Experimental demonstration of fault-tolerant state preparation with superconducting qubits

TL;DR

This paper demonstrates fault-tolerant state preparation in a five-qubit superconducting transmon device, showcasing the potential for reliable quantum computation despite operational errors.

Contribution

It provides the first experimental implementation of fault-tolerant state preparation using a minimal quantum code in superconducting qubits.

Findings

Successful fault-tolerant state preparation verified by quantum process tomography

Logical observables exhibit protected evolution consistent with fault-tolerance

Supports feasibility of scalable fault-tolerant quantum computing

Abstract

Robust quantum computation requires encoding delicate quantum information into degrees of freedom that are hard for the environment to change. Quantum encodings have been demonstrated in many physical systems by observing and correcting storage errors, but applications require not just storing information; we must accurately compute even with faulty operations. The theory of fault-tolerant quantum computing illuminates a way forward by providing a foundation and collection of techniques for limiting the spread of errors. Here we implement one of the smallest quantum codes in a five-qubit superconducting transmon device and demonstrate fault-tolerant state preparation. We characterize the resulting codewords through quantum process tomography and study the free evolution of the logical observables. Our results are consistent with fault-tolerant state preparation in a protected qubit subspace.

Figures

Tables

Table 1. Table 1: Two-qubit error per gate (EPG). Randomized benchmarking results for the two-qubit gate with four cross-resonance pulses. The two-qubit error-per-gate (EPG) and total sequence length are shown.

	$C{R}_1$	$C{R}_2$	$C{R}_3$	$C{R}_4$
Two-qubit	0.0380	0.0451	0.0330	0.0282
EPG	$\pm$ 0.0013	$\pm$ 0.0015	$\pm$ 0.0012	$\pm$ 0.0010
(length, ns)	(780)	(780)	(780)	(780)

Table 2. Table 2: Acceptance probability and fidelity of initial state preparation given that the state is in the codespace, s z ​ s x = 00 subscript 𝑠 𝑧 subscript 𝑠 𝑥 00 s_{z}s_{x}=00 . The largest errors are due to error on the gauge qubit.

Prepare	Accept	$|{\overline{0}}_p{\overline{0}}_g⟩$	$|{\overline{0}}_p{\overline{1}}_g⟩$	$|{\overline{1}}_p{\overline{0}}_g⟩$	$|{\overline{1}}_p{\overline{1}}_g⟩$
$|{\overline{0}}_p{\overline{0}}_g⟩$	0.7566	0.9726	0.0216	0.0040	0.0019
$|{\overline{0}}_p{\overline{1}}_g⟩$	0.7773	0.0245	0.9678	0.0037	0.0041
$|{\overline{1}}_p{\overline{0}}_g⟩$	0.7702	0.0028	0.0042	0.9673	0.0258
$|{\overline{1}}_p{\overline{1}}_g⟩$	0.7853	0.0033	0.0034	0.0224	0.9709
Prepare	Accept	$|{\overline{+}}_g{\overline{+}}_p⟩$	$|{\overline{+}}_g{\overline{-}}_p⟩$	$|{\overline{-}}_g{\overline{+}}_p⟩$	$|{\overline{-}}_g{\overline{-}}_p⟩$
$|{\overline{+}}_g{\overline{+}}_p⟩$	0.7897	0.9667	0.0065	0.0199	0.0069
$|{\overline{+}}_g{\overline{-}}_p⟩$	0.7707	0.0057	0.9632	0.0064	0.0247
$|{\overline{-}}_g{\overline{+}}_p⟩$	0.7799	0.0247	0.0069	0.9626	0.0058
$|{\overline{-}}_g{\overline{-}}_p⟩$	0.7731	0.0065	0.0253	0.0063	0.9619

Table 3. Table S1: Qubit and readout characterization. Qubit transitions ( ω 01 / 2 ​ π subscript 𝜔 01 2 𝜋 \omega_{01}/2\pi ), relaxation times ( T 1 subscript 𝑇 1 T_{1} ), coherence times ( T 2 subscript 𝑇 2 T_{2} ), readout resonator frequencies ( ω r / 2 ​ π subscript 𝜔 𝑟 2 𝜋 \omega_{r}/2\pi ), readout assignment errors ( ϵ r subscript italic-ϵ 𝑟 \epsilon_{r} ) obtained from readout parameters, ( P ​ ( 0 | 1 ) 𝑃 conditional 0 1 P(0|1) and P ​ ( 1 | 0 ) 𝑃 conditional 1 0 P(1|0) ), and individual and simultaneous single-qubit randomized benchmarking error per gate (EPG) results. Single-qubit gates are all 85 ns long. Anharmonicities of all qubits are around 330 MHz.

Qubit	${\text{D}}_1$	${\text{D}}_2$	${\text{D}}_3$	${\text{D}}_4$	${\text{S}}_1$
${\omega }_{01}/2\pi$ (GHz)	5.3503	5.3061	5.229	5.0748	5.1203
$T_1$	$50.4$	$70.3$	$77.7$	$68.5$	$68.0$
$\pm$ std ($\mu$s)	$\pm$ 4.8	$\pm$ 9.7	$\pm$ 9.8	$\pm$ 12.5	$\pm$ 9.6
$T_2$	$67.6$	$104.3$	$59.6$	$67.4$	$61.6$
$\pm$ std ($\mu$s)	$\pm$ 9.3	$\pm$ 23.3	$\pm$ 7.1	$\pm$ 17.3	$\pm$ 6.9
${\omega }_r/2\pi$ (GHz)	6.5250	6.4760	6.5742	6.5244	6.4295
${ϵ}_r$	0.0404	0.0246	0.0323	0.0321	0.0256
$p_0=P\left(0|1\right)$	0.0567	0.0402	0.0455	0.0573	0.0424
$p_1=P\left(1|0\right)$	0.0240	0.0090	0.0191	0.0069	0.0088
Indiv. RB	7.59	5.82	6.06	8.56	6.73
EPG (1e-4)	$\pm$ 0.12	$\pm$ 0.10	$\pm$ 0.10	$\pm$ 0.16	$\pm$ 0.10
Simul. RB	13.29	6.81	8.47	9.18	12.29
EPG (1e-4)	$\pm$ 0.25	$\pm$ 0.11	$\pm$ 0.13	$\pm$ 0.14	$\pm$ 0.26

Table 4. Table S2: Static ZZ strength (kHz). The strength is measured between each pair of qubits by running a pi-Hahn echo experiment, which is a Hahn echo experiment for each initial state of the spectator qubits in the computational basis, but with a varying angle for the final rotation in the Hahn echo sequence. Here ϵ z ​ z < 5 subscript italic-ϵ 𝑧 𝑧 5 \epsilon_{zz}<5 kHz.

	${\text{D}}_1$	${\text{D}}_2$	${\text{D}}_3$	${\text{D}}_4$	${\text{S}}_1$
${\text{D}}_1$	-	-49	${ϵ}_{zz}$	${ϵ}_{zz}$	-95
${\text{D}}_2$	-50	-	${ϵ}_{zz}$	${ϵ}_{zz}$	-29
${\text{D}}_3$	${ϵ}_{zz}$	${ϵ}_{zz}$	-	-77	-25
${\text{D}}_4$	${ϵ}_{zz}$	${ϵ}_{zz}$	-77	-	-43
${\text{S}}_1$	-94	-31	-25	-42	-

Table 5. Table S3: Initial state characterization . Acceptance probability, error on protected qubit state preparation ( ϵ p subscript italic-ϵ 𝑝 \epsilon_{p} ), and error on gauge qubit state preparation ( ϵ g subscript italic-ϵ 𝑔 \epsilon_{g} ) of initial state preparation given that the state is in the codespace, 0 ~ ​ 0 ~ ~ 0 ~ 0 \tilde{0}\tilde{0} , using two different CNOT gate sequence.

	Prepare	Accept	${ϵ}_p$	${ϵ}_g$
TPCX	$|{\overline{1}}_p{\overline{1}}_g⟩$	0.7206	0.0035	0.0137
	$|{\overline{+}}_g{\overline{+}}_p⟩$	0.2767	0.0225	0.0587
FPCX	$|{\overline{1}}_p{\overline{1}}_g⟩$	0.7853	0.0067	0.0257
	$|{\overline{+}}_g{\overline{+}}_p⟩$	0.7897	0.0134	0.0268

Equations

$$S_x$$

$$S_z$$

$${\bar{X}}_{L1}$$

$${\bar{X}}_{L2}$$

$$|\bar{0}\bar{0},\tilde{0}\tilde{0}⟩\propto |0000⟩+|1111⟩,$$

$$|\bar{0}\bar{1},\tilde{0}\tilde{0}⟩\propto |1100⟩+|0011⟩,$$

$$|\bar{1}\bar{0},\tilde{0}\tilde{0}⟩\propto |1010⟩+|0101⟩,$$

$$|\bar{1}\bar{1},\tilde{0}\tilde{0}⟩\propto |0110⟩+|1001⟩.$$

$${ϵ}_r=\frac{P(0|1)+P(1|0)}{2},$$

$$P_j(\mathrm{accept})=a_j+b_j\cos (\theta ),$$

$$P_j({\bar{X}}_{Lr}|\mathrm{accept})=\frac{c_j^{(r)}+d_j^{(r)}\cos (\theta )}{P_j(\mathrm{accept})}.$$

$$P(\bar{1}|\mathrm{accept})=(2-2e^{t/T_1}+e^{2t/T_1}{)}^{-1},$$

$$\rho (0)=\sum _{L_1,L_2,s_z,s_x}{p}_{L_1,L_2,s_z,s_x}{\rho }_{L_1,L_2,s_z,s_x},$$

$${\mathcal{E}}_{\gamma }(\rho )=A_0\rho A_0^{†}+A_1\rho A_1^{†},$$

$$a_{\text{A}}$$

$$b_{\text{A}}$$

$$a_{\text{B}}$$

$$b_{\text{B}}$$

$$c_{\text{A}}^{(1)}$$

$$d_{\text{A}}^{(1)}$$

$$c_{\text{B}}^{(1)}$$

$$d_{\text{B}}^{(1)}$$

$$c_{\text{B}}^{(2)}$$

$$d_{\text{B}}^{(2)}$$

Full text

The possibility of robust quantum computation rests on the fact that quantum information can be encoded in degrees of freedom that are difficult for local noise processes to change. Quantum codes with this potential have been demonstrated in many physical systems Chow et al. (2014); Corcoles et al. (2015); Takita et al. (2016); Cramer et al. (2016); Riste et al. (2015); Bell et al. (2014); Nigg et al. (2014); Reed et al. (2012); Ofek et al. (2016); Kelly et al. (2015). To make practical use of these codes, however, it is necessary not only to encode, decode, and observe errors, but to compute with faulty and inaccurate operations in a way that does not spread errors. The well-developed theory of fault-tolerant quantum computing reveals a steep experimental path toward this goal Gottesman (2010); Campbell et al. (2017). Recently, the question of what constitutes a minimal experimental demonstration of fault-tolerance was considered Gottesman (2016). Fault-tolerant state preparation was demonstrated soon thereafter using a quantum error detecting code with trapped atomic ions Linke et al. (2016). Here we go beyond that result, implementing fault-tolerant state preparation on a superconducting qubit system with supporting evidence including quantum state tomography of prepared codewords, acceptance and logical error probabilities with and without error insertion, and analysis of the measured logical observables under free evolution.

We implement one of the smallest quantum codes, a four qubit code encoding two qubits Leung et al. (1997), and characterize output states produced by fault-tolerant state preparation circuits. The circuits are fault-tolerant for only one of the two encoded qubits, which allows direct comparison of their error rates. The circuits are applied in a five-qubit transmon device with nearest-neighbor connectivity. This device is a nontrivial subset of a surface code lattice in the sense that it provides resources for detection of any single-qubit error. Although the connectivity and size places limits on the set of fault-tolerant circuits we can implement on the four-qubit code, we can use stabilizer measurements to prepare codewords in a way that is analogous to surface code state preparation.

Four qubit code – The four-qubit code Leung et al. (1997) encodes two logical qubits into four physical qubits and can detect any error that acts on one of those physical qubits. It is the smallest code that can detect a general error and is unique Calderbank et al. (1998). The four-qubit code is defined by the stabilizer group $S=\langle S_{x},S_{z}\rangle$ with stabilizers Gottesman (1997)

[TABLE]

Here $X=|0\rangle\langle 1|+|1\rangle\langle 0|$ and $Z=|0\rangle\langle 0|-|1\rangle\langle 1|$ are Pauli operators. The pair of encoded qubits are defined by logical operators

[TABLE]

The minimum distance of a stabilizer code is the minimum number of qubits acted on by any Pauli operator that commutes with $S$ but lies outside of it Gottesman (1997); Nielsen and Chuang (2000); in this case, that distance is two. Stabilizer codes are described by parameters $[[n,k,d]]$ where $n$ is the number of physical qubits, $k$ is the number of logical qubits, and $d$ is the minimum distance. Our code, thus, is a $[[4,2,2]]$ code.

The code space is spanned by four states

[TABLE]

On the left hand side, we order the labels $|L_{1}L_{2},s_{z}s_{x}\rangle$ where $s_{z}$ and $s_{x}$ are syndrome bits that record phase and bit-flip errors, respectively. The syndromes correspond to single-shot measurements of the observables $S_{x}$ and $S_{z}$, which have eigenvalues $(-1)^{s_{z}}$ and $(-1)^{s_{x}}$, respectively.

We define destabilizers $\tilde{Z}_{D}=Z_{4}$ and $\tilde{X}_{D}=X_{4}$ that commute with the logical operators and anticommute with corresponding stabilizers $S_{x}$ and $S_{z}$. The destabilizers change the values of the syndrome bits without affecting the logical qubits. The whole four-qubit Hilbert space is spanned by $16$ states $\{|L_{1}L_{2},s_{z}s_{x}\rangle\}$ where $L_{1}$ and $L_{2}$ take values over the four states of the logical qubits and $s_{z}$ and $s_{x}$ run over the four possible syndromes.

Implementation – The device consists of five fixed-frequency superconducting transmon qubits, four of which, $D_{i}$ with $i\in{1,2,3,4}$, are used as data qubits of the code (see Fig. 1). The central qubit, $S_{1}$, acts as a syndrome qubit, and it is coupled to the four data qubits via two coplanar waveguide (CPW) resonators acting as quantum buses, with two data qubits on each bus. Each qubit is coupled to its own CPW resonator for control and readout. Readout signals are amplified via Josephson Parametric Converters (JPCs) Bergeal et al. (2010); Abdo et al. (2011). Device fabrication methods are described in previous work Chow et al. (2014); Corcoles et al. (2015). The device is the current IBM Quantum Experience device QX .

Calibration training data is obtained from 4000 single shot measurements of $2^{5}=32$ different five-qubit computational states. Each measurement is a time-varying voltage signal reflected from the readout resonator. This signal is demodulated and integrated, yielding a single value in I/Q space. An arbitrary qubit state is determined by comparing the Euclidean distances between its integrated signal and the mean of the integrated signals of the ground and excited states obtained from the calibration data for that qubit. The shortest distance determines the outcome. The readout assignment error $\epsilon_{r}$ is given by

[TABLE]

where $P(\bar{b}|b)$ is the probability of observing the incorrect outcome given that the correct outcome is $b$. Readout assignment error for each qubit is given in the Supplemental Material.

Single-qubit gates are characterized using Clifford randomized benchmarking (RB) Magesan et al. (2011) and simultaneous RB Gambetta et al. (2012). We find single qubit error per gate (EPG) of all five qubits to be lower than $\sim 9\times 10^{-4}$ and obtain crosstalk error of less than $\sim 6\times 10^{-4}$ from simultaneous RB results (see Supplemental Material for all measured single qubit errors). The two largest crosstalk errors are observed on D1 and S1, which is consistent with the fact that this pair of qubits has the largest static $ZZ$ interaction strength (see Supplemental Material).

Two-qubit controlled-NOT (CNOT)  Paraoanu (2006); Rigetti and Devoret (2010); Chow et al. (2011) gates are constructed using the microwave-based cross resonance (CR) interaction. Using a four-pulse echoed cross resonance gate Córcoles et al. (2013); Takita et al. (2016) as a two-qubit Clifford gate generator, we characterize four pairs of two-qubit gates through Clifford RB (Table 1). The decomposition of a four-pulse echoed cross resonance CNOT gate (FPCX) into single-qubit gates and CR interactions is drawn in Fig.  2 (a). The FPCX echoes all of the first and second order Z-terms from the cross-resonance Hamiltonian on the control, target, and the spectator qubits (SQ); the terms are ZII, IZI, IIZ, ZZI, ZIZ, IZZ. Here, the spectator qubits are the three other qubits in the five-qubit lattice that are neither the control nor the target qubit for each particular CNOT. The FPCX sequence here is similar to the four pulse sequence used in previous work Takita et al. (2016) but has extra pulses to echo the IZI term, which is typically smaller than the other terms. FPCX was necessary in order to correct for errors seen when using a two-pulse echoed cross resonance CNOT gate (TPCX) (See Supplemental Material for results using TPCX).

Fault-tolerant state preparation – The logical state $\ket{\bar{0}_{p}\bar{0}_{g}}$ is prepared by running the X-stabilizer ($S_{x}$) circuit and measuring the syndrome qubit (shown in Fig. 2). The logical qubit $L_{1}$, denoted here by p, is a fault-tolerantly prepared protected qubit, and $L_{2}$, denoted by g, is a gauge qubit that is not prepared fault-tolerantly. The other logical states in the $\ket{\bar{0}}$ and $\ket{\bar{1}}$ basis are prepared by applying logical bit-flips, Eq. 3, during the $\ket{\bar{0}_{p}\bar{0}_{g}}$ state preparation. From the $\ket{\bar{0}_{p}\bar{0}_{g}}$ state, we can prepare $\ket{\bar{+}_{g}\bar{+}_{p}}$ by applying Hadamard gates on the four data qubits. Note that this swaps the indices of the logical states, exchanging $\bar{X}_{L1}$ with $\bar{Z}_{L2}$ and $\bar{X}_{L2}$ with $\bar{Z}_{L1}$. The other logical states in the $\ket{\bar{+}}$ and $\ket{\bar{-}}$ basis are prepared by applying logical phase-flips, Eq. 3, after the Hadamard gates.

To characterize the state preparation circuit, we performed quantum state tomography on the four data qubits. The difference between the ideal and reconstructed state of $\ket{\bar{1}_{p}\bar{1}_{g}}$ is shown in Fig. 3. The boxed top left corner of the reconstructed state represents the projection onto the codespace $(\tilde{0}\tilde{0})$. Considering the corresponding state $\rho(\tilde{0}\tilde{0})$, the largest errors are coherent errors on the gauge qubit. The acceptance probability $\mathrm{tr}(\rho_{\tilde{0}\tilde{0}})$ and fidelity of the prepared state $\rho$ are obtained from $\rho_{\tilde{0}\tilde{0}}$. Results computed from state tomography data of additional prepared logical states are given in Table 2.

Error insertion – To study how error propagates through the $\ket{\bar{1}_{p}\bar{1}_{g}}$ state preparation circuit, we introduce a phase error $Z(\theta)$ on $S_{1}$ after the 1st (A), 2nd (B), or 3rd (C) CNOT gate [see Fig. 2 (b)]. Since the state preparation is done by syndrome measurement, we first post-select on the syndrome measurement reading c${}_{s}=1$, noting that the syndrome qubit starts from the excited state at the beginning of the circuit. Next, we compute $S_{z}$ in software and post-select on $\text{c}_{1}\oplus\text{c}_{2}\oplus\text{c}_{3}\oplus\text{c}_{4}=0$. The acceptance probability is given by P($\text{c}_{1}\oplus\text{c}_{2}\oplus\text{c}_{3}\oplus\text{c}_{4}=0|\text{c}_{s}=1$), and the state of the protected (gauge) qubit is determined from the parity of $\text{c}_{1}$ and $\text{c}_{2}$ ($\text{c}_{1}$ and $\text{c}_{3}$).

Phase errors propagate from target to control through a CNOT gate, hence a $Z$-error at locations A, B, or C appears as an $X$-error on {D2,D3,D4}, {D3,D4}, or {D4}, respectively. As we increase the error parameter $\theta$, the acceptance probability decreases for locations A and C but remains constant for location B (see Fig. 4 (a)). Fig. 4 (b) plots the state preparation errors as a function of $\theta$. As we increase $\theta$ at location B, error on the protected qubit remains constant, while the error on the gauge qubit increases. For errors inserted at locations A and C, the gauge qubit error is always larger than protected qubit error.

Although a distance two code can only detect one error on the data qubits, correlated two-qubit gate errors are also detectable because the circuit is fault-tolerant by construction. In particular, the two-qubit gates never act directly on pairs of data qubits, so two-qubit gate errors can only affect one data qubit at a time. To mimic this correlated error, we simultaneously introduce $Y(\theta)$ errors on the control and target qubits after each CNOT gate. Similar to single-qubit error insertion, the acceptance probability decreases as a function of $\theta$ (see Fig. 4 (c)) and lower errors are observed on the protected qubit versus the gauge qubit (see Fig. 4 (d)).

To understand the functional form of the error insertion data, we modeled error insertion in the ideal state preparation circuit followed by asymmetric readout errors with the same readout parameters for each qubit. For each error location, we find the acceptance probability and conditional logical error probabilities as a function of the error’s angle $\theta$ and the readout parameters $p_{0}=P(0|1)$ and $p_{1}=P(1|0)$. For single $Z(\theta)$ error insertion at location $j$, the acceptance probability has the form

[TABLE]

and the conditional logical error probabilities on logical qubit $r$ have the form

[TABLE]

Each coefficient is a function of $p_{0}$ and $p_{1}$ that is given in the Supplemental Material. The expressions for locations A and C are identical. Likewise, for each location, the expression for combined logical error $\bar{X}_{L1}\bar{X}_{L2}$ on the gauge and protected qubit is the same as the corresponding expression for $\bar{X}_{L1}$ alone.

The dashed curves in Figs. 4 (a) and (b) are fits to the functions given in Eq. 9 and 10, but we include a systematic offset parameter $\delta$ that is added to $\theta$, i.e. we replace $\cos(\theta)$ by $\cos(\theta+\delta)$. The offset $\delta_{j}$ for each location $j$ is determined from either the acceptance or error data based on which has the greatest curvature. The acceptance probability then has 2 remaining free parameters, $\tilde{a}_{j}$ and $\tilde{b}_{j}$. Once these are known, each error probability has 2 remaining free parameters $\tilde{c}^{(r)}_{j}$ and $\tilde{d}^{(r)}_{j}$.

This model has the benefit of being simple, but it does not include all of the major error sources and most importantly omits dissipation and systematic phase errors that occur during and between two-qubit gates. We have confirmed that the same general functional form is obtained when dissipation is introduced prior to measurement, which is not surprising, since it can be incorporated into readout error parameters of each qubit.

Non-exponential decay under free evolution – In this section we study the free evolution of $|\bar{1}\bar{1}\rangle$, post-selected to the codespace of the four-qubit code. Our goal is to observe how decoherence and fixed coupling terms between transmons act on logical states, particularly in the time interval immediately following fault-tolerant state preparation. The experiments are analogous to decay and spin-echo experiments on physical qubits.

Although we are working with encoded states, these decay experiments do not demonstrate a fault-tolerant memory. A fault-tolerant quantum memory would be implemented in this context by repeated syndrome measurements. Repeated syndrome measurements are not feasible in this device due to both technical limitations of measurement durations as well as exponentially decreasing acceptance probability as a function of the number of syndrome measurements. These limitations could be overcome by implementing a quantum error-correcting, rather than error-detecting, code and using a device that is designed for fast, repeated readout.

The results for the $|\bar{1}\bar{1}\rangle$ state, shown in Fig. 5, have several features that are evidence of short-time protection from local noise. First, the decay is non-exponential, exhibiting a slow initial decay rate that increases with time. The ideal functional form for either encoded qubit is

[TABLE]

where we have assumed the same $T_{1}$ for each qubit. A cross-over with the ideal physical decay curve $P(1)=\mathrm{exp}(-t/T_{1})$ occurs at $t=T_{1}\ln 2$, which is on the order of $T_{1}$. Second, due to fault-tolerant state preparation, the initial population is greater for the protected than the gauge qubit in the presence of error.

To explain how the observed results for the $|\bar{1}\bar{1}\rangle$ state differ from the ideal form, we construct a simplified model of the logical decay incorporating errors in the initial state and readout. The initial state is modeled as

[TABLE]

which is a 15 parameter mixture of joint eigenstates $\rho_{L_{1},L_{2},s_{z},s_{z}}$ of $\bar{Z}_{L1}$, $\bar{Z}_{L2}$, $S_{x}$, and $S_{z}$. The parameter values are assigned from state tomography data. Each qubit of this state undergoes independent amplitude damping described by the channel

[TABLE]

where $A_{0}=|0\rangle\langle 0|+\sqrt{1-\gamma}|1\rangle\langle 1|$ and $A_{1}=\sqrt{\gamma}|0\rangle\langle 1|$. Each qubit has a different damping parameter $\gamma=1-e^{-t/T_{1}}$ given by a value of $T_{1}$ that is fitted to experimental data. After damping, each qubit is projectively measured in the computational basis. The readout error process is modeled as an asymmetric binary channel with crossover probabilities $P(0|1)$ and $P(1|0)$. The crossover probabilities are assumed to be the same for each qubit and fitted to the experimental data. Finally, the noisy outcomes are post-processed as described earlier.

Conclusion – We demonstrate that even in small code lattices, fault-tolerant principles can result in short-time protection from local dissipation, matching and close to outperforming the evolution of the physical qubits. Due to fault-tolerant circuit design, we observed that one of the two encoded logical qubits has significantly reduced conditional logical error. Additionally, we include quantum state tomography data for prepared codewords, study error insertion, and analyze the decay of measured logical observables under free evolution. The latter shows evidence of short-time protection from local dissipation. A composite two-qubit gate, the four-pulse echoed cross resonance gate, compensated for systematic phase errors during state preparation. This work, which aims at testing the fundamentals of small codes, is conceived in essence as an effort to understand how noise propagates in larger systems. Repeated stabilizer measurements are needed to study time dependence of fault-tolerant storage.

Acknowledgements – We acknowledge Baleegh Abdo for providing the JPCs, Markus Brink for device fabrication, Jim Rozen, Jack Rohrs and Oblesh Jinka for help with the cryogenic setup and Easwar Magesan for discussions on tomography analysis. MT, ADC, and JMC acknowledge support from Intelligence Advanced Research Projects Activity (IARPA) under contract W911NF-16-0114. AWC and JMG acknowledge partial support from ARO under contract W911NF-14-1-0124.

I Supplementary material for ‘Experimental demonstration of fault-tolerant state preparation with superconducting qubits’

II Device parameters

III Four qubit code

IV Fault-tolerant state preparation

Fig. S2 plots absolute differences between the measured and ideal matrix elements of the reconstructed states for $|\bar{1}_{p}\bar{1}_{g}\rangle$ and $|\bar{+}_{g}\bar{+}_{p}\rangle$ states in $Z$ and $X$-basis, respectively; we ran state preparation circuits using two different CNOT sequences. Gate decomposition of the two CNOT sequences, the two-pulse CNOT gate (TPCX) and four-pulse CNOT gate (FPCX), are shown in Fig. S4. Two-qubit error per gate results of the two sequences are compared in Table  S4. As discussed in the main letter, first and second order Z-terms from the cross-resonance Hamiltonian on control qubit, target qubit, and spectator qubits (SQ) are ZII, IZI, IIZ, ZZI, ZIZ, IZZ. While the FPCX sequence echoes all these terms, TPCX does not echo out IZI, IIZ, and IZZ.

We observe different sensitivity to systematic phase errors depending on what codeword is prepared using TPCX; Fig. S4 shows each preparation circuit. We observed that $|\bar{0}_{p}\bar{0}_{g}\rangle$ and hence all states in the $|\bar{\pm}\rangle$ basis were significantly more sensitive to phase error than the other logical basis states $|\bar{0}_{p}\bar{1}_{g}\rangle$, $|\bar{1}_{p}\bar{0}_{g}\rangle$, and $|\bar{1}_{p}\bar{1}_{g}\rangle$. When the $|\bar{+}_{g}\bar{+}_{p}\rangle$ state is prepared using TPCX, we observe low acceptance probability (see Table  S4), and the dominant state observed from the reconstructed state is the $|\bar{+}_{g}\bar{+}_{p},\tilde{0}\tilde{1}\rangle$ (see Fig.  S2). We hypothesize that the largest errors are due to drive-activated phase shifts (AC Stark shifts) during the two-qubit gates, which may have comparable magnitudes $\theta$ and the same sign on each data qubit. The relative phase accumulates as $4\theta$ on $|\bar{0}_{p}\bar{0}_{g}\rangle$, but the error effectively cancels on the other states. The FPCX sequence effectively cancels this error.

V Error insertion

The fitting parameters for Fig. 4 of the main text are as follows. The offset parameters $\delta_{\textrm{A}}=-0.1369$ and $\delta_{\textrm{C}}=0.0278$ are determined from the acceptance probability, and the parameter $\delta_{\textrm{B}}=-0.2291$ is fitted from the conditional logical error probability. The acceptance probability has parameters $\tilde{a}_{\textrm{A}}=0.5044$ and $\tilde{b}_{\textrm{A}}=0.2632$ at location A, $\tilde{a}_{\textrm{B}}=0.7614$ and $\tilde{b}_{\textrm{B}}=0.0059$ at location B, and $\tilde{a}_{\textrm{C}}=0.4983$ and $\tilde{b}_{\textrm{C}}=0.2708$ at location C. The conditional logical error for the protected qubit has parameters $\tilde{c}^{(1)}_{\textrm{A}}=0.0626$ and $\tilde{d}^{(1)}_{\textrm{A}}=-0.0444$ at location A, $\tilde{c}^{(1)}_{\textrm{B}}=0.0189$ and $\tilde{d}^{(1)}_{\textrm{B}}=-0.0006$ at location B, and $\tilde{c}^{(1)}_{\textrm{C}}=0.0646$ and $\tilde{d}^{(1)}_{\textrm{C}}=-0.0466$ at location C. The conditional logical error for the gauge qubit has parameters $\tilde{c}^{(2)}_{\textrm{A}}=0.0697$ and $\tilde{d}^{(2)}_{\textrm{A}}=-0.0279$ at location A, $\tilde{c}^{(2)}_{\textrm{B}}=0.3847$ and $\tilde{d}^{(2)}_{\textrm{B}}=-0.3573$ at location B, and $\tilde{c}^{(2)}_{\textrm{C}}=0.0795$ and $\tilde{d}^{(2)}_{\textrm{C}}=-0.0395$ at location C. The tilde denotes parameters from fitting data to Eq. 9 and 10 of the main text.

For the simplified error insertion model described in the main text, the acceptance probability coefficients are

[TABLE]

The coefficients in the conditional logical error probability are

[TABLE]

The $L_{2}$ (gauge) logical error probability at location A is the same as the $L_{1}$ (protected) logical error. We find that $\tilde{p}_{0}=0.108$ and $\tilde{p}_{1}=0.043$ minimize the sum of the absolute differences between the model and fitted parameters.

VI Encoded $|\bar{+}\bar{+}\rangle$ lifetime

The experimental results for the $|\bar{+}\bar{+}\rangle$ state are shown in Fig. S3. There is rapid decay and coherent oscillation for both encoded qubits. This decay can be slowed by using echo sequences. The rapid decay and coherent oscillation is consistent with free evolution under the static ZZ terms in the Hamiltonian. Taking a ZZ strength $\eta$ of order $50$ kHz gives timescales $t_{\pi}=\pi/|\eta|=(2\times-50\mathrm{kHz})^{-1}=10\mu$s, which is consistent with the loss of coherence in Fig. S3.