	$1\cdot {10}^{-4}$
Re-cooling	400$\mu$s		100$\mu$s
Qubit reset	50$\mu$s	$5\cdot {10}^{-3}$	10$\mu$s	$5\cdot {10}^{-3}{}^{\star }$

Table 3. Table 3: Single qubit DES protected gates enacting net operation R ​ ( θ , 0 ) 𝑅 𝜃 0 R(\theta,0) , following  Chingiz2014 . Here, ϕ 1 = cos − 1 ⁡ ( − θ / 4 ​ π ) subscript italic-ϕ 1 superscript 1 𝜃 4 𝜋 \phi_{1}=\cos^{-1}(-\theta/4\pi) , k = arcsin ⁡ [ sin ⁡ ( θ / 2 ) / 2 ] 𝑘 𝜃 2 2 k=\arcsin[\sin(\theta/2)/2] , a 𝑎 a : amplitude noise; d 𝑑 d : detuning noise; s 𝑠 s : simultaneous amplitude and detuning noise. Unless otherwise noted the rabi rate, Ω Ω \Omega remains fixed during all segments. The Walsh modulated DCG sequence (WAMF)  Khodjasteh2010 ; SoareNatPhys2014 ; Ball2014 maintains constant segment durations, τ l subscript 𝜏 𝑙 \tau_{l} , and employs amplitude modulation of the Rabi rate as described below.

Composite pulse	Error model	(${\theta }_1,{\varphi }_1$)	(${\theta }_2,{\varphi }_2$)	(${\theta }_3,{\varphi }_3$)	(${\theta }_4,{\varphi }_4$)	(${\theta }_5,{\varphi }_5$)	(${\theta }_6,{\varphi }_6$)
SK1	$a$	($\theta$, 0)	(2$\pi ,-{\varphi }_1$)	(2$\pi ,{\varphi }_1$)	-	-	-
BB1	$a$	($\theta$, 0)	($\pi ,{\varphi }_1$)	(2$\pi ,3{\varphi }_1$)	($\pi ,{\varphi }_1$)	-	-
CORPSE	$d$	($2\pi +\theta /2-k,0$)	($2\pi -2k,\pi$)	($\theta /2-k,0$)	-	-	-
WAMF	$d$	($\Omega$, $\theta$, 0)	($\Omega /2$, $\theta /2$, 0)	($\Omega /2$, $\theta /2$, 0)	($\Omega$, $\theta$, 0)	-	-
Reduced CinSK	$s$	($2\pi +\theta /2-k,0$)	($2\pi -2k,\pi$)	($\theta /2-k,0$)	(2$\pi ,-{\varphi }_1$)	(2$\pi ,{\varphi }_1$)	-
Reduced CinBB	$s$	($2\pi +\theta /2-k,0$)	($2\pi -2k,\pi$)	($\theta /2-k,0$)	($\pi ,{\varphi }_1$)	(2$\pi ,3{\varphi }_1$)	($\pi ,{\varphi }_1$)

Table 4. Table 4: Current and anticipated metrics for different shuttling operations , carried out with 40 Ca + ions in a multilayer trap (see Sec.  III and lower inset of Fig.  1 ). The axial trap frequency is about 2 π × 2\pi\times 1.4 MHz, while the radial frequencies range around 2 π × 2\pi\times 3 MHz. The shuttling is carried out with one ion, while the other operations are carried out with two ions. Note that the 3.6  μ 𝜇 \mu s duration for low excitation shuttling is obtained with an amplitude- and phase-calibrated de-excitation kick fast_transport . Since shuttling duration will not be the bottleneck as compared to other operations, we can anticipate a slightly longer duration of 5  μ 𝜇 \mu s for similarly low excitation with a smaller calibration effort.

Operation	Shuttling (one	Separation	Rotationfast_rotation
	segment)fast_transport	/mergefast_ion_splitting
Duration	3.6$\mu$s	80$\mu$s	42$\mu$s
Excitation	0.1	6	0.3
axial (phonons)
Excitation	N/A	0.1	0.2
radial (phonons)
Anticipated duration	5$\mu$s	30$\mu$s	20$\mu$s
Anticipated excitation	0.2	1	0.2
axial (phonons)
Anticipated excitation	0.01	0.1	0.1
radial (phonons)

Table 5. Table 5: MS gate infidelities in different QEC cycle scenarios: We use our model of the MS gate ϵ = ϵ m + ϵ d + ϵ I italic-ϵ subscript italic-ϵ m subscript italic-ϵ d subscript italic-ϵ I \epsilon=\epsilon_{\rm m}+\epsilon_{\rm d}+\epsilon_{\rm I} in Eqs. ( 17 ), ( 19 ), and ( 20 ), to reproduce the current performance of 5-ion MS gates in Table  2 . Then, we use the model to predict the infidelities for the stabilizer mappings for the scenarios of QEC with a warmer ion background, and those with the expected improved conditions.

	Ion crystal reconfiguration				Spectroscopic
					decoupling
Approach
	1 species		2 species		2 species
	without re-cooling		with re-cooling		with re-cooling
5-ion MS	current	anticipated	current	anticipated	current	anticipated
infidelity	value	value	value	value	value	value
map $S_1^x$	$2.1\%$	$0.29\%$	$4.9\%$	$0.21\%$	$4.9\%$	$0.21\%$
map $S_1^z$	$5.2\%$	$0.46\%$	$4.9\%$	$0.21\%$	$4.9\%$	$0.21\%$
map $S_2^x$	$-$	$0.85\%$	$4.9\%$	$0.21\%$	$4.9\%$	$0.21\%$
map $S_2^z$	$-$	$1.5\%$	$4.9\%$	$0.21\%$	$4.9\%$	$0.21\%$
map $S_3^x$	$-$	$2.4\%$	$4.9\%$	$0.21\%$	$4.9\%$	$0.21\%$
map $S_3^z$	$-$	$3.2\%$	$4.9\%$	$0.21\%$	$4.9\%$	$0.21\%$

Table 6. Table 6: Resource overview for the different QEC protocols: We describe the number of various operations that conform each QEC cycle (i.e. measurement of all the six stabilisers ( 23 )) within the different trapped-ion protocols introduced in Sec.  VI . The number of basic operations can be obtained from the different schemes presented in the Figs.  15 - 22 . In the two rightmost columns, we present the total time required for each QEC cycle according to the current and anticipated values given in Tables  2 and  4 . Note, however, that some of the local single qubit gates require the use of refocusing techniques that combine various single-qubit gates, and that some operations can be done in parallel in different manipulation/processing regions of the ion trap. The latter can lead to a minimization of the overall time of the QEC cycle (see table  7 for the optimised QEC cycle times).

	2-ion MS	5-ion MS	Single-qubit	Meas.	Re-cooling	Split, shuttle	Rotation	Total time	Total time
	gate	gate	gate			and merge		(current)	(anticipated)
								(ms)	(ms)
Non-fault-tolerant trapped-ion QEC protocols
Shuttling-based, single-species	-	12	42	6	-	20	2	6.7	1.7
multi-qubit gate (A.1.)
Shuttling-based, two-species	-	12	42	6	6	6	2	6.8	1.4
multi-qubit gate (A.2.)
Shuttling-based, two-species	24	-	48	6	24	54	36	23.6	7.2
two-qubit gate (A.3.)
Hiding-based, two-species	-	12	150	6	6	-	-	6.3	1.1
multi-qubit gate (A.4.)
Fault-tolerant trapped-ion QEC protocols
Shuttling-based, two-species	54	-	84	24	54	190	150	71.2	22.4
DiVincenzo-Shor (B.1.)
Shuttling-based, two-species	54	-	78	24	54	190	144	71.0	22.2
DiVincenzo-Aliferis (B.2.)

Table 7. Table 7: Time required for one cycle of error correction with both current and future hardware , assuming the shuttling-based two species protocol. Parameter n 𝑛 n is the number rounds of error correction applied (usually n = 2 𝑛 2 n=2 , but n = 3 𝑛 3 n=3 when the initial two rounds disagree). Parameter m 𝑚 m is the number of additional attempts at preparing the GHZ ancilla state beyond the minimum, due to detection of error(s) when verifying the ancilla. On average, m = 0.66 𝑚 0.66 m=0.66 for current hardware and m = 0.02 𝑚 0.02 m=0.02 for future hardware.

Protocol	Total time (current) (ms)	Total time (anticipated) (ms)
non-FT	21.2	5.9
FT DVA	49.3$\times n$	14.3$\times n$
FT DVS	46.0$\times n$ + 3.7$\times m$	13.1$\times n$ + 1.0$\times m$

Equations

$$\mathcal{I}=2{\mathcal{P}}_{\mathrm{B}}({\rho }_L,{\tilde{\rho }}_L)-1$$

$${\mathcal{P}}_{\mathrm{B}}({\rho }_L,{\rho }_{\mathrm{QEC}})>{\mathcal{P}}_{\mathrm{B}}({\rho }_L,{\tilde{\rho }}_L).$$

$${\mathcal{P}}_{\mathrm{B}}({\rho }_L,{\rho }_{\mathrm{QEC}})>{\mathcal{P}}_{\mathrm{B}}(\rho ,\tilde{\rho }),$$

$$U_{\mathrm{MS},\varphi }(\theta )={\mathrm{e}}^{-\mathrm{i}\frac{\theta }{4}{S}_{\varphi }^2},S_{\varphi }=\sum _{i=1}^n(\cos \varphi X_i+\sin \varphi Y_i)$$

$$U_{\mathrm{R},\varphi }(\theta )={\mathrm{e}}^{-\mathrm{i}\frac{\theta }{2}{S}_{\varphi }},$$

$$U_{{\mathrm{R}}_j,z}(\theta )={\mathrm{e}}^{-\mathrm{i}\frac{\theta }{2}{Z}_j},$$

$$X_j(\theta )={\mathrm{e}}^{-\frac{\mathrm{i}\theta }{2}{X}_j},Y_j(\theta )={\mathrm{e}}^{-\frac{\mathrm{i}\theta }{2}{Y}_j},Z_j(\theta )={\mathrm{e}}^{-\frac{\mathrm{i}\theta }{2}{Z}_j},$$

$$X_{i,j}^2(\theta )={\mathrm{e}}^{-\frac{\mathrm{i}\theta }{2}{X}_i{X}_j},Y_{i,j}^2(\theta )={\mathrm{e}}^{-\frac{\mathrm{i}\theta }{2}{Y}_i{Y}_j},$$

$$R(\theta ,\varphi ,\tau ):=\prod _l{U}_{\mathrm{R},{\varphi }_{\mathrm{l}}}({\theta }_l)=\prod _l{\mathrm{e}}^{-\mathrm{i}\frac{{\theta }_l}{2}{S}_{{\varphi }_l}},$$

$${\mathcal{F}}_{av}=$$

$$\displaystyle\mathcal{F}_{av}(\tau)\approx\mathcal{F}_{\chi}=\frac{1}{2}\Big{\{}1+\exp[-\chi(\tau)]\Big{\}}$$

$$\chi (\tau )=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }\frac{d\omega }{{\omega }^2}\sum _{i=a,d}{S}_i(\omega )F_i(\omega ).$$

$${\epsilon }_i^{\mathrm{d}}(\rho )=(1-p_{\mathrm{d}})\rho +p_{\mathrm{d}}{\sigma }_i^z\rho {\sigma }_i^z,$$

$${\epsilon }_j^{\mathrm{MS}}(\rho )=(1-p_{\mathrm{MS}})\rho +\frac{p_{\mathrm{MS}}}{3}\sum _{\alpha \in {\Lambda }_{\alpha }}{\sigma }_j^{\alpha }\rho {\sigma }_j^{\alpha },$$

$$\begin{array}{rl}{\epsilon }_{\mathrm{MS}}(\rho )& =(1-p_{\mathrm{MS}})\rho +\frac{p_{\mathrm{MS}}}{105}\sum _{i\in {\Lambda }_{\mathrm{a}}}\sum _{\alpha \in {\Lambda }_{\alpha }}{\sigma }_i^{\alpha }\rho {\sigma }_i^{\alpha }\\ & +\frac{p_{\mathrm{MS}}}{105}\sum _{j_1,j_2\in {\Lambda }_{\mathrm{a}}}\sum _{\alpha ,\beta \in {\Lambda }_{\alpha }}{\sigma }_{j_1}^{\alpha }{\sigma }_{j_2}^{\beta }\rho {\sigma }_{j_1}^{\alpha }{\sigma }_{j_2}^{\beta },\end{array}$$

$$\begin{array}{rl}{\epsilon }_{\mathrm{MS}}(\rho )& =(1-p_{\mathrm{MS}})\rho \\ & +\frac{p_{\mathrm{MS}}}{1023}\sum _{\mathbf{j}\in {\Lambda }_{\mathrm{a}}}\sum _{\mathbf{\alpha }\in {\tilde{\Lambda }}_{\alpha }}{\sigma }_{j_1}^{\alpha }{\sigma }_{j_2}^{\beta }{\sigma }_{j_3}^{\gamma }{\sigma }_{j_4}^{\kappa }{\sigma }_{j_5}^{\zeta }\rho {\sigma }_{j_1}^{\alpha }{\sigma }_{j_2}^{\beta }{\sigma }_{j_3}^{\gamma }{\sigma }_{j_4}^{\kappa }{\sigma }_{j_5}^{\zeta },\end{array}$$

$${ϵ}_{\mathrm{m}}\approx \frac{\pi N(\delta -{\omega }_z)}{2{\omega }_z^2{t}_{\mathrm{g}}}0.8\left(\bar{n}+1\right)+\frac{{\pi }^{2}N(N-1){\eta }^4}{8N^2}\left(1.2{\bar{n}}^2+1.4\bar{n}\right),$$

$$t_{\mathrm{g}}({\bar{n}}_f)=t_{\mathrm{g}}(0)\left(1+\frac{{\eta }^2(2\bar{n}+1)}{N}\right).$$

$${ϵ}_{\mathrm{d}}\approx 2{\Gamma }_{\mathrm{d}}{t}_{\mathrm{g}}\sum _{i,j}{\mathrm{e}}^{-|z_i^0-z_j^0|/{\xi }_{\mathrm{c}}},$$

$${ϵ}_{\mathrm{In}}\approx {\Gamma }_{\mathrm{In}}{t}_{\mathrm{g}}{\eta }^2\left(\bar{n}+\frac{1}{2}\right)+{\Gamma }_{\mathrm{In}}{t}_{\mathrm{g}}\frac{{\eta }^2(N-1)}{4},$$

$${\epsilon }_h(\rho )=(1-p_{\mathrm{h}})\rho +\frac{p_{\mathrm{h}}}{3}\sum _{\alpha \in {\Lambda }_{\alpha }}{\sigma }_h^{\alpha }\rho {\sigma }_h^{\alpha },$$

$${\epsilon }_a(\rho )=(1-p_{\mathrm{b}})\rho +p_{\mathrm{b}}{\sigma }_a^x\rho {\sigma }_a^x,$$

$$\begin{array}{rl}& S_x^{(1)}=X_1{X}_2{X}_3{X}_4,S_x^{(2)}=X_2{X}_3{X}_5{X}_6,S_x^{(3)}=X_3{X}_4{X}_6{X}_7,\\ & S_z^{(1)}=Z_1{Z}_2{Z}_3{Z}_4,S_z^{(2)}=Z_2{Z}_3{Z}_5{Z}_6,S_z^{(3)}=Z_3{Z}_4{Z}_6{Z}_7.\end{array}$$

$$U_{\mathrm{MS},0}(-\pi /2)U_{{\mathrm{R}}_j,z}(-\pi /2)U_{\mathrm{MS},0}(\pi /2)={\mathrm{e}}^{\mathrm{i}\frac{\pi }{4}{Z}_j{\Pi }_{i\ne j}{X}_i},$$

$$\begin{array}{rl}{U}_x^{(i,j)}& :=|x_{+}⟩{⟨x_{+}|}_i\otimes {\mathbb{I}}_j+|x_{-}⟩{⟨x_{-}|}_i\otimes \mathrm{i}{X}_j=X_j(\frac{-\pi }{2})X_{i,j}^2(\frac{\pi }{2}),\\ U_y^{(i,j)}& :=|y_{+}⟩{⟨y_{+}|}_i\otimes {\mathbb{I}}_j+|y_{-}⟩{⟨y_{-}|}_i\otimes \mathrm{i}{Y}_j=Y_j(\frac{-\pi }{2})Y_{i,j}^2(\frac{\pi }{2}),\end{array}$$

$$\prod _{n=1}^4{U}_x^{(0,j_n)}{|0⟩}_0|\psi ⟩={|0⟩}_0\frac{1}{2}(1+S_x^{(m)})|\psi ⟩+{|1⟩}_0\frac{1}{2}(1-S_x^{(m)})|\psi ⟩.$$

$$\ket{\psi_{3}}\!=\!\textstyle{\frac{1}{\sqrt{2}}\!\big{(}\ket{{x_{+}}}_{a_{1}}\!\!\ket{y_{-}}_{a_{2}}\!\!\ket{x_{-}}_{a_{3}}\!\!\ket{y_{+}}_{a_{4}}\!\!\!+\ket{x_{-}}_{a_{1}}\!\!\ket{y_{+}}_{a_{2}}\!\!\ket{x_{+}}_{a_{3}}\!\!\ket{y_{-}}_{a_{4}}\!\!\big{)}},$$

$$\ket{\text{GHZ}}=\textstyle{\frac{1}{\sqrt{2}}\big{(}\ket{x_{+},x_{+},x_{+},x_{+}}+\ket{x_{-},x_{-},x_{-},x_{-}}\big{)}},$$

$$\ket{\tilde{\psi}_{3}}\!=\!\textstyle{\frac{1}{\sqrt{2}}\!\big{(}\!\ket{x_{+}}_{a_{1}}\!\!\ket{y_{+}}_{a_{2}}\!\!\ket{x_{+}}_{a_{3}}\!\!\ket{y_{+}}_{a_{4}}\pm\ket{x_{-}}_{a_{1}}\!\!\ket{y_{-}}_{a_{2}}\!\!\ket{x_{-}}_{a_{3}}\!\!\ket{y_{-}}_{a_{4}}\!\!\big{)}}.$$

$$|{\mathrm{B}}_{\pm }⟩=\frac{1}{\sqrt{2}}\left({|x_{+}⟩}_{a_1}{|y_{+}⟩}_{a_2}\mp {|x_{-}⟩}_{a_1}{|y_{-}⟩}_{a_2}\right).$$

Full text

1. I Introduction

2. II Assessing the progress on quantum error correction (QEC) by small quantum codes

3. III Trapped-ion experimental toolbox for QEC

4. III.1 Experimental Architecture

5. III.2 Gate Operations

6. III.3 Dynamic Error Suppression

7. III.4 Ion Crystal Reconfiguration Techniques

8. III.5 Readout and Electronic Control

9. IV Effective error models for elementary QEC operations in trapped ions

10. IV.1 Dephasing channel for idle qubits

11. IV.2 Depolarizing channel for stabilizer mappings

12. IV.3 Depolarizing channel

13. IV.4 Bit-flip channel for measurement and reset

14. V **Trapped-ion topological QEC and fault tolerance **

15. V.1 Basic properties of the 7-qubit color code

16. V.2 Trapped-ion alternatives to CNOT-based QEC

17. V.2.1 Mølmer-Sørensen (MS) gate error propagation

18. V.2.2 Non-fault-tolerant stabilizer readout

19. V.2.3 Fault-tolerant DiVincenzo-Shor stabilizer readout with 2-ion MS gates

20. V.2.4 Fault-tolerant DiVincenzo-Aliferis stabilizer readout with 2-ion MS gates

21. VI Trapped-ion protocols to assess the break-even point for beneficial QEC

22. VI.1 Non-fault-tolerant trapped-ion QEC protocols

23. VI.1.1 Shuttling-based, single-species, multi-qubit gate protocol

24. VI.1.2 Shuttling-based, two-species, multi-qubit gate protocol

25. VI.1.3 Shuttling-based, two-species, two-qubit gate protocol

26. VI.1.4 Hiding-based, two-species, multi-qubit gate protocol

27. VI.2 Fault-tolerant trapped-ion QEC protocols

28. VI.2.1 Shuttling-based, two-species, DiVincenzo-Shor scheme

29. VI.2.2 Shuttling-based, two-species, DiVincenzo-Aliferis scheme

30. VII Numerical studies of the performance of trapped-ion QEC protocols

31. VII.1 Computing resources and numerical approach

32. VII.2 Simulation results

33. VII.2.1 Shuttling- and hiding-based QEC with multi-qubit MS gates

34. VII.2.2 Shuttling-based two-species QEC with 2-qubit MS gates: exploring fault tolerance

35. VIII Conclusions and outlook

I Introduction

Solving hard computational problems by exploiting the quantum-mechanical laws of Nature is one of the goals of current scientific and technological research nielsen-book . To turn this idea into experimental reality, intense research efforts are currently devoted to scale existing small prototypes, which have served for proof-of-principle demonstrations qc_review , into larger quantum devices capable of processing information quantum-mechanically even in the presence of noise and processing errors (i.e. fault-tolerantly). This poses a significant challenge from both fundamental and technological perspectives.

Fundamentally, the quantum-mechanical features responsible for the advantage of these processors with respect to their classical counterparts, also give rise to a different behavior with respect to noise and errors, which excludes the straightforward application of classical error correction schemes. Despite these difficulties, the theory of quantum error correction (QEC) qec_shor ; calderbank-pra-54-1098 ; steane-prl-77-793 has shown a well-defined route for the development of large-scale quantum computers. The main ingredients of QEC to combat the detrimental impact of noise are: (i) encoding quantum information redundantly in ever-larger quantum registers, and (ii) detecting and correcting errors during a computation without altering the encoded quantum information. Exploiting these ingredients using particular QEC codes, it has been shown theoretically that it is possible to perform quantum computing sequences of arbitrary complexity fault-tolerantly if the noise/error of elementary operations is maintained below a certain threshold FTQC . The redundant encoding of the information in these QEC protocols, which is required to improve the level of protection against noise for a fault-tolerant computation, can be achieved by either (a) concatenating elementary codes in several layers steane-prl-77-793 , or (b) storing the information in topological features of registers of increasing size toric_code ; bombin-prl-97-180501 . The quest is therefore to implement these QEC ideas in quantum devices of increasing sizes.

The first experiments on QEC have implemented the 3-qubit 3_qubit_code_nmr and 5-qubit 5_qubit_code_nmr quantum codes in nuclear magnetic resonance. Starting from initial experiments on the 3-qubit code with trapped ions 3_qubit_code_nist_ions and superconducting circuits 3_qubit_code_superconducting , these two platforms have recently been employed to show repetitive error correction 3_qubit_code_repetitive_qec_ions , fault-tolerant error detection with a four-qubit code ft_error_detection_monroe , and small-scale versions of the topological color nigg-science-345-302 and surface surface_code_superconducting_line ; surface_code_superconducting_square codes. We note, however, that QEC is also being pursued in other platforms repetitive_QEC_NV . The theory of QEC, described in the paragraph above, defines a clear roadmap towards the demonstration of fault tolerance in large quantum processors. However, despite this remarkable progress, the hardware platforms are still far away from the sizes that are required to render the errors on the encoded data negligibly small. Hence, it would be desirable to define a set of intermediate QEC goals, which are necessary for the progress towards the fully-fledged fault-tolerant quantum computer, and can serve as a guiding principle in the experimental design by benchmarking the progress in building and scaling these smaller quantum codes.

A necessary condition for QEC is that the effect of a complete round of error detection and correction must be beneficial for the encoded qubit. This is a non-trivial condition since the effect of an attempt at error correction, while aiming to correct the existing errors, inevitably introduces risk of new ones. Accordingly, quantifying such a crossover into beneficial/useful QEC, and certifying that it is met in a particular QEC code, will translate into specific requirements on the fidelities of the various gates, measurements, and other internal processes that conform the QEC cycle. This can establish a set of goals that must be achieved by future hardware development. Once this is achieved, another necessary criterion is to verify if the encoding, followed by a complete round of error detection and correction, is beneficial in comparison to the degradation of an unprotected physical qubit subjected to the same sources of physical noise during the time required by the QEC cycle.

We note that the theoretical studies of the performance of different QEC strategies, quantified by the particular value of the threshold, depend crucially on the assumptions about the underlying platform capabilities and noise models. Using over-simplified noise models, or unrealistic platform capabilities, can lead to an overestimation or underestimation of the correcting power of a given QEC code. Therefore, if we are aiming at assessing and guiding the progress of a particular experimental platform by the above intermediate QEC goals, a very careful microscopic modelling of the noise and the operational errors is required. The objective of this work is to present a detailed study along these lines for trapped-ion quantum processors with current and anticipated future capabilities in the near term.

Summary of the results of this work

In this manuscript, we quantify the above intermediate goals for beneficial QEC by introducing a quantum-information protocol with a clear and intuitive operational meaning in Sec. II. This protocol can serve as a benchmark scenario to assess the progress of experimental QEC codes.

We focus on trapped-ion implementations of small QEC codes 3_qubit_code_nist_ions ; 3_qubit_code_repetitive_qec_ions ; ft_error_detection_monroe ; nigg-science-345-302 ; phase_opt_color_code , and use the above measure to assess theoretically the methodological and technological improvements that would be required to reach the break-even point for a logical qubit, i.e. to enter the regime of beneficial QEC. In order to reach this goal, it is of the utmost importance to choose and adapt QEC schemes according to the particular technological advantages and disadvantages of the hardware platform at hand. In particular, one must exploit the particular technological strengths and simultaneously mitigate the dominant sources of noise. This requires a detailed knowledge of the state of the art and foreseeable technological improvements, which we discuss in Sec. III. We present a thorough description of an experimental toolbox for QEC using a high-optical access segmented ion trap to manipulate dual-species ion crystals in a cryogenic environment (see Fig. 1). We consider a universal set of single-qubit and multi-qubit entangling gates schindler-njp-15-123012 that differs from the more standard CNOT-based approaches nielsen-book . The current and anticipated performance of these elementary operations, as discussed in detail below, is summarized in Table 2. These tools shall be combined with spectroscopic decoupling of a subset of ions (i.e. hiding-based approach), and with crystal-reconfiguration techniques (i.e. shuttling-based approach) summarized in Table 4, which include splitting ion crystals, shuttling ions across trap segments, and merging two sets of ions into a larger crystal. Together with the possibility of using a dual-species crystal for sympathetic re-cooling of the ions and stabilizer readout, this toolbox contains all the ingredients required for trapped-ion QEC. In addition to this knowledge of experimental capabilities, assessing the prospects of QEC also requires a detailed modelling of the main sources of noise and errors for these operations, which we address in Sec. IV.

Equipped with this toolbox, we develop different approaches for trapped-ion QEC in Sec. V. We start by describing how multi-qubit Mølmer-Sørensen (MS) gates PhysRevA.62.022311 ; molmer-prl-82-1835 can be exploited for efficient stabilizer readout mueller-njp-13-085007 , as experimentally demonstrated in nigg-science-345-302 . In the context of fault-tolerant QEC, however, different schemes based on one-qubit gates and two-qubit MS gates would be required. Since fault-tolerant QEC schemes have been typically conceived using single-qubit and two-qubit CNOT gates shor_ft_qec ; aliferis_ft_qec , it would be desirable to devise trapped-ion circuits that exploit MS gates directly, and to study how errors propagate on those circuits to demonstrate fault tolerance. We address these points by presenting a detailed description of a generic MS-based toolbox for QEC. We apply this toolbox to the 7-qubit topological color code with trapped ions, either using a non-fault-tolerant stabilizer readout with 7 data qubits and 1 additional ancillary qubit (i.e. 7+1-qubit scheme) based on multi-qubit/ sequential two-qubit MS gates, or by using fault-tolerant stabilizer readout via MS-based schemes that realize the equivalent of the CNOT DiVincenzo-Shor (DVS) protocol for 7+5 qubits shor_ft_qec , or the DiVincenzo-Aliferis (DVA) protocol for 7+4 qubits aliferis_ft_qec . Although we have focused on this particular code, we remark that this trapped-ion QEC toolbox for stabilizer readout can be generalized to any other stabilizer QEC code of interest, and scaled to larger-size codes in a modular fashion.

The MS-based stabilizer readout is used, in combination with some of the elementary operations of Tables 2 and 4, as a building block for the development of trapped-ion QEC protocols in Sec. VI. As already outlined above, we explore different scenarios according to varying experimental capabilities:

Shuttling-based protocol: Here, we consider trapped-ion crystals with either a single or two ion species, i.e. data and ancillary qubits being encoded in the same/different atomic species. We develop sequences of crystal-reconfiguration operations and stabilizer mappings to perform a full QEC cycle on a single logical qubit. We explore how the ability of crystal re-cooling by sympathetic cooling via the ancillary ion at intermediate stages affects the performance of the protocol. 2. 2.

Hiding-based protocol: Here, we consider the protocols realized in a static ion crystal. Qubits are selectively addressed by shelving inactive ions via spectroscopic de- and re-coupling pulses, and combined with stabilizer mappings to perform a full QEC cycle on a single logical qubit. We consider encoding of data and ancillary qubits in two different species, and the possibility to apply re-cooling after the readout.

These QEC protocols are complemented with the with the error model introduced in Sec. IV, which improves upon customary circuit-error models that consider a unique quantum channel for all elementary operations in a QEC cycle. This allows us to perform a detailed study that goes beyond standard, albeit not very realistic, assumptions: (i) we consider that the different gates (including the identity), the state preparation, and the measurements, do not take the same amount of time. (ii) We use distinct error channels affecting the different stages of the QEC protocols. For instance, idle qubits are subjected to dephasing in a trapped-ion setup, whereas single- and multi-qubit gates are subjected to depolarising noise. More importantly, (iii) the different channels are not all characterized by a unique error probability. Certainly, single- and multi-qubit gates do not have the same error in any known experimental platform. We use a microscopic modelling of the ion crystals to derive the particular expressions/values of the corresponding error rates for each operation. Therefore, our treatment does not only go beyond models that do not consider, or simplify, the occurrence of errors on the syndrome readout, but it also goes beyond the standard so-called circuit-level noise model, which typically makes these over-simplifications.

These sections set the stage for a large-scale numerical analysis that investigates the performance of such protocols in Sec. VII. The introduced criterion for beneficial/useful QEC is used to quantify the three essential requirements that will need to be met in forthcoming experiments for trapped-ion QEC: (i) sufficiently small natural physical error rates from fundamental error sources; (ii) to detect and dynamically correct errors at a fast enough rate; and (iii) sufficiently accurate realizations of unavoidably imperfect error-correction routines, so that there still remains an overall gain of applying (imperfect) QEC procedures.

Finally, in Sec. VIII, we present our conclusions.

II Assessing the progress on quantum error correction (QEC) by small quantum codes

While small quantum codes have already been demonstrated on different platforms 3_qubit_code_nmr ; 5_qubit_code_nmr ; 3_qubit_code_nist_ions ; 3_qubit_code_superconducting ; 3_qubit_code_repetitive_qec_ions ; ft_error_detection_monroe ; nigg-science-345-302 ; surface_code_superconducting_line ; surface_code_superconducting_square , it would be of interest to ultimately demonstrate fault tolerance on existing or near-future hardware. However, this would require showing the supremacy (i.e. reducing the error rate) of the small codes with respect to the best-possible un-encoded qubits on all representative quantum circuits belonging to a large set of protocols gottesman_small_codes . Depending on the platform, this comparison can be very stringent. For instance, ion-trap processors can use decoherence-free qubits dfs or the $\mu$-metal shielded Zeeman qubits Ruster2016 with very long coherence times, such that the error on the identity quantum circuit would be very hard to beat using any small QEC code. Additionally, single-qubit and two-qubit gates with bare qubits have, so far, the smallest achieved in-fidelities in any experimental platform 16Ballance ; PhysRevLett.117.060505 , and it also seems unlikely that small QEC codes, with their large overhead in complexity, will be capable of beating them. We thus believe that alternative criteria have to be established, which serve as reasonable guiding principles in the development of future technologies that improve upon existing QEC codes.

Break-even point for useful QEC

In this section, we introduce the criteria used in our work to judge whether a particular combination of hardware and quantum code can successfully perform QEC, sustaining thus a logical encoded qubit. As mentioned in the introduction, a first necessary condition that must be verified by any implementation of a QEC code is that the effect of a complete round of error detection and correction proves to be beneficial. In order to make it quantitative, we must define a measure for the quality, or integrity, of a logical qubit.

The fidelity of the logical encoded state subjected to noise/errors $\tilde{\rho}_{L}$ with respect to its ideal form $\rho_{L}=\ket{\psi_{L}}\bra{\psi_{L}}$, namely $\mathcal{F}=\langle\psi_{L}|\tilde{\rho}_{L}|\psi_{L}\rangle$, and in particular how it changes if we perform a QEC cycle on the imperfect $\tilde{\rho}_{L}$ or not, might first appear as a natural measure. However, one encounters difficulties. Consider a logical qubit $\tilde{\rho}_{L}$ that has completely decohered under the effect of independent depolarizing noise on the constituent $n$ physical qubits. The collective entity no longer contains any information about the initial logical state $\ket{\Psi_{L}}$. Its fidelity is $\mathcal{F}=2^{-n}$ since the system is in an equal mixture of all possible states regardless of the initial encoded state. The problem is that a round of error correction will seem to improve the quality of this logical qubit: it will map all the states in the mixture to either the logical zero $\ket{0}_{L}$, or the logical one $\ket{1}_{L}$, creating a mixture of these two $\rho_{\rm QEC}=\textstyle\frac{1}{2}(\ket{0}_{L}\bra{0}_{L}+\ket{1}_{L}\bra{1}_{L})$. Consequently the fidelity will rise to $\mathcal{F}_{\rm QEC}=\frac{1}{2}$ under an ideal QEC cycle (or close to this number for imperfect correcting circuits). Thus, if we were to select fidelity as our measure for the quality of a logical qubit, we would be faced with the unsatisfactory feature that a logical qubit that has been completely lost, and is free of any meaningful information, can seem to be partially recovered.

One might attempt to correct this issue by projecting the $n$-qubit state into the logical subspace, and only then computing the fidelity. Nonetheless, this leads to another unsatisfactory feature: one would find an equality in apparent performance between a device that maintains the logical qubit entirely in that subspace and one which allows a large component of the state to leave the subspace, regardless of the nature of the part of the state outside the proper subspace. As we will later remark, the approach we take in this work can be thought of as essentially a more sophisticated variant of this idea.

The alternative measure we will employ has a very clear and intuitive operational meaning. It is best illustrated with a quantum-information protocol that separates the role of encoding the logical qubit, from the task of reading it. We will use the labels Alice and Bob for two entities that have these roles (see Fig. 2). Now suppose that a random qubit state $\ket{\psi}=\alpha|{0}\rangle+\beta|{1}\rangle$ is selected, and Alice is instructed to prepare a logical qubit $\ket{\psi_{L}}=\alpha|{0}\rangle_{L}+\beta|{1}\rangle_{L}$ using the code of her choice. This logical qubit of $n$ physical qubits $\rho_{L}$ is then subjected to some noise channel, which may have any form, including correlated noise (e.g. spatially or temporally correlations arising from global fluctuating magnetic fields with a characteristic correlation time) and coherent noise (e.g. resulting from over-rotations in single-qubit gate operations). The now-imperfect logical qubit $\tilde{\rho}_{L}$ is presented to Bob, along with the following classical information: “The original state was either $\ket{\psi}=\alpha|{0}\rangle+\beta|{1}\rangle$ or $\ket{\psi_{\bot}}=\beta^{*}|{0}\rangle-\alpha^{*}|{1}\rangle$.” In other words, Bob is given a choice of two states, the true qubit state and the state which is orthogonal to it. Now Bob is challenged to make his best guess as to which state the $n$ physical qubits encode: it is a problem of state discrimination with the imperfect logical qubit as the resource. He may use any physically allowed process in his analysis, and in particular he can perform error correction, decode to a single physical qubit, and destructively measure it.

For simplicity, we will assume that Alice and Bob are ideal agents in the following, i.e. encoding and analysis of the logical qubit takes place perfectly. Then the probability $\mathcal{P}_{\rm B}(\rho_{L},\tilde{\rho}_{L})$ that Bob guesses correctly will vary only with the quality of the received qubit: an error-free logical qubit will score $\mathcal{P}_{\rm B,max}=1.0$, since Bob will certainly succeed in his state discrimination task, whereas (for example) a logical qubit which has undergone complete depolarization will score $\mathcal{P}_{\rm B,min}=0.5$ since Bob can only guess randomly. We define the integrity $\mathcal{I}$ of our memory as simply a scaled probability,

[TABLE]

The scaling thus provides us with the natural limits of unity for a perfect memory, and zero for a memory which provides Bob with no useful information whatsoever.

For any given decoherence model one can find the probability that Bob will guess correctly given a fully decohered logical qubit. If, as in the following analysis, the decoherence is restricted to a specific channel, then Bob’s performance can be higher, i.e. there will be instances in the random selection of the qubit state to be encoded by Alice that happen to be robust against the specific channel. In the cases we will be concerned with in the following section, restricting to a purely dephasing environment is an excellent approximation, in which case if we happen to select $|{0}\rangle$ or $|{1}\rangle$ for Alice to encode, the effect of decoherence on the encoded qubit will not degrade Bob’s capacity to differentiate: he need only measure all qubits in the $z$-basis and determine whether the observed pattern belongs to the set of states associated with $|{0}\rangle_{L}$ or $|{1}\rangle_{L}$. Conversely, if we had happened to select $|{+}\rangle$ or $|{-}\rangle$ for Alice to encode, then after full dephasing Bob will not be able to gain any value from his analysis and his probability of guessing correctly will be $0.5$. His performance when Alice randomly selects qubit states, sampled uniformly over the Bloch sphere, is found to be $\mathcal{P}_{\rm min}=0.75$ after full dephasing. Therefore when we plot the average $\mathcal{P}$ for any degree of pure dephasing we will find it varies in the range $0.75\leq\mathcal{P}_{\rm B}(\rho_{L},\tilde{\rho}_{L})\leq 1$.

Armed with this notion of the integrity of a qubit as, essentially, the extent to which its state can be read out by Bob, we now ask the question of whether the QEC cycle is beneficial or harmful by allowing for an imperfect round of error detection and correction prior to Bob’s guess $\tilde{\rho}_{L}\to\rho_{\rm QEC}$. The full protocol for our measure, where the code to be used (e.g. surface code, 2D color code, etc) is to be specified, is described in Table 1. According to our criterion, the round of imperfect error correction is now deemed to be beneficial if Bob’s probability of subsequently discriminating the state correctly is higher when we indeed perform a round of QEC, versus simply opting not to do so, and allowing the environment to act for time $\tau$ uninterrupted, namely

[TABLE]

The break-even point for a beneficial QEC occurs when Eq. (2) is satisfied. For convenience of exposition we may imagine that a third party, besides Alice and Bob, is responsible for the cycle of error correction: since this individual is effectively a flawed assistant for Bob, we use the name Igor after the famous fictional lab assistant (see Fig. 2). Then, our criterion for successful error correction can be summarized as, “Is Igor a help or a hindrance to Bob?”.

It is worth noting that Alice’s encoding protocol is predetermined and may not vary with the particular choice of qubit she is instructed to encode. Similarly Igor, who does not have the classical description of the encoded qubit, will always perform the same procedure as he attempts to correct it. Moreover, in all the analysis presented in this paper, we also fix Bob’s protocol: he simply performs his own (perfect) round of error correction, then decodes the logical qubit to a single qubit and measures that qubit in the basis of his choice. His optimal basis choice, for all cases considered here, is simply $\{\ket{\psi},\ket{\psi_{\bot}}\}$ and Bob makes his ‘guess’ according to the outcome. Thus only the final step, the measurement, depends on the choice of encoded qubit which was issued to Alice. It is interesting to observe that with this choice of Bob’s protocol, Bob is effectively mapping the state of the $n$ qubits into the logical subspace (with his round of perfect error correction) and then making a guess with a probability of success given by the fidelity of the corrected logical qubit. Then our concept of integrity relates directly to the fidelity after the encoded qubit is mapped into the logical subspace via the process of error correction.

Notice that this protocol naturally generalizes to multiple rounds of error correction, i.e. multiple times when the imperfect Igor can attempt to help. We simply wait a time $\tau/(m+1)$, have Igor perform his cycle, and repeat until $m$ cycles are performed. After a final wait of $\tau/(m+1)$ so that a total time $\tau$ has elapsed, Bob receives the $n$ physical qubits. For a sufficiently high performing Igor, and a long enough time $\tau$, it will be beneficial to have multiple rounds. Note that in the numerical simulation of the protocols, as discussed at later stages of this paper, we will take into account the finite duration that applications of the QEC cycles require.

Provided that this criterion has been fulfilled, and that QEC is proven beneficial, we can turn to the second desirable property of QEC, namely that encoding, error detection and error correction, are beneficial in comparison to the degradation of an unprotected physical qubit $\rho=\ket{\psi}\bra{\psi}\to\tilde{\rho}$. For the particular task at hand, this amounts to proving that

[TABLE]

and would essentially demonstrate that the encoded logical qubit outperforms the quantum memory built with a single unprotected physical qubit of the same sort as those used to form the logical qubit.

III Trapped-ion experimental toolbox for QEC

III.1 Experimental Architecture

The proposed setup consists of a 1D segmented high-optical-access (HOA) ion trap fabricated by Sandia National Laboratories hoa_source , and operated in a cryogenic environment (see Fig.1). We consider 40Ca*+* ions for hosting the qubits, and 88Sr*+* ions for providing the capabilities for sympathetic cooling, and mixed-species readout for syndrome extraction. We consider that ions undergoing the quantum logic operations can be separated and shuttled across the segmented trap array by using high-speed (diabatic), low-excitation protocols in order to minimize cross-talk on neighboring qubits. The required pulsed control of the qubits, system synchronization, measurement and fast-feedback as required for QEC, can be achieved by a custom-engineered high-speed controller.

The choice of the trap is motivated by the requirements for the realization of a QEC code, which demand high-fidelity quantum operations on the order of more than 10 ions. Therefore, a micro-fabricated segmented ion trap that enables multiple trapping zones and versatile ion crystal reconfigurations is required. This increases the complexity of the trap to a level that can, to date, only be satisfied by a quasi-planar trap structures, which reduce the trapping depth such that precautions against ion loss have to be taken. This can be mitigated by lowering the pressure of the vacuum environment by operating the experiment at cryogenic temperatures.

The encoded qubit will be realized in 40Ca*+* ions which allow for an optical as well as a ground state qubit. The chosen species enables high-fidelity state detection of the optical qubit due to its simple electronic structure. The optical qubit is formed by the $4S_{1/2}(m_{f}=-1/2)$ ground state and $3D_{5/2}(m_{f}=-1/2)$ metastable excited state. The excited state has a lifetime of 1.1s which sets the upper limit on the qubit storage time schindler-njp-15-123012 . Quantum operations are performed with a laser nearly resonant to this transition at a wavelength of about 729nm. It is also possible to encode a qubit in the two $4S_{1/2}(m_{f}=\pm 1/2)$ Zeeman substates. The coherence of this qubit is only limited by magnetic field fluctuations, where recent improvements resulted in a coherence time of more than one second Ruster2016 . State manipulation of this qubit is performed by Raman lasers close to the $4S_{1/2}$ to $4P_{1/2}$ transition. Due to the complexity of the QEC algorithm, a second ion species for sympathetic cooling and stabilizer readout will also be explored. For this, 88Sr*+* ions can be used.

The quality of quantum operations is limited by different processes for the optical and the spin qubit. For the optical qubit, the absolute phase noise of the laser driving the transition limits the achievable coherence, whereas the spin qubit is only sensitive to the differential phase noise in the two Raman laser beams. For the spin qubit, off-resonant excitation of the $4P_{1/2}$ state is a process reducing the gate fidelity, which can only be mitigated by increasing simultaneously the intensity and detuning of the Raman laser beams. However, dynamic control of light at 397nm is more challenging than control at a longer wavelength of 729nm.

It is expected that the encoding and QEC of a single logical qubit with a low-distance code can be implemented in a single segmented linear trap with the ion crystal reconfiguration techniques outlined below. However, multiple logical qubits will likely require a more capable architecture in which ion reconfiguration can be performed more efficiently using three- or four-way junctions. This allows multiple processing regions where syndrome measurements can be performed in parallel, which is also crucial for an extensible QEC architecture.

III.2 Gate Operations

(i) State of the art.– The experimentally-available set of operations considered in this work consists of (i) global laser-driven Mølmer-Sørensen (MS) entangling operations PhysRevA.62.022311 ; molmer-prl-82-1835 , which can be expressed as

[TABLE]

where $\phi$ is controlled by the laser phase, and $\theta$ by its intensity and pulse duration. Here, $X_{i}=\sigma_{i}^{x}$ and $Y_{i}=\sigma_{i}^{y}$ are Pauli matrices. Additionally, the global laser beams can also drive (ii) global rotations around the Bloch sphere of an individual qubit with a rotation axis in the equatorial plane

[TABLE]

which are also controlled via the phase, intensity, and pulse duration of the laser beam. Finally, (iii) addressed ac-Stark shifts result in rotations around the $z$-axis on the Bloch sphere of an individual qubit

[TABLE]

where $\theta$ is controlled by the intensity of the off-resonant laser beam, its detuning, and the pulse duration. Here, $Z_{i}=\sigma_{i}^{z}$ is one of the Pauli matrices. This gate set is described in detail in schindler-njp-15-123012 , and a numerical method to find an efficient decomposition of an arbitrary quantum algorithm in a sequence of these gates is presented in Martinez2016 . In sections below, we shall use extensively the following single-qubit operations

[TABLE]

which can be obtained either directly from the available set $Z_{j}(\theta)=U_{{\rm R}_{j},z}(\theta),$ or by means of dynamic error suppression sequences (see Sec III.3). In addition, we shall also use MS gates (4) of $X$-type or $Y$-type acting on a pair of ions/qubits $i$ and $j$. These MS gates will be defined as

[TABLE]

which are obtained (up to a global phase) from $U_{\rm MS,\phi}(\theta)$ in Eq. (4) by setting $\phi=0$ and $\phi=\frac{\pi}{2}$ respectively, and can be implemented by using spectroscopic decoupling techniques or ion-crystal reconfigurations steps, such that the MS laser beams only couple to the ion pair $i,j$. Using this notation, $\theta=\frac{\pi}{2}$ MS gates are ”fully-entangling”, as they map the computational basis states of $N$ qubits to GHZ states (up to local unitary rotations) of $N$ qubits. For instance, for two ions $X^{2}_{i,j}(\pi/2)=(\mathbb{I}-{\rm i}X_{i}X_{j})/\sqrt{2}$ and one finds $X^{2}_{i,j}(\pi/2)\ket{0}_{i}\ket{0}_{j}=(\ket{0}_{i}\ket{0}_{j}-{\rm i}\ket{1}_{i}\ket{1}_{j})/\sqrt{2}$. Throughout this manuscript, we will use the term multi-qubit MS gates to refer to MS gates acting on more than 2 qubits.

The entangling MS gates are performed on the axial center-of-mass mode of the ion string. This has the advantage that only a single loop in phase-space has to be closed to erase unwanted spin-motion entanglement. However, this mode cannot be used for longer ion strings as the ion string approaches a zig-zag configuration. This scheme allows for operations generating a maximally entangled GHZ state of up to 8 ions, which can be implemented in about 50$\mu$s for optical qubits with state fidelities of {98.6, 95.7, 81.7}% for {2,4,8} ions monz-prl-106-130506 . If the limiting factor on the gate quality is phase-noise on the laser driving the qubit transition, this could be improved by a laser with smaller phase noise or by switching to ground-state Zeeman qubits which are only susceptible to the phase difference of the two Raman lasers. Recently, high fidelity entangling operations for two ground-state hyperfine qubits have been demonstrated, reaching infidelities below $10^{-3}$ 16Ballance ; PhysRevLett.117.060505 .

Carrying out high-fidelity dual-species QIP protocols, such as 40Ca+ and 88Sr+ in our case, is generally more difficult than single-species experiments. Complicating factors include a more complex motional mode structure and cooling requirements. However, dual-species entangling gates have already been achieved 15Tan ; 15Ballance , showing Bell-state infidelities of $2\cdot 10^{-2}$ and $2\cdot 10^{-3}$. Moreover, we have also achieved preliminary dual-species operations for QEC with 40Ca+ and 9Be+ using global optical beams, including experimental approaches for handling dual-species crystals, as well as initial dual-species gates for $XX$ and $ZZ$ stabilizer readout of a two-ion 9Be+ crystal with a single 40Ca+ ion. Preliminary infidelity estimates obtained from Bell state preparation are listed in Table 2; SPAM (state preparation and measurement) errors are included in the errors given.

(ii) Experimental input for the noise model.– To assess the performance of a QEC procedure with a reasonably-sized ion register, one requires a simplified, yet sufficiently realistic, error model. In several studies, circuit noise is assumed to affect equally the single- and two-qubit gates of the QEC protocol. However, in many experimental setups, the leading source of noise affecting idle qubits, single-qubit gates and entangling operations can be very different, requiring thus more elaborate noise models. The chosen error model in this work includes perfect gate operations followed by a depolarizing channel on the active qubits and inactive qubits are affected by dephasing noise. We use microscopic calculations and experimental results to set the parameters of this error model, which has the advantage that it can be numerically simulated efficiently even for large qubit registers by using parallelizable Monte Carlo techniques, providing quantitative target gate fidelities. More details on this error model can be found in section IV.

(iii) Expected performance.– The required parameters for the chosen noise model can be fixed by the knowledge of (i) the state infidelity for the gates, which describes the strength of the depolarizing noise, and (ii) the duration of the gates, which are used to estimate the effect of dephasing on idling qubits. Table 2 shows a summary of current and anticipated gate operations for these parameters. The current coherence time on the ground-state qubit is 2 seconds. By improving the magnetic field stability it is expected that this can be extended to 10 seconds. The current coherence time on the optical qubit is 200 ms, which is anticipated to be improved to the limit given by the spontaneous decay from the metastable excited state to about 2 seconds.

III.3 Dynamic Error Suppression

In developing a trapped-ion experimental toolbox for QEC we are motivated to explore how the capabilities on hand in the laboratory may be crafted to maximize compatibility with the stringent demands on quantum error correction. Our primary objectives are twofold: (1) ensure gate errors are suppressed to rates as low as practicable relative to fault-tolerance thresholds; and (2) ensure compatibility of the underlying error model with the mathematical assumptions of fault-tolerance in QEC. In our experiments we therefore routinely turn to open-loop control protocols applied at the physical layer and designed to improve gate performance in advance of QEC.

The strict requirements of fault-tolerance on qubit error rates have motivated the development of error-suppressing physical-layer control techniques BallNPJQuantum2016 ; Viola1998 ; Viola1999 ; Zanardi1999 ; Vitali1999 ; Viola2003 ; Byrd2003 ; Khodjasteh2005 ; Yao2007 ; Uhrig2007 ; Gordon2008 ; Khodjasteh2009dcg ; Khodjasteh2009 ; Khodjasteh2010 ; Liu2010 ; Biercuk_Filter ; CPMG_ions known as dynamic error suppression (DES). In these feedback-free protocols, temporal modulation of the qubit control field is employed in order to effectively average away decoherence induced by environmental fluctuations or control imperfections. These protocols are considered an important complement to QEC Preskill_Layered ; Khodjasteh2005 ; JonesPRX2012 , both because of their potential to improve the resource-efficiency of QEC, and the fact that these protocols work in the presence of noise with strong temporal correlations, a regime which violates most error models underpinning the functionality of QEC. In fact, even in the presence of strong qubit decoherence DES can extend the effective qubit lifetime by decoupling from slowly varying noise sources. We expect that, in general, targeted application of DES will be implemented at the physical level for qubit manipulation.

(i) Relevant control protocols.– We consider control protocols with diverse historical origins, but a common framework for implementation. Physical qubit operations consist of multiple elementary control operations, which are sequentially applied in such a way that the desired target operation (quantum gate) is realized while simultaneously reducing the net sensitivity to error. We treat control protocols taking the form of an $n$-segment sequence of unitaries, executed over the time period $[0,\tau]$. This implies a partition of the sequence duration $\tau$ into $n$ subintervals $I_{l}=[t_{l-1},t_{l}]$, $l\in\{1,...,n\}$, such that the $l$th control unitary has duration $\tau_{l}=t_{l}-t_{l-1}$. The total operation can thus be expressed as

[TABLE]

where we have used the rotations in Eq. (5), and defined $\theta_{l}\equiv\Omega_{l}\tau_{l}$ in terms of the the time-independent Rabi rate $\Omega_{l}$ during the $l$th time interval $[t_{l-1},t_{l}]$. The resultant rotation generator generates a rotation of the Bloch vector through an angle $\theta_{l}$ about an arbitrary axis $\bm{l}=\left(\cos(\phi_{l}),\hskip 2.84544pt\sin(\phi_{l}),\hskip 2.84544pt0\right)$.

The assignment of the relevant control parameters for each segment, $\{\Omega_{l},\tau_{l},\phi_{l}\}$, may be determined through a variety of techniques. “Composite-pulse” constructions address a combination of static pulse-length and off-resonance control errors, and are generally implemented via piecewise-constant phase modulation. Representative sequences include the so-called SK1 and BB1, correcting for pure amplitude errors Brown2004 ; Wimperis1994 , CORPSE for pure detuning errors Wimperis1994 ; Cummins2003 , and both reduced CinSK (CORPSE in SK1) and reduced CinBB (CORPSE in BB1) Bando2013 for simultaneous errors. Dynamically corrected gate (DCG) protocols are constructed similarly (via different underlying mathematics - recently unified in  Chingiz2014 ; ViolaGeneral ; Ball2014 ), but employ piecewise-constant amplitude and phase modulation of the applied segmented control fields. Representative approaches include the Walsh family of DCGs SoareNatPhys2014 ; Ball2014 .

The approach of producing composite sequential operations achieved through modulation of a control field can be extended to the implementation of two-qubit Mølmer-Sørensen gates. Here, one may exploit phase-modulation of the driving field used to generate the effective spin-spin coupling via an intermediate bosonic mode GreenPhiM . Application of a piecewise-constant, phase-modulation pattern to the driving field permits simultaneous decoupling of “spectator” bosonic modes and the suppression of temporal fluctuations in control amplitude without the need to consider nonlinearities in optical instruments associated with amplitude modulation. The analytic framework in which these gates may be defined rests on a mathematical underpinning similar to that used for the construction of single-qubit DES strategies. This approach is particularly effective in achieving high-order suppression of residual spin-motional entanglement in the ion chain.

DES protocols adapted for idle periods, and known as dynamic decoupling Biercuk_Filter are also commonly implemented to correct for a variety of error sources. For instance, we employ spectroscopic decoupling to store idle qubits temporarily in Zeeman sub-levels that are not affected by the lasers responsible for QEC gates nigg-science-345-302 . This decoupling is achieved by a sequence composed of $N_{\rm p}=9$ pulses that can be applied to a set of $\ell$ idle ions to be hidden/un-hidden, labeled by $h_{1},\cdots h_{\ell}\in\{1,\cdots,N\}$. The composite pulse sequence is designed in a way which, to lowest order, echoes out addressing errors due to residual light intensity on neighboring ions nigg-science-345-302 .

(ii) Evaluating control performance.– The operational fidelity for an imperfect operation is given by $\mathcal{F}_{av}(\tau)=\frac{1}{4}\langle|\text{Tr}(\tilde{U}(\tau))|^{2}\rangle$, following GreenFF ; ToddNJP , where the error propagator, $\tilde{U}(t)$, captures the influence of noise and approaches the identity in the limit of vanishing errors. Calculating the fidelity requires the error propagator to be expressed as an infinite series using the so-called Magnus Expansion as $\tilde{U}(\tau)=\exp[-{\rm i}\Phi(\tau)]$, where the effective error operator $\Phi(\tau)=\sum_{\mu=1}^{\infty}\Phi_{\mu}(\tau)$ at the end of the interaction has expansion terms taking the form of time-ordered integrals over nested commutators of the so-called toggling-frame Hamiltonian. Considering unitary errors, it is convenient to define the error vectors $\bm{a}_{\mu}(\tau)$ by re-expressing the operators $\Phi_{\mu}(\tau)=\bm{a}_{\mu}(\tau)\cdot\bm{\sigma}$ in the basis of Pauli operators ToddNJP . Then, one can expand the exponential in the error propagator to obtain the fidelity in the small noise limit

[TABLE]

with $a_{\mu}^{2}:=\bm{a}_{\mu}(\tau)\bm{a}_{\mu}(\tau)^{T}$ the norm square of the error vector. This expression contains a collection of terms with equal magnitude arising from different orders of the Magnus expansion (e.g. $a_{2}^{2}$ vs $a_{1}^{4}$). An expression for the leading order fidelity which keeps terms only to $\bm{a}_{1}(\tau)$ but approximates the full expansion SoareNatPhys2014 is given by

[TABLE]

where we have defined $\chi(\tau):=\langle a_{1}^{2}\rangle$. We may conveniently move to the Fourier domain via the formalism of the filter-transfer function using

[TABLE]

Here, we have introduced the noise power spectral densities in the amplitude ($i=a$) and dephasing ($i=d$) quadratures, $S_{i}(\omega)$, describing the statistical properties of the environmental noise process afflicting the control operations (see the second column of Table 3 for different examples).

According to this discussion, the key quantities describing the effect of the control modulation are then $F_{i}(\omega)$; these objects characterize the spectral properties of the applied control, and can be calculated analytically for any piecewise-constant sequence Biercuk2011 ; ToddNJP , thus providing a simple quantitative means to compare control protocols of interest in the presence of generic, multi-axis time-dependent noise. Because the net infidelity for an operation is given via an overlap integral of the noise power spectral density $S_{i}(\omega)$, and $F_{i}(\omega)$ for the control, we may describe these objects using the language of filter-design and refer to them as filter transfer functions.

The filter order characterizes the performance of a filter transfer function by performing a Taylor expansion of the filter-transfer function about $\omega=0$. Assuming noise with dominant spectral weight at low frequencies, the approximation $F(\omega)\propto(\omega)^{2p}$ holds for some $p$ associated with the most significant power law expansion term. The associated control protocol thus defines a high-pass filter with filter order $p-1$. This parameter takes on particular relevance in determining the efficacy of a selected control protocol subject to broadband noise. This filter order must be distinguished from the Magnus order of error cancellation associated with quasistatic errors, which can be understood from the DC limit of our filter-function formalism for constant noise fields. A pulse sequence for which the Magnus expansion terms fulfill $\Phi^{\text{(DC)}}_{1}=...=\Phi^{\text{(DC)}}_{\mu-1}=0$ is then said to compensate static errors to Magnus order $(\mu-1)$ (see Refs. ToddNJP ; Ball2014 ). The residual error is then dominated by terms proportional to the $\mu$th power in the magnitude of the error scaling. This distinction is particularly important when considering more general expressions for the fidelity beyond leading order, in which contributions to the error from multiple Magnus orders appear in the Fidelity, and are captured through the exponentiated form of $\mathcal{F}_{\chi}$in Eq. (11).

(iii) Expected performance and protocol selection.– The tools outlined above and detailed in publications including references GreenFF ; ToddNJP ; Chingiz2014 ; SoareNatPhys2014 ; GreenPhiM suggest an efficient suppression of gate error rates due to noise processes exhibiting strong temporal correlations. Given realistic error models for dephasing noise and slow control-amplitude drifts, factors of error suppression exceeding $\sim 100\times$ are projected using state-of-the-art systems, and substantiated using both numeric simulations and analytic calculations Chingiz2014 . Key implementation challenges relate to the calibration of the requisite control phases and amplitudes, generally achieved through rf modulation protocols such as $IQ$ or $\Phi M$. The addition of time segments to a basic gate operation or complex modulation patterns introduces new paths for error. Those errors which are systematic may be efficiently suppressed by judicious choice of DES strategy. Stochastic errors may accumulate as a result of the more complex protocol, but due to their independence they scale only approximately linearly with added gate time under DES (a proxy measure for complexity). Therefore on balance DES has the potential to provide substantial benefits.

Taking into consideration the discussion above, we determine a critical path to selection of appropriate modulation protocols. We first observe that high-order error suppression in the Magnus expansion does not imply high-order time-domain noise filtering and vice versa. This has been validated using experiments on trapped ions SoareNatPhys2014 , and formalized rigorously in Ref. ViolaGeneral . Given the “whitening” effect of DES protocols on low-frequency-dominated noise, it is naively expected that the residual errors under DES will exhibit lower correlations than would otherwise be achieved. However, the order of error cancellation in the Magnus expansion is the primary determinant of correlations between residual errors that can cause failure of QEC protocols. Accordingly, the choice of a DES strategy will involve a determination first of the requisite Magnus order of error cancellation to suppress residual error correlations, next a determination of the high-frequency behavior of system noise, and finally consideration of how added complexity in high-order DES strategies introduces new pathways for error due to poor pulse calibration. Demonstrations of the suppression of residual error correlations using analyses of randomized benchmarking validate this general approach and will be the subject of a forthcoming manuscript.

III.4 Ion Crystal Reconfiguration Techniques

(i) State of the art.– Since the seminal proposal for the Quantum CCD Kielpinski2002 , the advent of segmented ion traps and fast multichannel arbitrary waveform generators has enabled the demonstration of ion shuttling operations ROWE2002 . These operations need to be performed fast on the timescale set by gate operations. This is required to avoid excessive overhead and decoherence from qubit dephasing, as well as anomalous heating of the ion crystal. On the other hand, motional excitations from shuttling must also be avoided in order not to compromise the phonon-mediated MS entangling operations. Thus, the required waveform generators have to fulfill the requirements of (i) analog update rates below typical trap frequencies, (ii) simultaneous and real-time update of many (10-80) channels, and (iii) superior signal integrity, i.e. low noise at trap frequencies, low glitch impulse areas and low digital crosstalk. Designs for such devices have been reported in BAIG2013 ; BOWLER2013 .

With segmented traps and waveform generators available, inter-segment shuttling of single ions within few trap periods has been reported for 9Be+ BOWLER2012 and for 40Ca+ fast_ion_splitting . While fast separation has also been reported in BOWLER2012 , the realization for 40Ca+ from fast_ion_splitting has been more challenging due to the low transient minimum trap frequency resulting from the increased mass. Recently, a fast rotation of two 40Ca+ ions with low resulting excitation, which can be used for reordering the qubit register, has also been demonstrated fast_rotation . These experimental results are summarized in Table 4.

(ii) Role of ion-string length on crystal reconfiguration.– The extent to which shuttling operations have to be employed for logical qubit encoding, syndrome readout, error correction and gate operations, depends on the experimental capabilities to store and coherently manipulate ion crystals of intermediate size. For ion strings of increasing size, addressing errors increase, and the presence of more spectator vibrational modes decreases the fidelities of entangling MS gates PhysRevA.62.022311 . Additionally, for segmented micro-traps, the ions are confined in smaller potential wells with increased anharmonicities. Furthermore, the more complex geometry does not always allow for precise micromotion compensation in all spatial directions at a reasonable experimental effort. These two effects can give rise to decreased confinement stability, presumably via parametric resonances. As a consequence, de-crystallization and trap loss occur at increased rates, such that this can represent a serious obstacle. To our knowledge, these effects have not been thoroughly investigated or quantitatively characterized.

Therefore, the following particular points have to be addressed by future experimental investigations: (a) the actual extent to which the speed of low-excitation shuttling operations can be increased, see Table 4. (b) The extensibility of low-excitation separation/merging and reordering operations beyond two ions, and to mixed-species scenarios. (c) The scaling of the attainable fidelities of entangling gates with the ion register size. (d) The actual decrease of duration/increase of fidelity of laser-addressed hide/unhide operations, which -as shuttling operations- serve the task of selecting a subset of ions for QEC. (e) The impact of decay from the metastable state for hidden qubits on the overall error rates, which is to be determined from simulations.

(iii) Experimental input for the effective noise model.– In order for simulation results to provide guidance towards the best strategy for logical qubit operation, we need to establish a noise model that captures the essential mechanisms how shuttling operations contribute to errors, but keeps the complexity and computational requirements reasonably small. We thus chose the following model: each shuttling operation contributes with a fixed amount of energy to the radial and axial degrees of freedom of each ion involved in the operation. Despite the fact that the energy is mostly contributed in the form of a coherent oscillator displacement, we assume that there is no fixed phase relation between consecutive displacements corresponding to different shuttling operations. Therefore, the shuttling operations lead to momentum kicks, which heat up the ions. We do not distinguish different collective modes and rather keep track of the mean motional energies of each ion. For merging of ion strings, we assume instantaneous thermalization, such that the total energies are equally distributed among the ions. Whenever entangling gate operations are carried out, we take the motional excitation into account to estimate gate imperfections according to the infidelity estimates discussed in Sec. IV, which consider the excitation on spectator vibrational modes, as well as the excitation of the gate-mediating bus mode.

(iv) Expected performance for shuttling operations.– The anticipated improvements are due to ongoing efforts such as (a) filter un-distortion: the distortion induced by second-order low-pass filters on the segment supply lines are partially undone by correcting for the filter transfer function, at the expense of control voltage amplitude as a resource. This increases the degree of control. (b) Increased control voltage range: the larger segment voltages generated by a second generation waveform generator will increase the minimum confinement throughout separation/merge operations, and enable crystal reordering at larger radial trap frequencies. (c) Ramp generation: software for automated voltage ramp generation will find optimized voltage ramps, possibly employing control techniques such as shortcuts-to-adiabaticity or optimal control Furst2014 . According to these improvements of an existing setup with 40Ca+ ions, the anticipated key metrics for the different elementary shuttling operations are shown in Table 4.

III.5 Readout and Electronic Control

Maintaining a logical qubit via QEC will require repetitive ancilla readout (see Fig. 5) and reset, feedback on the logical qubits, and likely sympathetic re-cooling of the crystal. Achieving high single-qubit readout fidelity generally requires a trade-off between minimizing the dark and bright state histogram overlap, and minimizing decay from the excited to the ground state (as well as repumping of dark states in hyperfine qubits). Additionally, state discrimination must be performed in real time; post-processing techniques cannot be used to enhance fidelities. A control system, the M-ACTION, has been designed to address these challenges in the context of maintaining a logical qubit: structured around a fast CPU communicating to FPGAs, this control system minimizes real-time processing delay, allows rapid prototyping of algorithms in C++, and can feed back to hardware with low latency.

State of the art.– Carrying out experiments on most trapped-ion control systems has typically involved describing the experiment on a PC in a simple domain-specific language, running a simple compiler to produce real-time bytecode, and executing this on a peripheral device such as an FPGA board or PC card schindler-njp-15-123012 ; 15Mount running a simple finite-state machine. This approach does not support arbitrary feedback requiring non-trivial calculation within latencies comparable to other ion-trapping operations, i.e. well below 10 $\mu\,$s.

An alternative approach is to design the system to have significant low-level processing power directly at the FPGA board; this allows more complex real-time decisions and calculations without being limited by communication bandwidth, and will be essential for QEC and other protocols requiring feedback. This design principle has been implemented in the M-ACTION system 16NegnevitskyPrivateCommunication used in a number of recent experiments on calcium ions 17Leupold ; 16Kienzler . The system uses a chip consisting of an FPGA tightly coupled to two physical ARM Central Processing Units, which allows standard C++ to be compiled. Thus the numerical libraries of C++ can be fully utilized in decision processes, allowing low-latency decisions during experimental sequences 17Leupold .

Control electronics including synthesizers generating qubit drive fields must be linked to a stable master clock serving both for synchronization of distributed control electronics, and provision of a stable phase reference against which qubit coherence is measured. This is vital because the common decoherence mechanism of dephasing represents a relative measure of the phase coherence of two effective oscillators, as outlined in BallNPJQuantum2016 . Common approaches to the provision of stable references include the use of an atomic frequency standard with good long-term stability, such as commercial Rubidium and Cesium clocks, followed by a quartz oscillator providing superior broadband phase noise. Both long-term stability and short-term phase noise represent critical sources of error; analyses have demonstrated that the use of common lab-grade synthesizers serving as system master clocks can produce error rates nearing the percent level in less than 100$\,\mu$s. Such error rates are easily suppressed by more than four orders of magnitude through appropriate selection of the master clock. In future systems with multiple master clocks it will be essential to ensure that slowly varying drifts between clocks are minimized to maintain a fixed laboratory reference frame for operations BallNPJQuantum2016 .

Building blocks and expected performance.– In the planned QEC scheme, a common step is to map a syndrome onto an ancilla qubit, read out its state, and re-initialize the ancilla along with cooling the ion chain. The dominant source of readout infidelity in both 40Ca+ and 88Sr+ optical qubits will likely be background counts for short readout times, which increase the dark and bright histogram overlap. This can be counteracted by increasing the photon collection time or efficiency, such that more photons are collected and the dark and bright histograms become more separated. Spontaneous decay from the D states is another source of infidelity for longer readout times, exceeding $10^{-4}$ after roughly 100  $\mu$s for 40Ca+, and dominating for very large detection times.

Assuming a reasonable collection efficiency of 0.6 %, a background count rate of 104/s, and considering the possible beam intensities similar to those available for 40Ca+, the delay incurred in reading out the 88Sr+ ancilla will be 100–300 $\mu$s with an infidelity of below 10*-3*. A readout infidelity of 10*-4* in 150 $\mu$s for the optical transition in 40Ca+ has been achieved using Bayesian schemes that incorporate photon arrival times in the state estimate, and attempt to identify spontaneous decays 08Myerson . By increasing the photon collection efficiency and thus reducing the detection time to below 20 $\mu$s; this will be attainable with a background count rate of $2\times 10^{3}$/s and collection efficiency of 3.5 %.

After readout, the ion chain can be re-cooled using EIT cooling on the radial modes 16Lechner and sideband cooling on the bus mode used for multi-qubit gates. EIT cooling takes several hundred microseconds depending on the geometry, ion level structure and initial temperature. The initial temperature depends strongly on the fluorescence lasers: on resonance they will cause significant heating, whereas by red-detuning and by weakening them, Doppler cooling will occur at the expense of photon counts. Thus an optimum may be found between readout time, readout fidelity and heating, such that the total readout and cooling time is minimized.

After EIT cooling, the mean phonon number will already be below $\bar{n}=1$, thus few sideband-cooling pulses are required. Since cooling times will be at least several hundred microseconds, there is significant time available for classical computation (determining if/where an error has occurred) and feedback latency (preparing the error correction pulse/pulses) in M-ACTION. If the chain were cooled only once per several readouts, however, these classical delays could become the bottleneck. Computation for a 7-qubit code, even when using Bayesian readout, should take 5 $\mu$s, and feedback latency is around 50 $\mu$s. Ongoing efforts to reduce this will lead to a latency of 1.5 $\mu$s.

The corrective operation when an error is detected, which involves a single-qubit rotation on a processing qubit, is an optional step that can be avoided by altering future gates on that qubit to take the error into account. This requires more classical computation, however we do not anticipate the computation time being a problem for a single logical qubit. It could, however, result in an increasingly broad tree of sequences. These must either be pre-computed and pre-loaded onto the FPGA hardware, or loaded onto the hardware in real-time. Pre-loading the sequences will require more memory on the hardware and is infeasible beyond approximately 5–10 feedback cycles, whereas real-time loading will take up to 1 ms for tens of pulses. A scalable solution will be real-time loading using either a more efficient encoding scheme or a high-bandwidth communication link.

IV Effective error models for elementary QEC operations in trapped ions

In Sec. VI, we will introduce specific trapped-ion protocols to assess the progress of QEC. In this section, we build on our previous discussion of the state-of-the-art, and future developments in trapped-ion technology, to model the noise on the elementary operations of these QEC protocols by certain quantum channels. As already noted previously in this manuscript, several works on QEC use circuit error models with a unique quantum channel affecting equally all the elementary operations of the QEC cycle. In this work, we go beyond these assumptions, and develop a more involved model with several distinct channels, the parameters of which can be set by microscopic calculations and/or experimental measurements. This model contains certain simplifications/limitations, which we comment upon in due course.

IV.1 Dephasing channel for idle qubits

During the QEC cycles, there are several operations where the internal states of a subset of qubits is not affected. More specifically, these operations are (i) crystal reconfiguration, leaving all the qubit states unchanged, (ii) single-qubit rotations and MS entangling gates, which leave the spectator qubits unchanged, and (iii) ancillary qubit measurement and re-cooling where the data qubits remain idle. In all of these processes, the idling qubits suffer mainly dephasing due to their coupling to the environment, e.g. fluctuating magnetic fields, which can be modeled by the identity followed by a dephasing channel acting on the particular subset of $m$ idle qubits $i_{1},i_{2},\cdots i_{m}\in\{1,\cdots,N\}$. To simplify the model, we will assume that the noise channel fulfills the i.i.d. criterion, i.e. it is temporally and spatially uncorrelated. This leads to the usual dephasing channel as described in nielsen-book , but applied to the set of idle qubits ${\varepsilon_{\mathrm{d}}}(\rho)=\varepsilon^{\rm d}_{i_{1}}\circ\varepsilon_{i_{2}}^{\rm d}\circ\cdots\circ\varepsilon^{\rm d}_{i_{m}}(\rho)$, where

[TABLE]

is a Kraus map and $p_{\mathrm{d}}$ is the probability for a single phase flip. It would also be interesting to study spatially-correlated dephasing, which does not necessarily imply a faster decoherence as occurs for GHZ states monz-prl-106-130506 , but also enables almost decoherence-free subspaces in certain codes nigg-science-345-302 .

One can easily estimate the phase-flip probability by calculating the time-evolution of a single qubit subjected to a fluctuating shift of the transition frequency, which is modeled by a stochastic process. Assuming a Markovian regime, one finds $p_{\mathrm{d}}=\frac{1}{2}(1-\mathrm{e}^{-\Gamma_{\mathrm{d}}t_{\mathrm{i}}})\approx\frac{\Gamma_{\mathrm{d}}}{2}t_{\mathrm{I}}$, where $t_{\mathrm{I}}$ is the time interval where the qubit remains idle, and $\Gamma_{\mathrm{d}}$ is the rate of dephasing. This leads to a dephasing time $T_{2}=1/\Gamma_{\mathrm{d}}$, as measured in Ramsey-interferometry experiments where $\langle X_{i}(t_{\rm I})\rangle=\langle X_{i}(0)\rangle{\rm e}^{-t_{\rm I}/T_{2}}$ (see Sec. III.2), which yields $p_{\mathrm{d}}=t_{\mathrm{I}}/2T_{2}$.

IV.2 Depolarizing channel for stabilizer mappings

During the QEC cycles, the stabilizer readout is accomplished by mapping the syndrome information of the data qubits onto ancillary qubits. As described below, this can be accomplished by the combination of two multi-qubit MS gates, or by a sequence of two-qubit MS gates. We will model the leading error of this mapping using a depolarizing channel, as described for instance in nielsen-book , after each stabilizer mapping in the QEC protocol. We have explored three types of depolarizing channels affecting $n$ active qubits involved in the MS gates (e.g. $n=5$ active qubits for QEC using multi-qubit MS gates, formed by 4 data and 1 ancillary qubits labeled by $j_{1},j_{2},j_{3},j_{4},j_{5}\in\{1,\cdots,N\}$):

(i) Independent depolarizing noise: The first error model consists of independent depolarizing channels ${\varepsilon_{\rm MS}}(\rho)=\varepsilon^{\rm MS}_{j_{1}}\circ\varepsilon^{\rm MS}_{j_{2}}\circ\cdots\circ\varepsilon^{\rm MS}_{j_{n}}(\rho)$ acting on each of the active qubits

[TABLE]

where $p_{\rm MS}$ is the probability for a MS depolarizing error, and $\Lambda_{\alpha}=\{x,y,z\}$. We note that this error model underestimates the occurrence of multiple-qubit errors during the entangling gate, and can thus overestimate the correcting power of the QEC. Therefore, we have also explored other channels.

(ii) Two-qubit depolarizing noise: Provided that the $N$-ion MS gate (4) can be understood as an all-to-all interaction between qubit pairs, and is thus local-unitary equivalent to applying CNOTs between all $N(N-1)/2$ ion pairs, an error model that considers single- and two-qubit errors with the same error probability may be more realistic. This will certainly be the case for the QEC schemes based on sequences of 5-qubit MS gates, where the noise is described by the quantum operation

[TABLE]

where $p_{\rm MS}$ is the probability for a MS depolarizing error, we have introduced the set of $n$ indexes for the active ions $\Lambda_{\rm a}$, and the sum over multiple ion indexes excludes coincidences of the pair of indexes. For the 5-ion MS gate, local-unitary equivalent to 10 CNOTs, with each pair of ions potentially undergoing 15 possible Pauli errors (6 single-qubit and 9 two-qubit Pauli operators), this results in the pre-factor 1/105.

(iii) Multi-qubit depolarizing noise: As a worst-case scenario for the schemes based on multi-qubit MS gates, we have also explored a model where any 5-qubit error can occur with the same error probability due to a faulty 5-ion MS gate. This can be described by the quantum operation

[TABLE]

where $p_{\rm MS}$ is the probability for a MS depolarizing error to occur, and we have introduced $\bm{j}=(j_{1},j_{2},j_{3},j_{4},j_{5})$, and $\bm{\alpha}=(\alpha,\beta,\gamma,\kappa,\zeta)$ in $\tilde{\Lambda}_{\alpha}=\{0,x,y,z\}$. Here, the sum over multiple ion indices excludes coincidences of any indexes, we have introduced $\sigma^{0}=\mathbb{I}_{2}$ as the identity operation, and the sum over possible Pauli errors excludes the global identity $\alpha=\beta=\gamma=\kappa=\zeta=0$ (i.e. no error), thus giving rise to a total of $4^{5}-1=1023$ possible Pauli error configurations.

To estimate how the error probability of the above depolarizing channels $p_{\rm{MS}}$ depends on the different experimental parameters, we will calculate the state fidelity of the ideal MS gate followed by each of the depolarizing channels in Eqs.(14) and (15), $\mathcal{F}=\langle{\Psi_{\rm t}}|\epsilon_{\rm MS}\left(U_{\rm MS}|{\Psi_{0}}\rangle\langle{\Psi_{0}}|U_{\rm MS}^{\dagger}\right)|{\Psi_{\rm t}}\rangle$, where the ideal gate produces a GHZ-type state $|{\Psi_{\rm t}}\rangle=U_{\rm MS,\phi}(\pi/2)|{\Psi_{0}}\rangle$.

For the (i) independent depolarizing channel in Eq (14), one obtains $\mathcal{F}=(1-p_{\rm MS})^{5}+\frac{1}{30}(1-p_{\rm MS})^{3}p_{\rm MS}^{2}+\frac{1}{15}(1-p_{\rm MS})p_{\rm MS}^{4}\approx 1-5p_{\rm MS}$ for $p_{\rm MS}\ll 1$. In this case, only the processes where no error occurs contribute to the fidelity at the lowest order in $p_{\rm MS}$, such that the error probability is simply $p_{\rm MS}=(1-\mathcal{F})/5$. For the (ii) two-qubit depolarizing channel that includes one- and two-qubit errors with the same probability (15), one finds $\mathcal{F}=1-\frac{95}{105}p_{\rm MS}$, such that $p_{\rm MS}=105(1-\mathcal{F})/95$. In this case, processes with no error and two Z-type errors leave the GHZ state invariant, and contribute with the same order in $p_{\rm MS}$. Finally, for the (iii) multi-qubit depolarizing channel that includes all 5-qubit errors with the same probability (16), one finds $\mathcal{F}=1-\frac{1008}{1023}p_{\rm MS}$, such that $p_{\rm MS}=1023(1-\mathcal{F})/1008$. In this case, processes with no error, and two/four $Z$-type errors leave the GHZ state invariant, and contribute with the same order in $p_{\rm MS}$.

The probability of the depolarizing channel can then be extracted by comparing to the GHZ infidelity $\epsilon=1-\mathcal{F}$ obtained by a microscopic Hamiltonian modeling the evolution of the trapped-ion MS gate PhysRevA.62.022311 . In this way, one can include possible sources of noise and experimental imperfections that lead to evolutions that depart from the ideal MS gate PhysRevA.62.022311 ; molmer-prl-82-1835 . We now discuss three different sources of infidelity.

(a) Gate infidelity due to the motional excitations.– The employed MS gate utilizes a bi-chromatic laser-ion interaction that drives simultaneously the blue and red motional sidebands corresponding to the centre-of-mass (CoM) axial mode. This acts as a bus mode that mediates an all-to-all qubit-qubit interaction (4) capable of generating the aforementioned GHZ states. The motional excitation of this mode, as well as the presence of additional vibrational modes of the ion chain, lead to an infidelity caused by two main sources: (i) off-resonant couplings to the sidebands of spectator modes, and (ii) fluctuation of the effective Rabi frequency of the laser-ion coupling due to the motional excitations of bus and spectator modes, i.e. Debye-Waller factors. If one assumes, as argued in Sec. III.4, that there is a fast equilibration after ion-reconfiguration operations, resulting in a thermal vibrational state with a mean phonon number that increases after each particular crystal reconfiguration according to Table 4, it is possible to estimate the infidelity of the $N$-ion fully-entangling MS gate $\mathcal{F}=1-\epsilon_{\rm m}$ as

[TABLE]

where $\omega_{z}$ is the CoM axial mode frequency with mean phonon number $\bar{n}$, $\delta$ is the symmetric detuning of the bi-chromatic laser beams with respect to the electronic transition, $t_{\rm g}$ is the gate time, and $\eta=k_{\rm L}/\sqrt{2m\omega_{z}}$ is the single-ion axial Lamb-Dicke parameter. The first term in this equation represents the infidelity due to unclosed phase-space trajectories of the spectator modes, whereas the second one is due to the decrease of the Rabi frequency due to the thermal background of all modes, i.e. Debye-Waller factor. Note that, although all the modes of the ion crystal participate in the infidelity, the error can be bounded with quantities that are characterized by the mean phonon number of the CoM mode.

In order to apply this error estimate (17) to the different steps of the QEC cycle based on crystal reconfiguration, we assume that the gate time $t_{\rm g}(\bar{n}_{f})$ after a number of crystal reconfigurations that increase the mean phonon number to $\bar{n}_{f}$ is modified with respect to the optimized gate time $t_{\rm g}(\bar{n}_{0})$ in Table 2, where $\bar{n}_{0}\approx 0$ after laser cooling. This is accomplished by modifying the laser detuning, such that phase-space trajectories of the CoM mode are still closed for the modified time

[TABLE]

Therefore, the gates become slightly slower the higher the mean phonon number is. Note that this delay will not result in an appreciable change in the dephasing (13) that idle qubits suffer for each stabilizer readout, and one can assume that $2t_{\rm g}(0)$ is the dephasing time for idle qubits during each $N$-ion stabilizer mapping (see Eq. (13) above).

This motional infidelity $\epsilon_{\rm m}$ can become the leading source of error in the QEC protocol where the ancillary (readout) and physical ions are of the same species, and are shuttled/merged/split/rotated during the QEC cycle to extract the syndrome. Accordingly, re-cooling of the motion of the ion crystal via the ancillary ion cannot be exploited, as the scattered light would perturb the quantum state encoded in the data qubits. In this situation, the motional excitation of the ion string resulting from the different reconfiguration steps can become very large (see Table 4), yielding a motional infidelity that overcomes other possible sources of noise.

(b) Gate infidelity due to magnetic-field and laser-intensity fluctuations.– Another possible source of noise in the MS gate is caused by (i) fluctuations between the qubit frequency and the laser frequency (ii) and laser-intensity fluctuations. We model these two sources of noise by stochastic processes that yield fluctuations of the qubit frequencies, and of the laser coupling strengths to the motional sidebands of the MS scheme, respectively. If one assumes that the time correlations of these processes decay much faster than the gate time (i.e. Markovian assumptions), then the gate infidelity can be expressed as $\mathcal{F}\approx 1-(\epsilon_{\rm d}+\epsilon_{\rm I})$, where $\epsilon_{\rm I}$ is the error due to intensity fluctuations, and $\epsilon_{\rm d}$ is the dephasing error due to e.g. fluctuating magnetic fields experienced by the qubits during the gate. Such an error can be approximated by

[TABLE]

where $\Gamma_{\rm d}$ is the rate of dephasing leading to a dephasing time $T_{2}=1/\Gamma_{\rm d}$, $z_{i}^{0}$ are the positions of the ions in the string, and $\xi_{\rm c}$ a typical length scale over which magnetic-field fluctuations are correlated, i.e. for $\xi_{\rm c}=0$, we have local noise and $\epsilon_{\rm d}\approx 2t_{\rm g}N/T_{2}$, whereas for $\xi_{\rm c}\gg|z_{1}^{0}-z_{N}^{0}|$, we have global magnetic-field fluctuations and $\epsilon_{\rm d}\approx 2t_{\rm g}N^{2}/T_{2}$ would be the dephasing rate affecting a GHZ state. We note that, for the different mappings of the syndrome information into the ancilla qubits, the actual dephasing error will lie between these two limits, and its particular value will depend on the collective state of the qubits at the instants of time where the MS gates are applied. To simplify the description, we consider a conservative, worst-case scenario, and use an error rate $\epsilon_{\rm d}\approx 2t_{\rm g}N^{2}/T_{2}$ consistent with the values reported in Table 2.

Finally, intensity fluctuations during the gate will have two effects. On the one hand, the time-dependence of the fluctuations can lead to a residual spin-motion entanglement due to imperfect closure of the phase-space trajectories of the bus CoM mode. On the other hand, the acquired phase that depends on the area of the enclosed trajectory may also differ from the one required to generate the desired GHZ state. These two sources of error are accounted for, in corresponding order, by means of the following expression

[TABLE]

where $\Gamma_{\rm In}$ is the rate of intensity fluctuations, obtained through its zero-frequency power spectral density, and sets a typical time-scale for the effects of intensity noise $T_{\rm In}=1/\Gamma_{\rm In}$. We will adjust this parameter to be consistent with the fidelities reported in experiments (see Table 2).

In the QEC protocols based on two species, we use a different ion species for the ancillary and data qubits. Thus, we can exploit the ancillary ion for intermediate sympathetic re-cooling of the ion crystal. In any case, the population of vibrational modes remains small prior to the stabilizer mapping via MS gates. Hence, the (i) error due to thermal motional excitation in Eq. (17) will not be leading, but instead will contribute together with other sources of error. We consider also (ii) dephasing and (iii) intensity fluctuations during the gate as additional sources of gate infidelity following Eqs. (19) and (20). Therefore, the MS-gate infidelity that can be used to extract the probability of the depolarizing channels corresponds to $\epsilon=\epsilon_{\rm m}+\epsilon_{\rm d}+\epsilon_{\rm I}$.

IV.3 Depolarizing channel

Due to the use of spectroscopic decoupling protocols (III.3), in the noise model used in this study, we do not consider residual errors on the neighboring ions, and will model the error in this decoupling process by independent single-qubit depolarizing channels acting on each of the qubits being hidden or unhidden ${\varepsilon_{\mathrm{h}}}(\rho)=\varepsilon_{h_{1}}\circ\varepsilon_{h_{2}}\circ\cdots\circ\varepsilon_{h_{m}}(\rho)$, where

[TABLE]

and $p_{\rm{h}}$ is the error probability. Taking the current and expected single-qubit gate infidelities $\epsilon_{1}$ reported in Table 2 into account, we can estimate the probability of the depolarizing channel for spectroscopic decoupling as $p_{\rm h}=9\epsilon_{1}$.

IV.4 Bit-flip channel for measurement and reset

The QEC cycles also require a measurement and reset of the ancillary qubit. Faulty measurement/reset can be modeled by a bit-flip channel, as introduced in nielsen-book , that acts on the set of $n_{a}$ ancillary qubits $\{a_{1},a_{2},\cdots a_{n_{a}}\}\in\{1,\cdots,N\}$, namely ${\varepsilon_{\mathrm{b}}}(\rho)=\varepsilon_{a_{1}}\circ\varepsilon_{a_{2}}\circ\cdots\circ\varepsilon_{a_{n_{a}}}(\rho)$, where

[TABLE]

where $p_{\rm b}$ is the probability for a bit-flip error to occur. We estimate the value of this probability through the infidelities for measurement $p_{\rm b}=\epsilon_{\rm meas}$ and qubit reset $p_{\rm b}=\epsilon_{\rm res}$ reported in Table 2.

V **Trapped-ion topological QEC and fault tolerance **

V.1 Basic properties of the 7-qubit color code

We will focus on the development of trapped-ion QEC protocols to implement a logical qubit based on the 7-qubit color code, assessing their potential to be useful for QEC by the operational measure descried in Sec. II. The 7-qubit code constitutes an enabling building block of two main routes towards fault-tolerant quantum computation (FTQC). On the one hand, it is equivalent (up to local unitaries) to the 7-qubit Steane code steane-prl-77-793 ; nielsen-book and can, as such, be used as an elementary unit to achieve more and more robust logical qubits by means of concatenation. On the other hand, it constitutes the smallest, though functionally complete, representative of the class of 2D topological color codes bombin-prl-97-180501 , for which logical qubits of increasing robustness can be achieved by using codes defined in lattices of increasing size. We note that topological codes typically display higher error thresholds in comparison to concatenated ones, offering thus a practical and very promising route towards large-scale QEC.

One of the goals of our study is the identification of the accuracy requirements of the experimental building blocks used to realize complete QEC cycles on the logical qubit. For instance, a series of limitations on the experimental approach used in nigg-science-345-302 , such as e.g. a large overhead in spectroscopic decoupling operations, has already been identified. By a numerical analysis of the operational measures in Eqs. (2)-(3), we aim at deriving quantitative estimates on the experimental requirements to make a QEC based on this protocol beneficial. Moreover, the measure will also allow us to benchmark the performance of other protocols that avoid spectroscopic decoupling. In this sense, exploring a variety of protocols for this code is an ideal testbed for the development of key tools, which would be equally required in the implementation of other small- and medium-size quantum codes, such as e.g. the 9-qubit Bacon-Shor-code (see e.g. RevModPhys-Terhal-87-307-2015 ) or the rotated 9-qubit surface code of distance 3 Horsman-njp-2012 .

Let us thus briefly summarize a few central properties of the 7-qubit color code, which are relevant for the QEC protocols developed and studied below. This code allows one to store and manipulate $k=1$ logical qubit, which is redundantly encoded in entangled states distributed over $n=7$ physical qubits. The code has a logical distance of $d=3$, and thus allows one to detect and correct at least $t=1$ arbitrary error (phase and/or bit flip error) on any of the 7 physical qubits. The code belongs to the family of CSS codes calderbank-pra-54-1098 ; steane-prl-77-793 , and thus allows one to independently detect and correct bit- and phase-flip errors. Errors are identified by measuring the corresponding error syndrome, which is deduced from the sets of three four-qubit Z and X-type stabilizer operators associated to the three plaquettes of the code (see Fig. 3). If we denote the set of Pauli matrices of the physical qubits as $\{X_{i}=\sigma_{i}^{x},Y_{i}=\sigma_{i}^{y},Z_{i}=\sigma_{i}^{z}\}_{i=1}^{n}$, the stabilizers are

[TABLE]

Logical states are encoded in the code space, which is defined as the simultaneous eigenspace of eigenvalue +1 of the set of all $s=6$ stabilizer generators (23), such that $k=n-s=1$ coincides with the number of encoded qubits. Logical qubits employing larger distance codes, and thus allowing for the correction of multiple errors, can be constructed by encoding a logical qubit in larger lattice structures involving more physical qubits bombin-prl-97-180501 . An interesting representative of this procedure is the distance-5 color code (of 4-8-8-type bombin-prl-97-180501 ), which encodes a single logical qubit in 17 physical qubits arranged on a 2D lattice structure of 8 plaquettes.

A distinguishing feature, as compared e.g. to Kitaev’s surface code dennis-j-mat-phys-43-4452 ; raussendorf-prl-98-190504 , is that the color code bombin-prl-97-180501 permits a transversal realization of the entire group of Clifford gate operations nielsen-book . Thus, the realization of a logical Clifford gate on the logical qubit amounts to a bit-wise application of the corresponding gates on all physical qubits $X_{L}=\prod_{i}X_{i},Z_{L}=\prod_{i}Z_{i}$, and similarly the other logical single-qubit Clifford gate operations, such as the Hadamard $H_{L}$ and K-gate $K_{L}$. This property not only does facilitate the practical implementation of logical gate operations but, more crucially, also prevents an uncontrolled propagation of errors through the quantum hardware – a central requirement to ultimately reach the FTQC regime. A universal set of logical gate operations can be achieved by complementing the Clifford operations with a single non-Clifford gate. For 2D color codes such an additional gate operation, e.g. the T-gate nielsen-book by magic-state injection bravyi-pra-71-022316 involves a quantum state teleportation process between the register of system qubits and an ancilla qubit.

V.2 Trapped-ion alternatives to CNOT-based QEC

In this section, we develop trapped-ion alternatives to CNOT-based schemes for the readout of the 4-qubit stabilizer operators of the color code (23). This is the essential operation in a QEC cycle, which consists of measuring all X- and Z-type stabilisers, and performing conditional operations on the physical qubits. These readout schemes are also essential to encode a particular qubit state: starting from $\otimes_{i}\ket{0}$, one would measure all of the X-type stabilisers, perform conditional operations to project onto the code subspace, and apply a single-qubit rotation at the logical level followed by the required QEC cycles.

In this section, we start by describing rules for the propagation of errors in circuits involving Mølmer-Sørensen (MS) gates. These rules are used to understand the properties of schemes that work with a single ancillary ion (i.e. non-fault-tolerant schemes) and use either 5-ion (4) or 2-ion (8) MS entangling gates. The motivation to study these schemes is to gain insight on how important it is to avoid the direct occurrence of multi-qubit errors in QEC protocols nigg-science-345-302 . By using 5 or 4 ancillary qubits and sequences of 2-ion MS gates, it is also possible to implement a trapped-ion version of the CNOT-based schemes for fault-tolerant stabilizer readout by DiVincenzo-Shor (DVS) shor_ft_qec , and DiVincenzo-Aliferis (DVA) aliferis_ft_qec , respectively. The main goal of exploring these schemes is to assess under which conditions, and in which experimental parameter regimes, such fault-tolerant protocols might offer advantages in reaching the break-even point for useful QEC. Let us remark that the techniques hereby presented can be easily generalized to any other stabilizer of a different QEC code. Therefore, they will be an essential ingredient of future trapped-ion efforts for QEC.

V.2.1 Mølmer-Sørensen (MS) gate error propagation

For the construction of fault-tolerant quantum circuits, it is important to understand how errors propagate from one qubit to others by means of the entangling gate operations. The circuit identities in the left panel of Fig. 4 show the well-known propagation of $X$ and $Z$-type single qubit errors through CNOT gates. Analogous error propagation properties can be derived for fully-entangling 2-ion $X^{2}_{i,j}(\pi/2)$ and $Y^{2}_{i,j}(\pi/2)$ MS gates (8) (see right panel of Fig. 4). Errors of the same type as the basis of the MS gate commute with the gate operation, e.g. $X^{2}_{i,j}(\pi/2)X_{i}=X_{i}X^{2}_{i,j}(\pi/2)$, and thus do not propagate to the second qubit. On the contrary, errors of a type different from the MS-gate basis, e.g. a $Z_{i}$ occurring before a MS gate $X^{2}_{i,j}(\pi/2)$, are converted into an error of the type that is complementary to the error type, i.e. into a $Y_{i}$ error in this case. In addition, this triggers the creation of an error on the second qubit of the type of the gate, i.e. here an $X_{j}$ error. This can be seen from the identities $X^{2}_{i,j}(\pi/2)Z_{i}=\frac{1}{\sqrt{2}}(\mathbb{I}-{\rm i}X_{i}X_{j})Z_{i}={\rm i}Z_{i}X_{i}X_{j}\frac{1}{\sqrt{2}}(\mathbb{I}-{\rm i}X_{i}X_{j})=-Y_{i}X_{j}X^{2}_{i,j}(\pi/2)$ which, in an analogous fashion, also hold for $Y$-type MS gates and the other types of errors. Note that, in contrast to the CNOT gate operation, the MS gate is symmetric in the indices of the two qubits, i.e. these propagation rules hold equally for errors arriving on the second qubit of the gate, e.g. $Y^{2}_{i,j}(\pi/2)Z_{j}=Y_{i}X_{j}Y^{2}_{i,j}(\pi/2)$.

V.2.2 Non-fault-tolerant stabilizer readout

In this section, we start by reviewing the scheme for stabilizer readout of the 7-qubit code nigg-science-345-302 , which uses a single ancilla ion and multi-qubit MS gates. We then extend this scheme to a protocol that is based on 2-ion MS gates, which will be used to explore how important it is to avoid the direct occurrence of multi-qubit errors from the multi-ion MS gates.

(i) Multi-qubit MS stabilizer readout.– The readout of the stabilizers (23) can be accomplished by mapping the syndrome information of the data qubits onto a single ancillary qubit using two global $5$-ion MS gates (4), and an intermediate single-qubit rotation via a local ac-Stark shift (6), since

[TABLE]

as shown in mueller-njp-13-085007 . By applying this sequence to the ancillary ion and a particular subset of four qubits $\{j_{1},j_{2},j_{3},j_{4}\}$ belonging to a particular plaquette stabilizer (23), one can map all stabilizers \big{\{}S_{\alpha}^{(1)},S_{\alpha}^{(2)},S_{\alpha}^{(3)}\big{\}}_{\alpha=x,z} onto the ancillary qubit. For instance, for the first $X$-type stabilizer, one finds $U_{\rm MS,0}(-\pi/2)U_{{\rm R}_{0},z}(-\pi/2)U_{\rm MS,0}(\pi/2)={\rm exp}({\rm i}\frac{\pi}{4}Z_{0}S_{x}^{(1)})$. The stabilizer information can then be measured by performing a Ramsey-type sequence on the ancillary qubit (see Fig. 5). The ancilla qubit is initialized in $\ket{0}_{0}$, and after the pulse $U_{{\rm R}_{0},\phi}=U_{\rm R,\phi}(\theta/2)U_{{\rm R}_{0},z}(\pi)U_{\rm R,\phi}(-\theta/2)U_{{\rm R}_{0},z}(-\pi)$, one maps the stabiliser information into the ancilla using the above scheme. Finally, after applying the pulse on the ancilla qubit $U_{{\rm R}_{0},\phi+\pi/2}=U_{\rm R,\phi+\pi/2}(\theta/2)U_{{\rm R}_{0},z}(\pi)U_{\rm R,\phi+\pi/2}(-\theta/2)U_{{\rm R}_{0},z}(-\pi)$, one measures it in the computational basis $M_{0}^{z}$, obtaining $\pm 1$ when the ancilla qubit is found in state $\ket{0}_{0}$ or $\ket{1}_{0}$, respectively. These outcomes correspond to the $\pm 1$ eigenvalue information of the corresponding stabiliser.

(ii) Two-qubit MS stabiliser readout.– Let us now introduce a circuit for the readout of a 4-qubit stabiliser based on 2-ion MS gates. Such a circuit could be constructed by compiling the known circuits based on 4 CNOTs, using the fact that a 2-ion MS gate is equivalent to a 2-qubit CNOT, up to local unitary operations Nebendahl-PRA-2009 . Alternatively, one can construct such circuits minimizing the building blocks by noticing that

[TABLE]

where $\ket{x_{\pm}}=(\ket{0}\pm\ket{1})/\sqrt{2}$, and $\ket{y_{\pm}}=(\ket{0}\pm{\rm i}\ket{1})/\sqrt{2}$. This identity shows that a combination of a two-qubit MS gate between an ancillary-data qubit pair $(0,j)$, and a single-qubit $\pi/2$-pulse on the data qubit $j$, acts essentially as a rotated version of a CNOT gate. Thus, applying sequentially this pair of operations between the ancilla qubit and the four plaquette qubits, realizes the mapping of certain stabiliser $S^{(m)}_{x}=X_{j_{1}}X_{j_{2}}X_{j_{3}}X_{j_{4}}$ onto the ancilla qubit. Considering that the ancilla qubit is initially in $\ket{0}_{0}$, while the data qubits are in an arbitrary state $\ket{\psi}$, we find

[TABLE]

Hence, the two possible values of the stabilizer $\pm 1$ can be directly inferred by measuring the ancilla qubit $M_{0}^{z}$ in the $Z$-basis, such that the initial and final Ramsey pulses of the multi-qubit readout of Fig. 5 are not required any longer. Similarly, $Z$-type stabiliser operators can be measured by interchanging the basis of the data (plaquette) qubits from $X$ to $Z$ before and after applying the above sequence of gates, $Y_{j}(\pi/2)X_{j}Y_{j}(-\pi/2)=-Z_{j}$. The resulting circuits for measuring $X$- and $Z$-type stabiliser operators are shown in Fig. 6.

Note, however, that using a single ancillary qubit does not allow for a fault-tolerant measurement of the stabilizers for the 7-qubit code. The reason is that a single-qubit error can propagate to the set of data qubits and lead to two errors therein. These will then, in the subsequent round of QEC, lead to a logical error (see Fig. 7 for details). Similar effects would occur for the previous scheme based on 5-ion MS gates, also forbidding a fault-tolerant behavior.

V.2.3 Fault-tolerant DiVincenzo-Shor stabilizer readout with 2-ion MS gates

In this section, we develop a fault-tolerant version of the stabiliser readout using 2-ion MS gates. Let us consider the CNOT-based approach by DiVincenzo-Shor (DVS) shor_ft_qec , in which a 4-qubit ancilla GHZ-type state (”cat state”) is prepared, verified, and subsequently coupled transversally, and thus fault-tolerantly, to the respective four data qubits. This scheme requires, besides the 7 data qubits of the color code, 5 additional ancilla qubits: 4 for the ancilla GHZ state (indices $a_{1}$, $a_{2}$, $a_{3}$ and $a_{4}$) and one extra ancilla qubit ($a_{0}$) for verifying the GHZ state in a measurement. The preparation, verification, transversal coupling, and decoding for the readout of a single stabiliser can be accomplished by 12 CNOT gates, and a couple of Hadamards. Since the CNOT for two ion qubits in a larger register can be constructed using a single MS-gate and 4 single-qubit rotations Y_{a_{n}}\big{(}\frac{-\pi}{2}\big{)}X_{a_{n}}\big{(}\frac{-\pi}{2}\big{)}X_{j_{n}}\big{(}\frac{-\pi}{2}\big{)}X_{a_{n},j_{n}}^{2}\big{(}\frac{\pi}{2}\big{)}Y_{a_{n}}\big{(}\frac{\pi}{2}\big{)}={\rm e}^{{\rm i}\pi/4}(\ket{0}\bra{0}_{a_{n}}\otimes\mathbb{I}_{j_{n}}+\ket{1}\bra{1}_{a_{n}}\otimes X_{j_{n}}), the straightforward translation of the DVS protocol onto a trapped-ion hardware would require 12 MS gates, and 50 single-qubit rotations. Let us now discuss, step by step, an alternative MS-based approach that reduces the total number of gates (see Figs. 8 and 9).

(i) Preparation of the ancilla GHZ state: The 4-qubit GHZ state can be prepared by a sequence of 2-ion $XX$ and $YY$ MS gates acting on the ancilla qubits initially prepared in $\ket{\psi_{0}}=\ket{0_{a_{1}},0_{a_{2}},0_{a_{3}},0_{a_{4}}}$. A single $XX$ MS gate leads to a Bell state of the ancilla qubits $a_{1}$,$a_{2}$, rewritten as X^{2}_{a_{1},a_{2}}\big{(}\frac{\pi}{2}\big{)}\ket{\psi_{0}}=\frac{1}{\sqrt{2}}\big{(}\ket{x_{+}}_{a_{1}}\!\ket{y_{-}}_{a_{2}}+\ket{x_{-}}_{a_{1}}\!\ket{y_{+}}_{a_{2}}\big{)}\ket{0_{a_{3}},0_{a_{4}}}=:\ket{\psi_{1}}. A subsequent $YY$ entangling gate applied to $a_{2}$ and $a_{3}$ extends this state into a 3-qubit GHZ-type state, namely Y^{2}_{a_{2},a_{3}}\big{(}\frac{\pi}{2}\big{)}\ket{\psi_{1}}=\frac{1}{\sqrt{2}}\big{(}\ket{x_{+}}_{a_{1}}\!\ket{y_{-}}_{a_{2}}\!\ket{x_{-}}_{a_{3}}+\ket{x_{-}}_{a_{1}}\!\ket{y_{+}}_{a_{2}}\!\ket{x_{+}}_{a_{3}}\big{)}\ket{0_{a_{4}}}=:\ket{\psi_{2}}. Finally, a $X$-type MS gate on $a_{3}$ and $a_{4}$ produces a GHZ-type state X^{2}_{a_{3},a_{4}}\big{(}\frac{\pi}{2}\big{)}\ket{\psi_{2}}=:\ket{\psi_{3}}, where we have introduced

[TABLE]

which can be converted into a standard GHZ-type state after its verification.

(ii) Verification of the GHZ state: This GHZ-type state $\ket{\psi_{3}}$ (27) can be verified by coupling the first $a_{1}$ and fourth $a_{4}$ ancilla qubits to the verification ancilla qubit $a_{0}$ via $X$- and $Y$-type MS gates which, together with local unitary rotations on the verification qubit, yield the operations $U_{x}^{(a_{1},a_{0})}$ and $U_{y}^{(a_{4},a_{0})}$ in Eq. (25), respectively. The expressions (25) show that, under these two operations, the ancilla verification qubit initially prepared in $\ket{0}_{a_{0}}$ is either not flipped at all (i.e. first component of the GHZ state $\ket{\psi_{3}}$), or flipped twice (i.e. second component of the GHZ state $\ket{\psi_{3}}$), gaining an additional phase shift that can be compensated with a local Z-rotation. Therefore, for a perfect preparation of the GHZ-type state U_{y}^{(a_{4},a_{0})}Z_{a_{0}}\big{(}\frac{-\pi}{2}\big{)}U_{x}^{(a_{1},a_{0})}\ket{0_{a_{0}}}\ket{\psi_{3}}=\ket{0_{a_{0}}}\ket{\psi_{3}}, and the verification qubit should ideally end up in $\ket{0}_{a_{0}}$ which can be checked by measuring $M_{0}^{z}=+1$ in the computational basis.

The state $\ket{\psi_{3}}$ is finally converted by local $Z$-rotations into the desired GHZ state Z_{a_{2}}\big{(}\frac{\pi}{2}\big{)}Z_{a_{3}}(\pi)Z_{a_{4}}\big{(}\frac{-\pi}{2}\big{)}\ket{\psi_{3}}=\ket{\text{GHZ}}, where we have introduced

[TABLE]

which will be coupled transversally to the data qubits. The circuit for the MS-based scheme used for the preparation as well as verification of the ancilla GHZ state is shown in Fig. 8.

(iii) Coupling of the ancilla GHZ state to the data qubits: To realize the readout of an $X$-type stabiliser operator, the verified 4-qubit ancilla GHZ state is then coupled transversally to the corresponding four data qubits. Again, using the operators in Eq. (25), it can be shown that the sequence of pairwise unitaries $U_{x}^{(4)}=\prod_{n=1}^{4}U_{x}^{(a_{n},j_{n})}$ leads to $\ket{\psi_{4}}=U_{x}^{(4)}\ket{\text{GHZ}}\ket{\psi}=\frac{1}{\sqrt{2}}(\ket{x_{+},x_{+},x_{+},x_{+}}\ket{\psi}+\ket{x_{-},x_{-},x_{-},x_{-}}S_{x}^{(m)}\ket{\psi}),$ where we have introduced an arbitrary basis state of the four data qubits $\ket{\psi}$. Accordingly, these sequential operations map the $\pm 1$ eigenvalue information of a stabiliser $S^{(m)}_{x}=X_{j_{1}}X_{j_{2}}X_{j_{3}}X_{j_{4}}$, and thus part of the error syndrome, onto the relative phase of the ancillas \ket{\psi_{4}}=\frac{1}{\sqrt{2}}\big{(}\ket{x_{+},x_{+},x_{+},x_{+}}\pm\ket{x_{-},x_{-},x_{-},x_{-}}\big{)}\ket{\psi}. This relative phase of $+1$ or $-1$ will result in even/odd parity of the combined outcome $(M_{1}^{z},M_{2}^{z},M_{3}^{z},M_{4}^{z})$ of a subsequent measurement of the four ancilla qubits in the $Z$-basis as shown in Fig. 9. In this figure, we also show the additional local rotations that must be performed for the measurement of $Z$-type stabilizers. As customary for stabilizer codes, the same circuit can be used as a building block to prepare the state of the encoded qubit.

To demonstrate the fault-tolerant nature of the constructed circuit, we must show that, if at most a single error (single-qubit error, two-qubit gate error, or measurement error) occurs anywhere in the circuit, it will not result in an uncorrectable error on the data qubits that would yield a logical error. Note that $X$ errors resulting at the output of the circuit are not dangerous as they can only result in a wrong relative sign of the GHZ state, which is later coupled to the data qubits. This can ultimately result in a stabiliser measurement error. Such stabiliser measurement errors are taken care of by applying several rounds (up to three) of readout to reliably decode the error syndrome. With respect to the fault-tolerance of the circuit, the key point is that such $X$ errors never propagate to data qubits since they commute with the $XX$ MS gates used during the coupling stage, both for $X$- and $Z$-type stabiliser readout.

In contrast, undetected phase flip errors ($Z$) generated during the preparation and verification of the GHZ state in Fig. 8 will propagate to the data qubit layer during the coupling step in Fig. 9, resulting in bit flip errors ($X$). If two bit flip errors are introduced into the code, this will result in an uncorrectable logical error. However, the preparation and verification circuit is constructed in such a way that any combination of two phase flip errors is detectable, as it will necessarily result in a $M_{0}^{z}=-1$ measurement of the ancilla verification qubit $a_{0}$. If this is the case, the GHZ state must be discarded, and another attempt at preparing and verifying the required ancilla GHZ state is started. It is tedious but straightforward to show that all dangerous two-qubit phase flip errors that can affect the data qubits are equivalent to a $Z_{3}Z_{4}$ error, and will be detected in the verification step through the outcome $M_{0}^{z}=-1$. Note that a $Z_{1}Z_{2}$ error is equivalent to a $Z_{3}Z_{4}$ error, as the resulting two bit flip errors in the code are equivalent up to an $S_{x}$-stabiliser. Three-qubit phase flip errors, e.g. $Z_{2}Z_{3}Z_{4}$, are equivalent to a single phase flip error $Z_{1}$ using the same argument, which propagates to the data layer, but only results in a single $X_{1}$ bit flip error. This is an allowed process, as single-qubit errors will be picked up and corrected for in the next QEC round to leading order.

V.2.4 Fault-tolerant DiVincenzo-Aliferis stabilizer readout with 2-ion MS gates

In this section, we develop a fault-tolerant MS-based version of the DiVicenzo-Aliferis (DVA) approach aliferis_ft_qec , which was originally introduced in terms of CNOT gates. Similar to the DVS scheme discussed above, an ancilla GHZ state (28) is initially prepared by a sequence of MS gates. The main difference of the DVA protocol is that its verification is postponed until the end of the readout step. Hence, the GHZ state is coupled transversally to the data qubits, after which the ancilla state is decoded and measured to obtain the stabiliser information. Importantly, the decoding circuit is constructed in such a way that it also allows one to detect unambiguously the occurrence of two single-qubit errors that have propagated to the data qubits potentially causing a logical error. If such a situation is detected, the corresponding two-qubit error correction operation is either physically applied to the data qubits, or used on a software-level to update the Pauli frame.

DiVincenzo and Aliferis argue that, by postponing the verification step involving measurements to the end, this scheme can be highly beneficial and avoid bottlenecks when the measurements of qubits are much slower than gate operations aliferis_ft_qec , which is typically the case of trapped-ion hardware (see Tables 2 and 4). Furthermore, only 4 ancilla qubits are needed, as compared to the 5 needed for the DVS scheme of Fig. 8. Another nice feature is that the verification is not of a stochastic nature, which can simplify considerably the time control and synchronization in a larger QEC protocol. Previous studies have aimed at a comparison of DVS and DVA CNOT-based schemes for the Steane code in a trapped-ion architecture Tomita-Gutierrez-PRA-2013 . Here, we develop similar schemes based on 2-ion MS gates, and take into account the specific architectural constraints of our experimental system.

*(i) DiVincenzo-Aliferis QEC based on CNOT gates.–*Figure 10 shows the standard circuit based on CNOT gates for the fault-tolerant measurement of a four-qubit stabiliser. If no error occurs at all, the measurement $M_{2}^{x}$ in the $X$ basis of the second ancilla qubit, which was initially prepared in $\ket{x_{+}}_{a_{2}}$, will reveal the desired $\pm 1$ stabiliser information after decoding. Additionally, the remaining ancilla qubits (”check qubits”) will end up in the state $\ket{0}_{a_{n}}$, and yield a $M_{1}^{z}=+1$, $M_{3}^{z}=+1$ and $M_{4}^{z}=+1$ outcome. For this circuit, it can be shown that all dangerous two-qubit errors on the ancilla qubits that propagate to the data qubits, and would induce a logical error, are equivalent to an $X_{a_{3}}X_{a_{4}}$ error. This error, which could result e.g. from a bit-flip error before the CNOT gate involving ancilla qubits $a_{3}$ and $a_{4}$ during the GHZ-state preparation (encoding), would propagate through two of the CNOT gates in the coupling step to the $j_{3}$ and fourth $j_{4}$ qubits. Note, however, that the circuit is constructed in such a way that these two errors, $X_{a_{3}}X_{a_{4}}$, propagate among the ancilla qubits during the decoding stage causing a bit flip on all three ancilla ”check” qubits, and thus yielding $M_{1}^{z}=-1$, $M_{3}^{z}=-1$ and $M_{4}^{z}=-1$. One can finally show that this outcome on the ”check” ancillas is uniquely associated to the occurrence of such an $X_{a_{3}}X_{a_{4}}$ error. In other words, any single error on a data qubit occurring during the decoding circuit cannot cause the same outcome, and cannot be thus confused with the $X_{a_{3}}X_{a_{4}}$ error. Thus, in case the $(-1,-1,-1)$ outcome is obtained, one can correct the two errors by applying $X_{j_{3}}X_{j_{4}}$ to the data qubits, be it physically or on a software book-keeping level.

Note that an additional useful feature is the fact that whenever any single or two of the ancilla ”check” qubits yields an outcome $M_{a_{n}}^{z}=-1$, this also signals that something has gone wrong, and the stabiliser information obtained from measuring the second ancilla qubit $M_{2}^{x}$ is better discarded.

(ii) DiVincenzo-Aliferis QEC based on MS gates.– Figure 11 displays the detailed circuit for a fault-tolerant DiVincenzo-Aliferis-type stabiliser measurement based on 2-ion MS gates. The encoding of the four ancilla qubits, i.e. the preparation of the four-qubit GHZ state (see Eq. (28)) by three 2-ion MS gates followed by three local $Z$-type rotations, occurs in the same way as for DVS scheme as discussed in Sec. V.2.3. Similarly, the coupling to the data qubits takes place transversally by a series of four $XX$ MS gates and four local $X$-rotations on the data qubits. Before the decoding step, the four ancilla qubits are ideally (if no error has occurred) in the state $\frac{1}{\sqrt{2}}(\ket{x_{+}x_{+}x_{+}x_{+}}\pm\ket{x_{-}x_{-}x_{-}x_{-}})$, depending on whether the data qubits are in a $+1$ or $-1$ eigenstate of the measured stabiliser. Under the two $Z$-rotations of Fig. 11, this state is transformed into \textstyle{Z_{a_{2}}\big{(}\frac{\pi}{2}\big{)}Z_{a_{4}}\big{(}\frac{\pi}{2}\big{)}}\frac{1}{\sqrt{2}}(\ket{x_{+}}_{a_{1}}\!\!\ket{x_{+}}_{a_{2}}\!\!\ket{x_{+}}_{a_{3}}\!\!\ket{x_{+}}_{a_{4}}\pm\ket{x_{-}}_{a_{1}}\!\!\ket{x_{-}}_{a_{2}}\!\!\ket{x_{-}}_{a_{3}}\!\!\ket{x_{-}}_{a_{4}})=:\ket{\tilde{\psi}_{3}}, which is a locally equivalent GHZ-type state

[TABLE]

From this state, it can be readily seen that the subsequent $X$-type MS gate on ancillas $a_{1}$, $a_{4}$ and the $Y$-type MS gate on ancillas $a_{2}$, $a_{3}$ decouple the third and fourth qubit, leaving the first two ancilla qubits in one of two possible Bell-type states \textstyle{Y^{2}_{a_{2},a_{3}}\big{(}\frac{\pi}{2}\big{)}X^{2}_{a_{1},a_{4}}\big{(}\frac{\pi}{2}\big{)}\ket{\tilde{\psi}_{3}}=\ket{{\rm B}_{\pm}}\otimes\ket{1}_{a_{3}}\otimes\ket{0}_{a_{4}}}, where

[TABLE]

Let us now discuss how to access the stabiliser eigenvalue information, simultaneously tracking possible dangerous errors. The first two ancilla qubits being in the Bell-type state (30) are measured, revealing the $\pm 1$ stabiliser eigenvalue by the odd/even parity of the outcome $(M^{z}_{1},M^{z}_{2})$. The third and fourth ancilla qubits act as checks, and are expected to yield $M_{3}^{z}=-1$ and $M_{4}^{z}=+1$ in the ideal case without any errors. Similar to the case of the DVS scheme discussed above, it can be shown that all dangerous two-qubit errors in the circuit of Fig. 11 are equivalent to a $Z_{a_{3}}Z_{a_{4}}$ error at the end of the encoding circuit. One example of such dangerous error histories is shown in Figs. 12 to 14 for the encoding, transversal coupling, and decoding steps of the DVA-type scheme, respectively. As shown in these figures, this type of $Z_{a_{3}}Z_{a_{4}}$ error (see Fig. 12), which has propagated to two $X_{j_{3}}X_{j_{4}}$ errors in the code layer (see Fig. 13), is unambiguously signaled by the $M_{3}^{z}=+1,M_{4}^{z}=-1$ outcome of the third and fourth ancilla qubit (see Fig. 14).

VI Trapped-ion protocols to assess the break-even point for beneficial QEC

Based on the description of the experimental capabilities in Sec. III, the development of an effective error model composed of different quantum channels in Sec. IV, and the trapped-ion toolbox for QEC in Sec. V, we can now present the different trapped-ion protocols to asses the break-even point for beneficial QEC (2)-(3) in the 7-qubit color code. This code, being equivalent to the 7-qubit Steane code steane-prl-77-793 , has been subject to a series of studies assessing its QEC performance and error thresholds (we refer the reader to Ref. Terhal2009 for a comparative study of this code and other small-scale QEC codes). Depending on the modelling of the noise, typical error thresholds lie around the $10^{-4}$ level Terhal2009 ; Aho2006 . Additionally, we would like to remark that previous work has also explored the performance of this code in a trapped-ion architecture Tomita-Gutierrez-PRA-2013 , considering CNOT-gate and shuttling-based protocols in a two-dimensional array of coupled traps, paying particular attention to the influence of available ancilla resources.

As advanced in the introduction, the QEC protocols that we explore in this work exploit optimised MS-gate-based circuitry, and are embedded in a single linear segmented trap. The various protocols we investigate are based on either on (i) spectroscopic decoupling, i.e. storing idling qubits temporarily in Zeeman sub-levels that do not belong to the computational subspace, or on (ii) ion reconfiguration, i.e. combination of shuttling, splitting, merging, and rotations of the ion crystal to physically move idle qubits to the storage region. For all the different protocols within these two types, we have developed three conceptual layers:

Real-space representation: A sketch of the sequence of operations in real space and time, which is particularly useful to visualize the effect of ion crystal reconfigurations in the shuttling-based protocols. 2. 2.

Circuit representation: Circuit diagrams which show the entirety of elementary operations that should be applied in the real experiment. Besides the gate operations, measurements, etc., these contain a list of the ion crystal reconfiguration building blocks (splitting, shuttling and merging operations). 3. 3.

Quantum channel representation: Circuit diagrams containing information about the sequence of quantum channels, which reflect the effective noise models used to describe imperfect operations in Sec. IV, as well as the dependences of channel parameters on the experimentally relevant metrics (e.g. gate times and infidelities, coherence times). This underlies the numerical simulations of the following section.

In the following, we shall describe in detail these three layers for the first shuttling-based protocol, and restrict to the real-space representation for the remaining ones.

VI.1 Non-fault-tolerant trapped-ion QEC protocols

In this section, we discuss the protocols for a QEC cycle with an 8- or 9-ion crystal, where $7$ data qubits are used to implement the aforementioned 7-qubit color code, and 1 additional ancillary qubit is exploited for syndrome extraction by measuring the 6 stabilizers in Eq. (23) (see Fig. 3). For one of the protocols, we will require an extra ion to implement laser cooling of the ion crystal. By using a single ancillary qubit, this QEC protocol cannot be made fault-tolerant (see Sec. V.2.2). However, even in this case, it is very instructive to use it to benchmark the experimental progress towards fault-tolerant quantum computation according to the criteria in Eqs. (2)-(3). As shown in Sec. VII, one could already prove the non-trivial beneficial character of our QEC protocol by improving the experimental hardware according to Tables 2 and 4. Moreover, this is the less-demanding possible scenario for future trapped-ion experiments along the lines of nigg-science-345-302 .

We consider both shuttling- and hiding-based approaches, as well as stabiliser mappings based on either multi-qubit MS gates (cf. Fig. 5) or on a sequence of 2-qubit MS gates (cf. Fig. 6). The criteria for beneficial QEC exposed in Eqs. (2)-(3) of Sec. II, can be assessed through the quantum-information protocols detailed in Table 1. In order to implement this protocol numerically, we now present the explicit scheduling of different shuttling- and hiding-based approaches for a single QEC cycle in a trapped-ion 7-qubit color code.

VI.1.1 Shuttling-based, single-species, multi-qubit gate protocol

Let us start by considering a shuttling-based approach to the multi-qubit mapping of each of the data-qubit stabilizers onto the ancillary qubit (see Fig. 5), both of which belong to the same atomic species. Since all the ions in this protocol are of the same species, and the lasers responsible for the entangling MS gates (4) act globally on the ion chain, one needs to combine storage and processing/manipulation zones in the trap, and reconfigure the ion crystal in order to apply sequentially the readout of all six stabilizers of the code (23).

In Fig. 15, we depict the real-space representation of this protocol which utilizes a single arm of the segmented ion trap of Fig. 1. In this case, we will only make use of two storage zones ${S}_{1}$ and ${S}_{2}$, surrounding a central manipulation region $M_{2}$, within one arm of the trap. Initially, Alice encodes an arbitrary state $\ket{\psi}=\alpha|{0}\rangle+\beta|{1}\rangle$ in the 7 data qubits, and hands them to Bob, who can ask his assistant Igor to perform an imperfect round of QEC before he decides if the original state was either $\ket{\psi}$ or $\ket{\psi_{\bot}}=\beta^{*}|{0}\rangle-\alpha^{*}|{1}\rangle$. In this protocol, Igor has an additional ancillary ion of the same species, which can be used for the readout of the stabilizer information. He distributes the 7 data qubits within the two storage zones forming two separate ion crystals, and locates the ancillary ion in the manipulation zone (see Fig. 15). The steps of such a QEC cycle are described by different operations in the figure (time evolution occurs downwards), in which these crystals are split, such that the data qubits of a given stabilizer can be shuttled to the manipulation region and merged with the ancillary ion. At this stage, Igor applies the stabilizer mapping in Fig. 5 by shining the corresponding lasers onto the ions of the manipulation zone. After splitting the chain, and shuttling the data qubits back to the storage zone, the ancillary ion can be measured by state-dependent fluorescence, such that Igor can collect the syndrome information. Note that the scattered photons do not affect the information stored in the physical qubits, as these have been shuttled to the distant storage regions.

These steps must be repeated for each of the stabilizers (23) of the 7-qubit color code. As depicted in Fig. 15, the last two stabilizers mappings require a reordering of the ion crystals in the storage zones, which can be accomplished by rotating the crystal, which effectively implements a mirror image about the symmetry axis of the code, such that the roles of plaquettes 2 and 3 are interchanged (See Fig. 3). In this way, one can repeat the same crystal-reconfiguration operations of the second set of stabilizers, and finalize the syndrome extraction. We note that similar combinations of rotations and split/shuttle/merge operations will be a crucial building block of all shuttling-based QEC protocols in the following sections.

In Fig. 16, we describe the circuit representation of this protocol, where one can keep track of the sequential order of the different elementary operations (time evolving from left to right in this figure). At each time-step, a given box describes which operation takes place on which set of ions (see the caption of Fig. 16 for a detailed account).

In order to explore the performance of Igor’s QEC cycle under realistic experimental errors and sources of noise, we need to (i) consider the time intervals during which these operations take place, as idle qubits will be subjected to a dephasing noise (13) that degrades the information stored in the code. Moreover, we also need to (ii) consider that the ion reconfigurations excite the motional modes of the ion crystal, affecting the fidelity of the MS gates (17) in subsequent stabilizer mappings, which enter in the depolarizing channels of Eqs. (14)-(16). This goes beyond standard noise models in the literature, all of which assume that the gate errors are non-increasing with the depth of the circuit. Finally, we need to (iii) take into account that the measurement and reset of the ancillary ion are also faulty, which is accounted by the bit-flip channel (22). Accordingly, the circuit representation is translated onto the quantum channel description of Fig. 17, where the above errors are represented as particular Markovian error channels (see Sec. IV) with error probabilities that depend on the history of previous operations, and on the total time required for each of the steps (see the caption of Fig. 17 for the details). This circuit of ideal gates interspersed with dephasing, depolarizing, and bit-flip channels, is the one that is numerically simulated in Sec. VII to estimate the break-even point that determines when the QEC cycle becomes useful.

VI.1.2 Shuttling-based, two-species, multi-qubit gate protocol

Let us now reconsider the same QEC cycle with a two-species ion crystal, where the $7$ data qubits and the additional ancillary qubit are of a different atomic species. This has two important implications: (i) it reduces the number of crystal reconfigurations that must be implemented by Igor, since the measurement of the ancillary qubit does not need to be performed on an isolated qubit. Even if the physical ions are in the same processing region as the ancilla ion, the photons scattered while the ancilla ion is being measured will not be absorbed by them, and thus the encoded state shall not be affected. On the other hand, the photon recoil onto the ion crystal can induce motional excitations that would compromise the fidelities of subsequent MS entangling gates. However, (ii) the use of two species allows Igor to apply sympathetic re-cooling of the ion crystal, which minimizes the number of motional excitations due to recoil or crystal reconfigurations prior to any MS gate in the stabilizer mappings, and thus reduces the motional error (17) during the entangling gates.

The real-space representation of this 2-species protocol is depicted in Fig. 18. As advanced previously, the number of crystal reconfiguration operations is highly reduced with respect to the one-species protocol in Fig. 15. Additionally, intermediate re-cooling steps can also be introduced previous to any stabiliser mapping, such that the fidelity of the corresponding MS entangling gates is not compromised as the QEC cycle proceeds. The circuit and quantum channels representation are somewhat similar to those of the one-species protocol in Figs. 16 and 17, and will not be presented in detail. However, we remind that they are also important, as they provide a scheduling of the physical operations that would have to be applied in an experiment, and also contain the relevant information for the numerical modeling of Sec. VII.

VI.1.3 Shuttling-based, two-species, two-qubit gate protocol

We now consider a similar shuttling-based two-species QEC protocol, but relying on sequential 2-qubit MS gates for the stabilizer mapping (see Sec. V.2.2). We recall that using the combination of a MS gate between a pair of ancilla $a_{0}$ and data $d_{j}$ qubits, followed by a rotation on the data qubit, one can implement the conditional gates $U_{x}^{(a_{0},d_{j})}$ in Eq. (25). The sequential combination of these gates, together with possible rotations on data qubits $Y_{d_{j}}(\pm\pi/2)$, leads to a mapping of the stabilizer information into the ancilla qubit (see Fig. 6), which can be obtained from its fluorescence $M_{0}^{z}$.

In order to reduce the complexity of all the crystal reconfigurations required to perform the sequential gates of the QEC cycle, Igor shall make use of all of the five regions of the segmented trap in Fig. 1 (i.e. two manipulations zones $M_{1}$ and $M_{2}$ interspersed between three storage regions $S_{1},S_{2},S_{3}$). Moreover, since the ancilla qubit is not measured after each of the conditional gates $U_{x}^{(a_{0},d_{j})}$, Igor needs to re-cool the crystal several times during each stabiliser readout without affecting the ancilla and data qubits. Therefore, we consider that the ancilla and data ions are of the same species, but equip Igor with an extra ion of a different species $c$ for sympathetic re-cooling of the crystal. Hence, Igor has $7+1+1$ ions for his imperfect round of QEC, which he distributes within the segmented trap as depicted in the starting configuration of Fig. 19. The data qubits of a given stabilizer are shuttled one-by-one from the storage regions onto the manipulation zone $M_{2}$, where sympathetic re-cooling represented by blue rectangles is applied prior to the data-ancilla mapping $U_{x}^{(a_{0},d_{j})}$ or $Y_{d_{j}}(+\pi/2)U_{x}^{(a_{0},d_{j})}Y_{d_{j}}(-\pi/2)$ for an $X$- or $Z$-type stabiliser readout, which are labelled as $Map(a_{0},d_{j})$ and represented by red ellipses. After all the four data qubits of a particular stabiliser have been coupled to the ancilla ion, Igor can proceed to isolate the ancilla-cooling pair of ions in the manipulation zone, and collect the state-dependent fluorescence of the ancilla ion, inferring in this way the stabilizer information.

As can be seen in this figure, the complexity of the crystal-reconfiguration operations increases considerably, such that idle qubits will suffer more environmental dephasing during the QEC cycle. Therefore, although the multi-qubit errors (16) of the collective stabiliser mapping of the previous subsections are avoided in this scheme, and the fidelity for 2-qubit MS gates is higher than that of 5-multi-qubit ones (see Table 2), we do not expect any big improvement of the non-fault-tolerant protocol. A possible advantage can only take place if a fully fault-tolerant scheme is implemented.

VI.1.4 Hiding-based, two-species, multi-qubit gate protocol

An alternative approach to the shutting-based protocols presented above is to work with a static ion crystal, but equip Igor with spectroscopic decoupling capabilities. Accordingly, Igor can take out a particular subset of ions from a given stabilizer mapping by spectroscopically decoupling them, and coherently shelving the physical qubit’s population in electronic states that do not couple to the lasers driving gate operations on the qubit transition (see also Secs. III.2 and IV.3).

We consider the scenario of a two-species encoding. This allows one to measure the ancilla qubit after the stabiliser mapping, without affecting the state of non-hidden data qubits. In addition, the ancilla qubit, encoded in an ion species different from the data qubits, enables recooling of the entire ion string after each fluorescence ancilla measurement, minimizing thus motional excitations due to photon recoil.

The scheme for a complete QEC cycle involves an overall number of 12 hiding and un-hiding operations, each of which can be realized by a composite pulse sequence (9 single-ion pulses detailed in Sec. III.3) that maps the state of data qubits to a set of storage $D$ states coherently nigg-science-345-302 . Rather than displaying the real-space configuration, it is more instructive in this case to give the circuit representation, since no crystal reconfiguration is used, and all ions reside in the same manipulation zone. In Fig. 20, green boxes depict the hiding/un-hiding pulses, which determine which qubits are involved in the respective stabilizer represented by red boxes. Idle qubits undergo dephasing with a strength according to the time durations of the respective mapping, measurement, recooling and (un-)hiding operations acting on the other qubit(s).

VI.2 Fault-tolerant trapped-ion QEC protocols

In this section, we discuss the trapped-ion protocols for fault-tolerant QEC with the 7-qubit color code. As emphasized in previous sections, we need to go beyond a single ancillary qubit by either focusing on the DVS or DVA schemes described in Secs. V.2.3 and V.2.4, respectively. We will consider only shuttling-based approaches, since the number of spectroscopic decoupling pulses for a hiding-based scheme would increase dramatically for these fault-tolerant protocols. Therefore, a fully hiding-based fault-tolerant protocol with a static ion crystal is likely to perform worse than the shuttling-based alternatives.

VI.2.1 Shuttling-based, two-species, DiVincenzo-Shor scheme

We first consider the trapped-ion implementation of the DVS scheme, which combines the ancilla encoding and verification of Fig. 8 with the subsequent transversal coupling to data qubits of Fig. 9 for each of the code stabilizers. To implement a cycle of QEC using these scheme, Igor must be equipped with 5 additional ancillary ions $a_{0},\cdots,a_{4}$. Furthermore, to simplify all the required crystal reconfigurations, we equip Igor with a couple of cooling ions $c_{1},c_{2}$ of a different atomic species such that sympathetic re-cooling can be implemented prior to any entangling MS gate. Igor distributes the 7 data qubits, together with the ancillary and cooling ions, according to the stating configuration of Fig. 21. In the left panel of this figure, prior to the ancilla measurement $M_{0}^{z}$, we depict the different operations that Igor must apply for the ancilla encoding and verification. If this part of the protocol is successful $M_{0}^{z}=+1$, one relabels the ancillary qubits, and proceeds with the rest of the scheme. The set of instructions that follow this relabelling correspond to the transversal coupling to the data qubits of Fig. 9, and ends in the measurement of the remaining ancillas $(M_{1}^{z},M_{2}^{z},M_{3}^{z},M_{4}^{z})$, the parity of which allows Igor to infer the $S_{\alpha}^{(1)}$ stabiliser information. After re-ordering of the data qubits to move the ions of another stabiliser to the storage region $S_{2}$, one can repeat the same procedure in Fig. 9 for the next stabilizer, and proceed to complete a full round of QEC. Note that most of these re-ordering operations can be implemented during the measurement period, such that no extra dephasing occurs. To take advantage of the fault-tolerant nature of the scheme, Igor should run two such full rounds of stabiliser readout. If the results coincide, he should then apply a decoder to determine which error has occurred, and apply a particular single-qubit $X$- or $Z$-type rotation to the corresponding data qubit to correct for it. If the measurement results do not match, Igor should apply the full readout scheme once more, and use the decoder on the third set of stabiliser values.

VI.2.2 Shuttling-based, two-species, DiVincenzo-Aliferis scheme

Let us now describe the trapped-ion implementation of the DVA scheme, in which the ancilla encoding, coupling, and verification must occur along the prescription of Fig. 11. In this case, it suffices to equip Igor with 4 additional ancillary ions $a_{1},\cdots,a_{4}$, and a couple of cooling ions $c_{1},c_{2}$ of a different atomic species. The distribution of these ions within the different zones of the segmented trap is specified in the real-space representation of Fig. 22, where we also list the operations that Igor must perform for the readout of the $S_{\alpha}^{(1)}$ stabilizer. In contrast to the DVS scheme, the verification step takes place in the final measurement step, and depends on the outcome $(M_{3}^{z},M_{4}^{z})$ of the check ancilla qubits. If Igor obtains $(+1,-1)$, this signals that two errors have propagated into the code space, and Igor must apply $X$- or $Z$-type rotations to the $(j_{3},j_{4})$ data qubits. Simultaneously, Igor uses the parity of $(M_{1}^{z},M_{z}^{z})$ to infer the eigenvalue information of the stabiliser. Once again, he proceeds with the readout of the remaining stabilizers in a modular fashion, which require an intermediate re-ordering of the data qubits (i.e. bringing the ions belonging to the stabiliser to be measured into the storage zone $S_{2}$). Once this is achieved, the sequence of operations is again described by the real-space representation of Fig. 22.

Once again, to take full advantage of the fault-tolerant nature of the scheme, Igor must perform two or three rounds of stabiliser readout and then apply a minimum-weight decoder to determine which error has occurred, and how to correct it.

VII Numerical studies of the performance of trapped-ion QEC protocols

VII.1 Computing resources and numerical approach

Having established our criterion for beneficial QEC in Eqs. (2) and (3), and described the different trapped-ion protocols together with their quantum channel description in Figs. 15-22, let us now describe our numerical approach to assess the performance of these QEC schemes.

Our strategy for numerical analysis is to perform exact simulation of the physical system using pure states in a Monte Carlo method. The results of the simulation are achieved by averaging over the output of at least ten thousands of individual runs; in each run, at each opportunity for an error event the question of whether it occurs (and when relevant, the error’s severity) is resolved by drawing a random number. Once sufficiently many results are aggregated, one obtains the same data as would result from a single numerical run using a density matrix. The advantages are twofold: firstly, the memory requirements of the pure state simulation are more modest, allowing for simulation of on the order of 30-40 qubits. A direct simulation of 30 qubits with the density matrix approach would require a matrix of $2^{60}\approx 10^{18}$ elements, which is infeasible. This will become crucial when considering fault-tolerant schemes, and also when upgrading these protocols to include larger-distance codes or instances with more than one logical qubit. The second advantage of the Monte Carlo approach is that it is trivial to parallelize, thus one can fully make use of cluster computing resources.

The hardware used for this work is a cluster of approximately 400 nodes, each of which is based on a motherboard with two Intel E5-2640v3 CPUs, and between 64 and 256GB of memory. The nodes are connected by Intel TruScale QDR Infiniband, and in principle they can be efficiently used to collectively model a quantum system. However, in practice, the efficient use is to operate the nodes in parallel and independently of one another, aggregating results afterwards according to the Monte Carlo paradigm as outlined above.

VII.2 Simulation results

In this subsection, we present the simulation results for all the different QEC protocols discussed above. One type of QEC cycles is based on the multi-qubit MS gate (see Figs. 15 and 18), whereas the other is based on the sequential 2-qubit MS gates, including both the non fault-tolerant (see Fig. 19) and the fault-tolerant approaches (see Fig. 21 and 22. Generally, we will use the same Alice-Igor-Bob framework for assessment of the beneficial character of QEC, which has been discussed earlier: Alice prepares a perfect instance of the encoded qubit, and this logical qubit is then subjected to a period of environmental exposure during which Igor may perform one or more cycles of error correction, before Bob assesses the integrity of the qubit by attempting to determine the encoded state (from one of the two choices). For the simulations with multi-qubit MS gates, the initial encoded state was chosen randomly, while for those simulations with 2-qubit MS gates, since they proved to show more encouraging results, each time we let Alice encode only the $|+\rangle$ state, which is the most vulnerable state under dephasing environmental noise. Therefore, Bob’s success probability $\mathcal{P}_{\rm B}$ shown in the following two sections has two ranges: the first is from 1 to 0.75, while the second varies from 1 to 0.5, but both cases correspond to the complete decay of the qubit from full coherence to total decoherence.

For each of these cases, two sets of figures are presented depending on the choice of trapped-ion parameters: First, we present a set of figures drawn from the best fidelities that have been reported to date in relevant experiments. An exception is the single-species shuttling-based scheme and the two-species shuttling-based DVS scheme, where the performance of the MS gates after a few stabilizer mappings is expected to be so bad that one can directly discard the approach (see Tables 2-5). Second, we present a set of figures for the future parameters that we anticipate will be possible to reach in the near future. We refer to these two cases as the “Current” and the “Future” (or “anticipated”) performance, respectively.

VII.2.1 Shuttling- and hiding-based QEC with multi-qubit MS gates

The figures 23-27 in this section constitute the results of our first set of simulations, i.e. those involving MS gates with multiple qubits (5 qubits, specifically). Note that for these simulations, the qubit reset fidelity is assumed to be $5\cdot 10^{-3}$, see Table 2. The results shown in each figure correspond to one of the trapped-ion QEC protocols described in detail in the previous section, and we refer to them here as “single-species shuttling without cooling” (shuttling-based protocol 1, in Fig. 24), “dual-species shuttling with cooling” (shuttling-based protocol 2, in Figs. 25 and 26) and “hiding” (hiding-based protocol, in Figs. 23 and 27).

Each data point in our figures is the averaged result of at least $40,000$ runs. Each curve is formed from $200$ data points and therefore involves eight million runs in total. For reference, the grey curves show how a single physical qubit would perform if used in the same setting described in Sec. II, i.e. Alice prepares it in a given state and Bob measures it to guess the state (versus the orthogonal state) after the state has been subjected to environmental noise. Moreover, the red curve in each figure shows the performance of the color code of $7$ physical qubits, but without the error correction provided by ‘Igor’ mid-way through the time $\tau$ during which the encoded state is subjected to environmental noise. The blue curves show how this changes when indeed Igor’s cycle is performed. Regrading the criteria for beneficial QEC discussed in Sec. II, we wish to see the blue curve above the red one such that criterion (2) is achieved, and ideally even above the grey one, implying that (at least for some choices of duration of the experiment) equation (3) is also fulfilled. If our simulations display such crossing, we can conclude that advantageous QEC could be achieved in the experiments, given that the particular performance of the different building blocks is realized. We note that in Fig. 26 there is an additional curve, in green, which shows the effect of applying Igor’s correction twice, at 33% and 67% of the Alice-Bob time interval; again, one hopes to see the curves associated with Igor surpass the red, or even the grey curves.

From our numerical simulations (see Fig. 23 for the hiding approach), we can conclude that the “Current” performance figures for the gate times, fidelities, and so on, would be in general insufficient to prove a beneficial QEC cycle: the curves representing the integrity of the logical qubit at the end of the period $\tau$ are strictly lower when error correction is applied mid-way, versus simply omitting to perform any such correction. This is principally caused by the complexity of the required circuits, and by the higher error rate of the entangling MS gates as compared to other building blocks of the protocol. Essentially, in these low-distance codes, the MS gates introduce more noise than can be removed by the QEC cycle. For the shuttling-based approaches with current parameters (not shown in figures), a similar poor performance is found (e.g. for the one-species scheme without re-cooling, the MS gates become so noisy after a few rounds (see Table 5), that we can directly discard it from reaching the break-even point (2)).

Remarkably, our results for the “Future” performance are far more encouraging. The least successful hardware variant is QEC protocol based on a single-species shuttling without re-cooling (cf. Fig 24). For this protocol, the crossing into beneficial effects of the QEC cycle (2) occurs only when the total time $\tau$ is such that the logical qubit receives considerable dephasing from the environment. Moreover, the beneficial effect can vanish entirely when one moves to a more pessimistic model for the MS gate noise including correlations (noise models (ii) and (iii) discussed in Sec. IV.2). Additionally, the desirable property of outperforming the unprotected physical qubit in our particular task of state discrimination through the QEC cycle (3) cannot be achieved. Fortunately, the results for the “two-species shuttling” (cf. Figs. 25 and 26) and the “hiding” protocols (cf. Fig. 27), using the “future” performance numbers, are considerably more encouraging. One sees that the crossing to a beneficial error correction cycle happens early and with a high value of the integrity. Importantly, this implies that multiple rounds of error correction can be beneficially applied, such that the logical qubit can be sustained for a longer time (see Fig. 26).

Let us now address if one could obtain still better results by implementing the QEC using sequential 2-qubit MS gates rather than multi-qubit MS gates. While this would necessitate more gates in total, each gate has a higher fidelity and, moreover, would propagate errors from the ancillary qubits onto the data qubits in a more restricted fashion. In addition to addressing this question numerically, the sequential 2-qubit MS gates will be an essential ingredient for the realization of fault-tolerant QEC, which we also explore in this section. The necessary analysis and scheduling for this second approach was described above in Section VI.2. We now describe the results of the corresponding set of simulations, which employ the 2-qubit MS gate as the entangling operation. For this second set of simulations, we also assumed the better value of $1\cdot 10^{-3}$ for the qubit reset fidelity from Table 2.

VII.2.2 Shuttling-based two-species QEC with 2-qubit MS gates: exploring fault tolerance

Before presenting the simulation results to explore the break-even point for the full fault-tolerant QEC protocols, let us first present a simpler analysis to verify that the DiVicenzo-Shor and DiVicenzo-Aliferis schemes with MS-gates of Sec. V are indeed fault tolerant in the formal sense. To achieve this, the periods $\tau$ of environmental exposure in the Alice-Igor-Bob scenario are removed. Instead, Alice presents the flawless logical qubit directly to Igor, who performs a redundant round of error correction, and then directly passes the logical qubit to Bob for the logical state discrimination. Hence, there is no effect of the environmental noise, except during the time when Igor is applying his imperfect error correction cycle. We introduce a parameter $\lambda$ to control the severity of the imperfections in the operations applied by Igor, which multiplies the error rates that have been identified as the expected hardware targets (see Tables 2-5). Setting $\lambda=1$ corresponds to assuming that all these targets are exactly met. By plotting Bob’s success probability $\mathcal{P}_{\rm B}$ as a function of $\lambda$, we should see a linearly descending curve for a non-FT protocol (because any single gate failure within Igor’s circuit can have the consequence of reducing $\mathcal{P}_{\rm B}$), whereas a characteristically inverted quadratic curve should arise for a true FT protocol (because it requires two or more gate failures to damage the integrity, the probability of which goes as $\lambda^{2}$). The results of our numerical simulations are shown in Figure 28. The predicted shape of the curve is indeed observed through our numerics, thus verifying that the analytically derived protocols based on MS gates, and their translation into the numerical simulation, are correctly fault tolerant.

It is relevant to note that our approach of requiring better-than-breakeven performance, when Igor enters the picture, is closely related to the concept of a pseudo-threshold where a higher level of concatenation outperforms the preceeding level (see e.g. Ref. Aho2006 ). Moreover, as we noted in the introduction to Section VI, thresholds in the range of $10^{-4}$ are expected in the context of fault tolerance Terhal2009 ; Aho2006 . Therefore the crossing seen in Fig. 28 and the corresponding gate fidelities (i.e. $1.75$ times those in Tables 2-5) are reassuringly close to expectations. The very recent suggestions of Chao and Reichardt ChaoReichardt2017 for smaller ancilla structures in fault tolerant circuits may provide some further boost to the transition.

Having thus verified the nature of the circuits, we can proceed to assess their performance when there are finite periods of exposure to the environment. First, we simulate using the current values of operational infidelities from Tables 2-5. The results are shown in Fig. 29. In this Figure, and the remaining figures in this section, the underlying protocol is that of 2-species shuttling with re-cooling (as described in previous sections). Typically, for the figures in this section, each data point is aggregated from one million numerical experiments, and each full curve involves about 100 data points; thus a curve represents approximately $10^{8}$ numerical experiments.

Figure 29 shows two reference lines for the criteria of beneficial QEC: the grey line indicates the performance with a single physical qubit (3), while the red line shows the performance with an encoded logical qubit when no error correction is performed by Igor (2). As one would expect, for very short periods of environmental exposure, the red line lies above the grey one, since the probability for two (or more) errors affecting the encoded data qubits is much smaller than the single-qubit error probability affecting the bare qubit. The yellow line shows the performance of the non-FT QEC protocol that employs sequential 2-qubit MS gates (cf. Fig. 19). As can be observed, while this line never beats the $\mathcal{P}_{\rm B}$ for the bare physical qubit (3), it does indeed exceed the performance of the encoded qubit (2) once the environmental exposure is severe. Note that this result indicates the superior performance of the QEC schemes with two-ion MS gates versus the previous ones based on multi-ion MS gates. We recall that for the corresponding multi-ion MS circuits, no clear advantage could be seen using current hardware performance i.e. there was no equivalent to the yellow-red crossing of Fig. 29. Let us remark that this result is non-trivial, since the complexity of the circuits using sequential 2-ion MS gates increases considerably with respect to the schemes that exploit multi-qubit MS gates (compare Fig. 19 to Fig. 18). In any case, the break-even point (2) is achieved when the integrity of the qubit has already decayed considerably. In order to take full advantage of QEC, improving on this feature, we will now consider the DV-S and DV-A schemes for fault-tolerant QEC.

The purple line of Fig. 29, just visible in the lower right corner of the plot, is the performance of the DiVicenzo-Aliferis (DV-A) protocol. The line does not exist over the majority of the graph simply because there is insufficient time to perform a full error correction cycle due to the long circuit depth associated to Fig. 22. The time required for Igor’s actions is summarised in Table 7. Note that for the FT approaches, it is necessary to evaluate each stabilizer more than once in order to control measurement errors; Igor performs the checks twice, and then a third time if the first two outcomes do not agree. For the Shor FT scheme, it is necessary to prepare and verify an ancilla state prior to stabilizer measurement. If the verification fails we must restart the ancilla preparation. In our simulations we allow for up to four such restarts; the probability that more are needed is negligible even for current technologies.

Notice that in Fig. 29, even when the total protocol time is large enough to permit Igor to act, the performance of the DV-A scheme is very poor, and none of the break-even points for useful QEC can be reached. The conclusion from this set of simulations is that a device built with the ‘current’ performance numbers could suffice for a basic demonstration of QEC, but could not possibly make a successful demonstration of a fault tolerant QEC code. Fortunately, this picture changes as we move to the future performance figures.

Figure 30 shows the results of repeating the simulations in Fig. 29 but now with the future anticipated values from Tables 2-5. One can see that the performance is profoundly improved. Now, the lines for the non-FT Igor, and both the DV-A and DV-S fault-tolerant protocols, lie almost on top of one another. Moreover, they both beat the “no-Igor” red line, fulfilling Eq. (2) for the whole period of time studied numerically. Additionally, they also lie above the “single physical qubit” grey line, fulfilling Eq. (3), over a wide range of values of the environmental exposure time. Finally, we note that for longer times, one could use multiple rounds of error correction. Although the non-FT curve and the two FT curves seem nearly identical, the latter do outperform the non-FT protocol for small levels of environmental error (see top left of the figure). This is consistent with our expectations from Fig. 28, where we learned that when Igor’s hardware performs exactly at the future anticipated level (i.e. $\lambda=1$) then the FT protocols are superior to the non-FT one.

The curves in Figure 30 are so encouraging, that it may turn out that the trapped-ion hardware development does not need to reach the expected values of Tables 2-5 in order to achieve the goal of beneficial QEC. In order to test this feature, we have tripled the error rates in all operations, and analyzed the performance for the QEC procedures. The results are shown in Fig. 31. We see that now there is a slight variation in performance with the non-FT Igor marginally superior to FT-DV-A, which in turn is marginally superior to FT-DV-S. (Note that it is to be expected that the FT circuits are now inferior to the non-FT circuit, since by tripling the error we are now at the far right of the range in Fig. 28.) In Fig. 31 we also see that the non-FT Igor is quicker to perform than DV-A which in turn is quicker than DV-S, since the curves are not plotted when there is insufficient Alice-Bob time interval for a complete Igor cycle. Additionally the figure shows that the crossing of the curves with error correction and the physical qubit occurs only slightly earlier in Fig. 31 (0.894) than that in Fig. 30. The conclusion from this figure is that, even using a system with three times greater operational infidelities with respect to the expected future estimates of our work, one could nevertheless support a strong demonstration of beneficial QEC. This is a very encouraging for the near-term development of trapped-ion QEC.

In our final set of simulations, shown in Figure 32, we consider multiple rounds of error correction considering the future expected levels of infidelity. These results show that, by several rounds of QEC, one can protect the logical qubit at a level that is superior to a single physical qubit over a sustained period of time. By the simple principle of selecting the number of Igor cycles according to the total Alice-Bob interval, we find that the coherence time of the logical qubit can reach values nearly twice as big as the raw physical qubit – a very significant alteration that should be easily observed and which demonstrates the ‘encoded qubit alive’ goal clearly. Of course, a factor of two is far from sufficient to achieve large scale quantum computing, but this is to be expected since we are employing only one layer of a small code. For further error suppression, one would either concatenate the Steane code (recursively replacing each data qubit with a full logical qubit, through at least three or four levels) or else one would scale using topological techniques. As the Steane code is smallest instance of the 2D Color Code, the latter would be an attractive option – but such considerations are beyond the scope of the present paper.

This concludes our review of our numerical simulations. It is worth reiterating that in every case we presume that Alice and Bob are perfect since we are interested in the integrity of the memory itself, separate from the creation or analysis of the logical qubit. One may wonder whether this presents difficulties for experimental tests, since in reality Alice, Igor and Bob are merely phases of a single experiment and presumably suffer the same error rates. An analysis of this point is beyond the scope of the present paper, but it has been considered in a subsequent work Xiaosi2017 . The encouraging conclusion is that, for a broad class of error models (including the typical ones), experimental evaluation of integrity is possible even with noisy Alice and Bob. The key observation is that typically it suffices for Alice to randomly choose between a fixed subset of possible states to send; then, if one can find a fault tolerant circuit for Alice to use to prepare each such state, and a corresponding fault tolerant analysis method for Bob, one finds that the noisy nature of their actions has relatively little impact on the measured integrity.

Finally, we remark that while the error models considered here have been stochastic, the framework we have introduced applies to any form of noise and therefore a full exploration of coherent and even non-Markovian noise is an interesting prospect. A recent paper has highlighted the potential for coherent errors to impact QEC performance in a qualitatively different way Barnes2017 .

VIII Conclusions and outlook

In this work, we have presented a detailed description of current/future experimental capabilities for the implementation of topological QEC with trapped-ion crystals. We have also described the characteristics of the main sources of noise and imperfections in the experiments. Based on this discussion, we have introduced a complete trapped-ion toolbox for QEC, including a discussion of fault-tolerant designs based on the characteristics of the trapped-ion set of available gate operations. Using this toolbox, we have presented different protocols to implement a QEC cycle based on the 7-qubit color code, which exploit either crystal reconfiguration, or spectroscopic decoupling/re-coupling techniques. We have derived effective error models for the different building blocks of these QEC cycles, which are in close connection with the experimental sources of noise, and go well-beyond the simplified standard approaches that use quantum channels affecting all QEC operations with the same probability. Using these effective models for the current and expected performance of the QEC building blocks, we have performed extensive numerical simulations to determine the experimental conditions that are required for these QEC protocols to become beneficial, a fundamental and necessary condition for any future implementation of QEC. Moreover, we have also assessed the conditions for the encoded logical qubit to outperform the physical unprotected qubit for a particular quantum-information task.

From this study, and in light of our numerical results, we can draw the following conclusions. The performance of the 7-qubit trapped-ion color code for the single- or two-species, shuttling- or hiding-based, QEC protocols with multi-qubit MS gates, and assuming the ”current” performance of the experimental building blocks, is inadequate to achieve the break-even point of beneficial QEC. Therefore, the trapped-ion hardware must be improved. Using the operational criteria (2)-(3) introduced in this work, we have been able to assess and quantify the required experimental improvements towards the QEC goal, and present realistic values of the different building blocks that must be achieved (see Tables 2-4). Our numerical results for the “future” expected improvements show that the crossing onto a beneficial QEC cycle can indeed be achieved with either the shuttling- or the hiding-based protocols with multi-qubit MS gates, and occurs at much earlier times and with a much higher value of the integrity of the encoded qubit. Therefore, we conclude that it will be of primary importance to incorporate and optimize the QEC building blocks towards the values introduced in Tables 2-4 for the success of trapped-ion implementations of the QEC color code with two-species ion crystals. Moreover, we have observed a clear advantage of the QEC schemes based on sequential 2-qubit MS gates, especially in the context of the fault-tolerant designs. Therefore, our studies show that the natural next step in the progress towards trapped-ion fault-tolerant QEC will be to upgrade the syndrome extraction routines according to the schemes hereby introduced. We finally note that for the presented protocols, not only is the necessary condition for a beneficial QEC cycle (2) fulfilled, but that we have also shown that the encoded qubit can perform better than the unprotected qubit (3). Moreover, we have shown that repetitive QEC cycles can sustain the integrity of the logical qubit for increasing periods of time, provided that the above break-even point is achieved.

Acknowledgements.

The research is based upon work supported by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via the U.S. Army Research Office Grant No. W911NF-16-1-0070. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the ODNI, IARPA, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the view of the U.S. Army Research Office. We also acknowledge support by U.S. A.R.O. through Grant No. W911NF-14-1-010. A.B. acknowledges support from Spanish MINECO Projects FIS2015-70856-P, and CAM regional research consortium QUITEMAD+. P. S., T. M. and R. B. acknowledge support from the Austrian Science Fund (FWF), through the SFB FoQus (FWF Project No. F4002-N16) and the Institut für Quanteninformation GmbH.