Protecting quantum entanglement from leakage and qubit errors via repetitive parity measurements

TL;DR

This paper demonstrates a method to detect and mitigate leakage errors in entangled qubit states during quantum error correction, using repeated parity measurements and hidden Markov models, enhancing the reliability of quantum information processing.

Contribution

It introduces a computationally efficient leakage detection technique based on hidden Markov models, compatible with real-time quantum error correction in larger systems.

Findings

Leakage can be detected using parity measurement records.

Bell states stabilized over 26 parity measurements with leakage mitigation.

Leakage identification is suitable for real-time quantum error correction.

Abstract

Protecting quantum information from errors is essential for large-scale quantum computation. Quantum error correction (QEC) encodes information in entangled states of many qubits, and performs parity measurements to identify errors without destroying the encoded information. However, traditional QEC cannot handle leakage from the qubit computational space. Leakage affects leading experimental platforms, based on trapped ions and superconducting circuits, which use effective qubits within many-level physical systems. We investigate how two-transmon entangled states evolve under repeated parity measurements, and demonstrate the use of hidden Markov models to detect leakage using only the record of parity measurement outcomes required for QEC. We show the stabilization of Bell states over up to 26 parity measurements by mitigating leakage using postselection, and correcting qubit errors using Pauli-frame transformations. Our leakage identification method is computationally efficient and thus compatible with real-time leakage tracking and correction in larger quantum processors.

Figures

Tables

Table 1. Table 1: Values of error rates used in the various HMMs in this work. All values are obtained by optimizing the likelihood of observing the given syndrome data except for the ancilla leakage rate (denoted ∗ ) which is directly obtained from the experiments (as noted in the text).

Error type	${𝑯}_{\mathrm{𝐙𝐙}}$-A	${𝑯}_{\mathrm{𝐙𝐙}}$-D	${𝑯}_{\mathrm{𝐙𝐙},\mathrm{𝐗𝐗}}$-A	${𝑯}_{\mathrm{𝐙𝐙},\mathrm{𝐗𝐗}}$-D
leakage [$p_{\mathrm{leak}}$]	${0.0040}^{\ast }$	0.0064	${0.0040}^{\ast }$	0.0064
seepage [$p_{\mathrm{seep}}$]	0.101	0.108	0.101	0.103
data-qubit error [$p_{\mathrm{data}}$]	0.042	0.050	0.045	0.030
during leakage [$p_{\mathrm{data},\mathrm{leaked}}$]	-	0.155	-	0.489
Y error (additional) [$p_{\mathrm{data},Y}$]	-	-	-	0.014
readout error [$p_{\mathrm{readout}}$]	0.011	0.004	0.027	0.014
ancilla error [$p_{\mathrm{ancilla}}$]	0.028	0.030	-	0.029
($M_A\left[m-1\right]=1$, $M_A\left[m\right]=1$) [$p_{\mathrm{anc},0,0}$]	-	-	0.001	-
($M_A\left[m-1\right]=1$, $M_A\left[m\right]=-1$) [$p_{\mathrm{anc},0,1}$]	-	-	0.021	-
($M_A\left[m-1\right]=-1$, $M_A\left[m\right]=1$) [$p_{\mathrm{anc},1,0}$]	-	-	0.044	-
($M_A\left[m-1\right]=-1$, $M_A\left[m\right]=-1$) [$p_{\mathrm{anc},1,1}$]	-	-	0.058	-
during leakage [$p_{\mathrm{ancilla},\mathrm{leaked}}$]	-	0.113	-	-

Table 2. Table S1: Measured parameters of the three-transmon device. † Q DH subscript 𝑄 DH Q_{\mathrm{DH}} is operated 30 30 30  MHz below its maximum frequency to avoid spurious interaction with a spurious two-level system. †† The Purcell mode and readout resonator mode of Q DH subscript 𝑄 DH Q_{\mathrm{DH}} have near-perfect hybridization (with qubit at ω op / 2 ​ π subscript 𝜔 op 2 𝜋 \omega_{\mathrm{op}}/2\pi ), making them indistinguishable. ††† Single-shot readout on the data qubits was not optimized. †††† Single-qubit gates are characterized using Clifford randomized benchmarking  Magesan  et al. ( 2012 ) ††††† Two-qubit gates are characterized using interleaved RB  Magesan  et al. ( 2012 ); Barends  et al. ( 2014 ) with a leakage-extraction modification  Rol  et al. ( 2019 ) .

Gate and Coherence Parameters	${𝑸}_{\mathrm{𝐃𝐋}}$	${𝑸}_{𝐀}$		${𝑸}_{\mathrm{𝐃𝐇}}$
operating qubit frequency, ${\omega }_{\mathrm{op}}/2\pi$ (GHz)	$5.02$	$5.79$		${6.88}^{†}$
max. qubit frequency, ${\omega }_{\max }/2\pi$ (GHz)	$5.02$	$5.79$		$6.91$
anharmonicity, $\alpha /2\pi$ (MHz)	$-306$	$-308$		$-331$
coherence time (at ${\omega }_{\mathrm{op}}/2\pi$), $T_2^{\mathrm{echo}}$ ($\mu$s)	$29.6\pm 2.7$	$21.7\pm 1.4$		$14.7\pm 0.9$
relaxation time (at ${\omega }_{\mathrm{op}}/2\pi$) $T_1$ ($\mu$s)	$25.3\pm 1.2$	$17.0\pm 0.6$		$25.6\pm 1.2$
Ramsey dephasing time (at ${\omega }_{\mathrm{op}}/2\pi$), $T_2^{\ast }$ ($\mu$s)	$24.5\pm 2.0$	$14.6\pm 1.2$		$5.9\pm 0.7$
average error per single qubit gate††††, $e_{\mathrm{SQ}}$ (%)	$0.08\pm 0.02$	$0.14\pm 0.016$		$0.21\pm 0.06$
resonance exchange coupling, $J_1/2\pi$ (MHz)	$17.2$		14.3
bus resonator frequency, $\sim {\omega }_{\mathrm{bus}}/2\pi$ (GHz)	$8.5$		$8.5$
error per CZ†††††, $e_{\mathrm{CZ}}$ ($\%$)	$1.4\pm 0.6$		$0.9\pm 0.16$
leakage per CZ†††††, $L_1$ ($\%$)	$0.27\pm 0.12$		$0.15\pm 0.07$
ZZ coupling (at ${\omega }_{\mathrm{op}}/2\pi$), ${\zeta }_{\mathrm{ZZ}}/2\pi$ (MHz)	$0.95$		$0.33$
Measurement Parameters	${𝑸}_{\mathrm{𝐃𝐋}}$	${𝑸}_{𝐀}$		${𝑸}_{\mathrm{𝐃𝐇}}$
readout pulse frequency, ${\omega }_{\mathrm{ro}}/2\pi$ (GHz)	$7.225$	$7.420$		$7.838$
readout resonator frequency, ${\omega }_{\mathrm{ro}}/2\pi$ (GHz)	$7.275$	$7.385$		$7.867$
Purcell resonator frequency, ${\omega }_{\mathrm{ro}}/2\pi$ (GHz)	$7.260$	$7.405$		$7.872$
qubit-RR coupling strength, $g_{01,\mathrm{RR}}/2\pi$ (MHz)	$202$	$188$		$135$
PF-RR coupling strength, $J_{\mathrm{RR},\mathrm{PF}}/2\pi$ (MHz)	$48$	$30$		$38$
dispersive shift qubit-RR, ${\chi }_{\mathrm{RR}}/\pi$ (MHz)	$-2.5$	$-5.3$		$-{2.8}^{††}$
dispersive shift qubit-PF, ${\chi }_{\mathrm{PF}}/\pi$ (MHz)	$-1.5$	$-4.7$		$-{2.8}^{††}$
critical photon number, $n_{\mathrm{crit}}$	2.3	2.7		$2.4$
intra-resonator photon number RR, $n_{\mathrm{RR}}$		$1.2$
quantum efficiency, $\eta$ ($\%$)		$48\pm 1.0$
Average assignment error, $e_a$ ($\%$)	${9.0}^{†††}$	$1.0\pm 0.1$		${16}^{†††}$
Measurement integration time, ${\tau }_{\mathrm{int}}$ (ns)	$600$	$600$		$600$

Equations

$$A(H)=2n_{\mathrm{p},H}-2\log \left[\underset{p_i}{\max }L(\{\vec{o}\}|H\{p_i\})\right],$$

$$A_{h,h}=P(H[m]=h|H[m-1]=h^{\prime }).$$

$$B_{o,h}=P(O[m]=o|H[m]=h).$$

$${\pi }_h^{(\mathrm{prior})}[1]=P(H[1]=h),$$

$${\pi }_h^{(\mathrm{post})}[m]=P(H[m]=h|O[1]=o_1,\dots ,O[m]=o_m),$$

$${\pi }_h^{(\mathrm{prior})}[m]$$

$$:=P(H[m]=n|O[1]=o_1,\dots ,O[m-1]=o_{m-1})$$

$$=\sum _{h^{\prime }}{A}_{h,h^{\prime }}{\pi }_{h^{\prime }}^{(\mathrm{post})}[m-1],$$

$${\pi }_h^{(\mathrm{post})}[m]=\frac{B_{o_m,h}{\pi }_h^{(\mathrm{prior})}[m]}{{\sum }_{h^{\prime }}{B}_{o_m,h^{\prime }}{\pi }_{h^{\prime }}^{(\mathrm{prior})}[m]}.$$

$$L_{\mathrm{comp},\mathrm{D}}=\sum _{h\mathrm{unleaked}}{\pi }_h^{(\mathrm{post})}[M].$$

$$L_{\mathrm{comp},\mathrm{A}}[M]=\frac{{\sum }_{h,h^{\prime }\mathrm{unleaked}}{B}_{o_m,h}{A}_{h,h^{\prime }}{\pi }_{h^{\prime }}^{(\mathrm{post})}[M-1]}{{\sum }_{h,h^{\prime }}{B}_{o_m,h}{A}_{h,h^{\prime }}{\pi }_{h^{\prime }}^{(\mathrm{post})}[M-1]}.$$

$$\displaystyle A=\left(\begin{array}[]{cc}1&0\\ 0&1\end{array}\right)+p_{\mathrm{leak}}\left(\begin{array}[]{cc}-1&0\\ 1&0\end{array}\right)$$

$$\displaystyle+p_{\mathrm{seep}}\left(\begin{array}[]{cc}0&1\\ 0&-1\end{array}\right).$$

$$\displaystyle B=\left(\begin{array}[]{cc}1&0\\ 0&1\end{array}\right)+p_{0,1}\left(\begin{array}[]{cc}-1&0\\ 1&0\end{array}\right)$$

$$\displaystyle+p_{1,0}\left(\begin{array}[]{cc}0&1\\ 0&-1\end{array}\right).$$

$$A=A_0+\sum _{\mathrm{err}}{p}_{\mathrm{err}}{D}_{\mathrm{err}}^{(\mathrm{A})},B=B_0+\sum _{\mathrm{err}}{p}_{\mathrm{err}}{D}_{\mathrm{err}}^{(\mathrm{B})}.$$

$$D_i^{(\mathrm{A})}=\frac{\partial A}{\partial p_i^{(\mathrm{A})}},D_i^{(\mathrm{B})}=\frac{\partial B}{\partial p_i^{(\mathrm{B})}}.$$

$$[A_0{]}_{(h_0,h_1//2),(h_0,h_1)}=1,[B_0{]}_{-1^{h_0+h_1},(h_0,h_1)}=1,$$

$$[D_{\mathrm{ancilla}}^{(A)}{]}_{(0,0),(0,0)}=-1,[D_{\mathrm{ancilla}}^{(A)}{]}_{(0,3),(0,0)}=1.$$

$$[D_{\mathrm{ancilla}}^{(A)}{]}_{(0,1),(0,3)}=-1,[D_{\mathrm{ancilla}}^{(A)}{]}_{(0,2),(0,3)}=1.$$

$$[D_{\mathrm{readout}}^{(A)}{]}_{(0,1),(0,2)}=-1,[D_{\mathrm{readout}}^{(A)}{]}_{(0,3),(0,2)}=1.$$

$$[D_{\mathrm{ancilla},\mathrm{leaked}}^{(A)}{]}_{(1,j),(1,k)}=[D_{\mathrm{ancilla}}^{(A)}{]}_{(0,j),(0,k)},$$

$$[D_{\mathrm{readout}}^{(A)}{]}_{(1,j),(1,k)}=[D_{\mathrm{readout}}^{(A)}{]}_{(0,j),(0,k)}.$$

$$[D_{\mathrm{ancilla}}^{(A)}{]}_{(1,h_1//2),(1,h_1)}=$$

$$[D_{\mathrm{ancilla}}^{(A)}{]}_{(1,h_1//2\oplus 5)}=$$

$$[D_{\mathrm{readout}}^{(A)}{]}_{(1,h_1//2),(1,h_1)}=$$

$$[D_{\mathrm{readout}}^{(A)}{]}_{(1,h_1//2\oplus 15)}=$$

$$[D_{\mathrm{data},\mathrm{Y}}^{(A)}{]}_{(1,h_1//2),(1,h_1)}=-1,[D_{\mathrm{data},\mathrm{Y}}^{(A)}{]}_{(1,h_1//2\oplus 3)}=1,$$

$$[B_0{]}_{1,(0,s)}=[B_0{]}_{-1,(1,s)}=[B_0{]}_{-1,(2,s)}=1,$$

$$[A_0{]}_{(a+s\mathrm{mod}2,s)(a,s)}=1\mathrm{if}a<2,[A_0{]}_{(2,s),(2,s)}=1.$$

$$[D_{\mathrm{readout}}^{(B)}{]}_{1,(0,s)}$$

$$=[D_{\mathrm{readout}}^{(B)}{]}_{-1,(1,s)}=[D_{\mathrm{readout}}^{(B)}{]}_{-1,(2,s)}=-1,$$

$$[D_{\mathrm{readout}}^{(B)}{]}_{-1,(0,s)}$$

$$=[D_{\mathrm{readout}}^{(B)}{]}_{1,(1,s)}=[D_{\mathrm{readout}}^{(B)}{]}_{1,(2,s)}=1.$$

$$[D_{\mathrm{data}}^{(A)}{]}_{(a,s),(a^{\prime },s)}=$$

Full text

I Introduction

Large-scale quantum information processing hinges on overcoming errors from environmental noise and imperfect quantum operations. Fortunately, the theory of QEC predicts that the coherence of single degrees of freedom (logical qubits) can be better preserved by encoding them in ever-larger quantum systems (Hilbert spaces), provided the error rate of the constituent elements lies below a fault-tolerance threshold Terhal (2015). Experimental platforms based on trapped ions and superconducting circuits have achieved error rates in single-qubit gates Barends et al. (2014); Harty et al. (2014); Ballance et al. (2016), two-qubit gates Barends et al. (2014); Ballance et al. (2016); Rol et al. (2019), and qubit measurements Jeffrey et al. (2014); Harty et al. (2014); Bultink et al. (2016); Heinsoo et al. (2018) at or below the threshold for popular QEC schemes such as surface Raussendorf and Harrington (2007); Fowler et al. (2012) and color codes Bombin and Martin-Delgado (2007). They therefore seem well poised for the experimental pursuit of quantum fault tolerance. However, a central assumption of textbook QEC, that error processes can be discretized into bit flips ($X$), phase flips ($Z$) or their combination ($Y=iXZ$) only, is difficult to satisfy experimentally. This is due to the prevalent use of many-level systems as effective qubits, such as hyperfine levels in ions and weakly anharmonic transmons in superconducting circuits, making leakage from the two-dimensional computational space of effective qubits a threatening error source. In quantum dots and trapped ions, leakage events can be as frequent as qubit errors Brown and Brown (2018); Andrews et al. (2019). However, even when leakage is less frequent than qubit errors as in superconducting circuits Barends et al. (2014); Rol et al. (2019), if ignored, leakage can produce the dominant damage to encoded logical information. To address this, theoretical studies propose techniques to reduce the effect of leakage by periodically moving logical information, and removing leakage when qubits are free of logical information Aliferis and Terhal (2007); Fowler (2013); Ghosh and Fowler (2015); Suchara et al. (2015). Alternatively, more hardware-specific solutions have been proposed for trapped ions Brown et al. (2019) and quantum dots Cai et al. (2019). In superconducting circuits, recent experiments have demonstrated single- and multi-round parity measurements to correct qubit errors with up to 9 physical qubits Ristè et al. (2013); Liu et al. (2016); Córcoles et al. (2015); Ristè et al. (2015); Kelly et al. (2015); Takita et al. (2016, 2017); Harper and Flammia (2019); Andersen et al. (2019). Parallel approaches encoding information in the Hilbert space of single resonators using cat Ofek et al. (2016) and binomial codes Hu et al. (2019) used transmon-based photon-parity checks to approach the break-even point for a quantum memory. However, no experiment has demonstrated the ability to detect and mitigate leakage in a QEC context.

In this report, we experimentally investigate leakage detection and mitigation in a minimal QEC system. Specifically, we protect an entangled state of two transmon data qubits ($Q_{\mathrm{DH}}$ and $Q_{\mathrm{DL}}$) from qubit errors and leakage during up to $26$ rounds of parity measurements via an ancilla transmon ($Q_{\mathrm{A}}$). Performing these parity checks in the $Z$ basis protects the state from $X$ errors, while interleaving checks in the $Z$ and $X$ bases protects it from general qubit errors ($X$, $Y$ and $Z$). Leakage manifests itself as a round-dependent degradation of data-qubit correlations ideally stabilized by the parity checks: \langle Z\texttt{\otimes}Z\rangle in the first case and \langle X\texttt{\otimes}X\rangle, \langle Y\texttt{\otimes}Y\rangle, and \langle Z\texttt{\otimes}Z\rangle in the second. We introduce hidden Markov models (HMMs) to efficiently detect data-qubit and ancilla leakage, using only the string of parity outcomes, demonstrating restoration of the relevant correlations. Although we use postselection here, the low technical overhead of HMMs makes them ideal for real-time leakage correction in larger QEC codes.

II Results

II.1 A mimimal QEC setup

Repetitive parity checks can produce and stabilize two-qubit entanglement. For example, performing a Z\texttt{\otimes}Z parity measurement (henceforth a $\mathrm{ZZ}$ check) on two data qubits prepared in the unentangled state $|\texttt{++}\rangle=(|0\rangle+|1\rangle)\otimes(|0\rangle+|1\rangle)/2$ will ideally project them to either of the two (entangled) Bell states $|\Phi^{+}\rangle=(|00\rangle+|11\rangle)/\sqrt{2}$ or $|\Psi^{+}\rangle=(|01\rangle+|10\rangle)/\sqrt{2}$, as signaled by the ancilla measurement outcome $M_{\mathrm{A}}$. Further $\mathrm{ZZ}$ checks will ideally leave the entangled state unchanged. However, qubit errors will alter the state in ways that may or may not be detectable and/or correctable. For instance, a bit-flip ($X$) error on either data qubit, which transforms $|\Phi^{+}\rangle$ into $|\Psi^{+}\rangle$, will be detected because $X$ anti-commutes with a $\mathrm{ZZ}$ check. The corruption can be corrected by applying a bit flip on either data qubit because this cancels the original error ($X^{2}=I$) or completes the operation X\texttt{\otimes}X, of which $|\Phi^{+}\rangle$ and $|\Psi^{+}\rangle$ are both eigenstates. The correction can be applied in real time using feedback Ristè et al. (2013); Liu et al. (2016); Negnevitsky et al. (2018); Andersen et al. (2019) or kept track of using Pauli frame updating (PFU) Knill (2005); Kelly et al. (2015). We choose the latter, with PFU strategy ”$X$ on $Q_{\mathrm{DH}}$”. Phase-flip errors are not detectable since $Z$ on either data qubit commutes with a $\mathrm{ZZ}$ check. Such errors transform $|\Phi^{+}\rangle$ into $|\Phi^{-}\rangle=(|00\rangle-|11\rangle)/\sqrt{2}$ and $|\Psi^{+}\rangle$ into $|\Psi^{-}\rangle=(|01\rangle-|10\rangle)/\sqrt{2}$. Finally, $Y$ errors produce the same signature as $X$ errors. Our PFU strategy above converts them into $Z$ errors. Crucially, by interleaving checks of type $\mathrm{ZZ}$ and $\mathrm{XX}$ (measuring X\texttt{\otimes}X), arbitrary qubit errors can be detected and corrected. The $\mathrm{ZZ}$ check will signal either $X$ or $Y$ error, and the $\mathrm{XX}$ check will signal $Z$ or $Y$, providing a unique signature in combination.

II.2 Generating entanglement by measurement

Our parity check is an indirect quantum measurement involving coherent interactions of the data qubits with $Q_{\mathrm{A}}$ and subsequent $Q_{\mathrm{A}}$ measurement Saira et al. (2014) (Fig. 1A). The coherent step maps the data-qubit parity onto $Q_{\mathrm{A}}$ in $120$ ns using single-qubit (SQ) and two-qubit controlled-phase (CZ) gates Rol et al. (2019). Gate characterizations Sup indicate state-of-the-art gate errors $e_{\mathrm{SQ}}=\{0.08\pm 0.02,0.14\pm 0.016,0.21\pm 0.06\}\%$ and $e_{\mathrm{CZ}}=\{1.4\pm 0.6,0.9\pm 0.16\}\%$ with leakage per CZ $L_{1}=\{0.27\pm 0.12,0.15\pm 0.07\}\%$. We measure $Q_{\mathrm{A}}$ with a $620$-ns pulse including photon depletion McClure et al. (2016); Bultink et al. (2016), achieving an assignment error $e_{\mathrm{a}}=1.0\pm 0.1\%$. We avoid data-qubit dephasing during the $Q_{\mathrm{A}}$ measurement by coupling each qubit to a dedicated readout resonator and a dedicated Purcell filter Heinsoo et al. (2018) (Fig. S1). The parity check has a cycle time of $740$ ns, corresponding to only $2.5\pm 0.2\%$ and $5.0\pm 0.3\%$ of the data-qubit echo dephasing times Sup .

The parity measurement performance can be quantified by correlating its outcome with input and output states. We first quantify the ability to distinguish even- ($|00\rangle,|11\rangle$) from odd-parity ($|01\rangle,|10\rangle$) data-qubit input states, finding an average parity assignment error $e_{\mathrm{a,ZZ}}=5.1\pm 0.2\%$. Second, we assess the ability to project onto the Bell states by performing a $\mathrm{ZZ}$ check on $|\texttt{++}\rangle$ and reconstructing the most-likely physical data-qubit output density matrix $\rho$, conditioning on $M_{\mathrm{A}}=\pm 1$. When tomographic measurements are performed simultaneously with the $Q_{\mathrm{A}}$ measurement, we find Bell-state fidelities $F_{|\Phi^{+}\rangle|M_{\mathrm{A}}=+1}=\langle\Phi^{+}|\rho_{M_{\mathrm{A}}=+1}|\Phi^{+}\rangle=94.7\pm 1.9\%$ and $F_{|\Psi^{+}\rangle|M_{\mathrm{A}}=-1}=94.5\pm 2.5\%$ (Fig. 1, C and D). We connect $|\Psi^{+}\rangle$ to $|\Phi^{+}\rangle$ by incorporating the PFU into the tomographic analysis, obtaining $F_{\mathrm{|\Phi^{+}\rangle}}=94.6\pm 0.9\%$ without any postselection (Fig. 1E). The nondemolition character of the $\mathrm{ZZ}$ check is then validated by performing tomography only once the $Q_{\mathrm{A}}$ measurement completes. We include an echo pulse on both data qubits during the $Q_{\mathrm{A}}$ measurement to reduce intrinsic decoherence and negate residual coupling between data qubits and $Q_{\mathrm{A}}$ (Fig. S3). The degradation to $F_{\mathrm{|\Phi^{+}\rangle}}=91.8\pm 0.5\%$ is consistent with intrinsic data-qubit decoherence under echo and confirms that measurement-induced errors are minimal.

II.3 Protecting entanglement from bit flips and the observation of leakage

QEC stipulates repeated parity measurements on entangled states. We therefore study the evolution of F_{\mathrm{|\Phi^{+}\rangle}}=(1+\langle X\texttt{\otimes}X\rangle-\langle Y\texttt{\otimes}Y\rangle+\langle Z\texttt{\otimes}Z\rangle)/4 and its constituent correlations as a function of the number $M$ of checks (Fig. 2A). When performing PFU using the first $\mathrm{ZZ}$ outcome only (ignoring subsequent outcomes), we observe that $F_{\mathrm{|\Phi^{+}\rangle}}$ witnesses entanglement ($>0.5$) during 10 rounds and approaches randomization ($0.25$) by $M=25$ (Fig. 2B). The constituent correlations also decay with simple exponential forms. A best fit of the form \langle Z\texttt{\otimes}Z\rangle[M]=a\cdot e^{-M/\upsilon_{ZZ}}+b gives a decay time $\upsilon_{\mathrm{ZZ}}=9.0\pm 0.9$ rounds; similarly, we extract $\upsilon_{\mathrm{XX}}=11.7\pm 1.0$ rounds (Fig. 2, C and D). By comparison, we observe that Bell states evolving under dynamical decoupling only (no $\mathrm{ZZ}$ checks, see Fig. S4) decay similarly ($\upsilon_{\mathrm{ZZ}}=8.6\pm 0.3$, $\upsilon_{\mathrm{XX}}=12.8\pm 0.4$ rounds). These similarities indicate that intrinsic data-qubit decoherence is also the dominant error source in this multi-round protocol.

To demonstrate the ability to detect $X$ and $Y$ but not $Z$ errors, we condition the tomography on signaling no errors during $M$ rounds. This boosts \langle Z\texttt{\otimes}Z\rangle to a constant, while the undetectability of $Z$ errors only allows slowing the decay of \langle X\texttt{\otimes}X\rangle to $\upsilon_{\mathrm{XX}}=33.2\pm 1.7$ rounds (and of \langle Y\texttt{\otimes}Y\rangle to $\upsilon_{\mathrm{YY}}=31.3\pm 1.9$ rounds). Naturally, this conditioning comes at the cost of the postselected fraction $f_{\mathrm{post}}$ reducing with $M$ (Fig. S5).

Moving from error detection to correction, we consider the protection of $|\Phi^{+}\rangle$ by tracking $X$ errors and applying corrections in post-processing. The correction relies on the final two $M_{\mathrm{A}}$ only, concluding even parity for equal measurement outcomes and odd parity for unequal. For this small-scale experiment, this strategy is equivalent to a decoder based on minimum-weight perfect matching (MWPM) Fowler et al. (2012); O’Brien et al. (2017), justifying its use. Because our PFU strategy converts $Y$ errors into $Z$ errors, one expects a faster decay of \langle X\texttt{\otimes}X\rangle compared to the no-error conditioning; indeed, we observe $\upsilon_{\mathrm{XX}}=11.8\pm 1.0$ rounds. Most importantly, correction should lead to a constant \langle Z\texttt{\otimes}Z\rangle. While \langle Z\texttt{\otimes}Z\rangle is clearly boosted, a weak decay to a steady state \langle Z\texttt{\otimes}Z\rangle=0.73\pm 0.03 is also evident (Fig. 2D). As previously observed in Ref. Negnevitsky et al. (2018), this degradation is the hallmark of leakage [see also Kelly et al. (2015); Liu et al. (2016)]. We additionally compare the experimental results to simulations using a model that assumes ideal two-level systems O’Brien et al. (2017) (no leakage) based on independently calibrated parameters of  Table S1 (Fig. S8 A to D). At $M=1$ model and experiment coincide for all correction strategies. At larger $M$ ‘first’ and ‘final’ correction strategies deviate significantly, consistent with a gradual build-up of leakage, which we now turn our focus to.

II.4 Leakage detection using hidden Markov models

Both ancilla and data-qubit leakage in our experiment can be inferred from a string $\vec{M}_{A}=(M_{\mathrm{A}}[m=0],\ldots,M_{\mathrm{A}}[m=M])$ of measurement outcomes. Leakage of $Q_{\mathrm{A}}$ to the second excited transmon state $|2\rangle$ produces $M_{\mathrm{A}}=-1$ because measurement does not discern it from $|1\rangle$. This leads to the pattern \vec{M}_{A}=(\texttt{\ldots}-1,-1\texttt{\ldots}) until $Q_{\mathrm{A}}$ seeps back to $|1\rangle$ (coherently or by relaxation), as it is unaffected by subsequent $\pi/2$ rotations (Fig. 3C). Leakage of a data qubit (Fig. 3B) leads to apparent repeated errors [signaled by \vec{M}_{A}=(\texttt{\ldots}+1,+1,-1,-1\texttt{\ldots})], as the echo pulses only act on the unleaked qubit. This is equivalent to a pattern of repeated error signals in the data-qubit syndrome $s_{\mathrm{D}}[m]:=M_{\mathrm{A}}[m]\cdot M_{\mathrm{A}}[m-2]$ — s_{\mathrm{D}}=(\texttt{\ldots},-1,-1,-1,\texttt{\ldots}). (We call $s_{\mathrm{D}}[m]=-1$ an error signal as in the absence of noise $s_{\mathrm{D}}[m]=+1$, while the measurements $M_{\mathrm{A}}[m]$ will still depend on the $\mathrm{ZZ}$ parity.)

Neither of the above patterns is entirely unique to leakage; each may also be produced by some combination of qubit errors. Therefore, we cannot unambiguously diagnose an individual experimental run of corruption by leakage. However, given a set of ancilla measurements $M_{\mathrm{A}}[0],\ldots,M_{\mathrm{A}}[m]$, the likelihood $L_{\mathrm{comp,Q}}(\vec{M}_{A})$ that qubit $Q$ is in the computational subspace during the final parity checks is well-defined. In this work, we infer $L_{\mathrm{comp,Q}}(\vec{M}_{A})$ by using a hidden Markov model (HMM) Baum and Petrie (1966), which treats the system as leaking out of and seeping back to the computational subspace in a stochastic fashion between each measurement round (a leakage HMM in its simplest form is shown in Fig. 3A, and further described in Secs. IV.2, IV.3 and IV.4). This may be extended to scalable leakage detection (for the purposes of leakage mitigation) in a larger QEC code, by using a separate HMM for each data qubit and ancilla. To improve the validity of the HMMs, we extend their internal states to allow the modeling of additional noise processes in the experiments (detailed in Secs. IV.5 and IV.6).

Before assessing the ability of our HMMs to improve fidelity in a leakage mitigation scheme, we first validate and benchmark them internally. A common method to validate the HMM’s ability to model the experiment is to compare statistics of the experimentally-generated data to a simulated data set generated by the model itself. As we are concerned only with the ability of the HMM to discriminate leakage, $L_{\mathrm{comp,Q}}(\vec{M}_{A})$ provides a natural metric for comparison. In Fig. 3, D and E, we overlay histograms of $10^{5}$ experimental and simulated experiments, binned according to $L_{\mathrm{comp,Q}}(\vec{M}_{A})$, and observe excellent agreement. To further validate our model, we calculate the Akaike information criterion Akaike (1974):

[TABLE]

where $L(\vec{o}|M)$ is the likelihood of making the set of observations $\{\vec{o}\}$ given model $H$ (maximized over all parameters $p_{i}$ in the model, as listed in Table 1.), and $n_{\mathrm{p},M}$ is the number of parameters $p_{i}$. The number $A(H)$ is rather meaningless by itself; we require a comparison model $H^{(\mathrm{comp})}$ for reference. Our model is preferred over the comparison model whenever $A(H)>A(H^{(\mathrm{comp})})$. For comparison, we take the target HMM $H$, remove all parameters describing leakage, and re-optimize. We find the difference $A(H)-A(H^{(\mathrm{comp})})=1.1\times 10^{5}$ for the data-qubit HMM, and $2.1\times 10^{4}$ for the ancilla HMM, giving significant preference for the inclusion of leakage in both cases. [The added internal states beyond the simple two-state HMMs clearly improves the overlap in histograms, Fig. S10, A and B. The added complexity is further justified by the Akaike information criterion Sup ].

The above validation suggests that we may assume that the ratio of actual leakage events at a given $L_{\mathrm{comp,Q}}$ is well approximated by $L_{\mathrm{comp,Q}}$ itself (which is true for the simulated data). Under this assumption, we expose the HMMs discrimination ability by plotting its receiver operating characteristic Green and Swets (1966) (ROC). The ROC (Fig. 3F) is a parametric plot (sweeping a threshold $L^{\mathrm{th}}_{\mathrm{comp,Q}}$) of the true positive rate $\mathrm{TPR}$ (the fraction of leaked runs correctly identified) versus the false positive rate $\mathrm{FPR}$ (the fraction of unleaked runs wrongly identified). Random rejection follows the line $y=x$; the better the detection the greater upward shift. Both ROCs indicate that most of the leakage ($\mathrm{TPR}=0.7$) can be efficiently removed with $\mathrm{FPR}\sim 0.1$. Individual mappings of $\mathrm{TPR}$ and $\mathrm{FPR}$ as a function of $L^{\mathrm{th}}_{\mathrm{comp,Q}}$ can be found in Fig. S9, A and B. Further rejection is more costly, which we attribute to these leakage events being shorter-lived. This is because the shorter a leakage event, the more likely its signature is due to (a combination of) qubit errors. Fortunately, shorter leakage events are also less damaging. For instance, a leaked data qubit that seeps back within the same round may be indistinguishable from a relaxation event, but also has the same effect on encoded logical information Fowler (2013).

We now verify and externally benchmark our HMMs by their ability to improve \langle Z\texttt{\otimes}Z\rangle by rejecting data with a high probability of leakage. To do this, we set a threshold $L^{\mathrm{th}}_{\mathrm{comp,Q}}$, and reject experimental runs whenever $L_{\mathrm{comp,Q}}(\vec{M}_{A})<L^{\mathrm{th}}_{\mathrm{comp,Q}}$. For both HMMs we choose $L^{\mathrm{th}}_{\mathrm{comp,Q}}$ to achieve $\mathrm{TPR}=0.7$. With this choice, we observe a restoration of \langle Z\texttt{\otimes}Z\rangle to its first-round value across the entire curve (Fig. 3G), mildly reducing $f_{\mathrm{post}}$ to $0.82$ (averaged over $M$). This restoration from leakage is confirmed by the ‘final + HMM’ data matching the no-leakage model results in Fig. S8, A to D. As low $L_{\mathrm{comp,Q}}(\vec{M}_{A})$ is also weakly correlated with qubit errors, the gain in \langle Z\texttt{\otimes}Z\rangle is partly due to false positives. Of the $\sim 0.13$ increase at $M=25$, we attribute $0.07$ to actual leakage (estimated from the ROCs). By comparison, the simple two-state HMM, leads to a lower improvement, whilst rejecting a larger part of the data (Fig. S10G), ultimately justifying the increased HMM complexity in this particular experiment.

II.5 Protecting entanglement from general qubit errors and mitigation of leakage

We finally demonstrate leakage mitigation in the more interesting scenario where $|\Phi^{+}\rangle$ is protected from general qubit error by interleaving $\mathrm{ZZ}$ and $\mathrm{XX}$ checks Negnevitsky et al. (2018); Andersen et al. (2019). $\mathrm{ZZ}$ may be converted to $\mathrm{XX}$ by adding $\pi/2$ $y$ rotations on the data qubits simultaneous with those on $Q_{\mathrm{A}}$. This requires that we change the definition of the syndrome to $s_{\mathrm{D}}[m]=M_{\mathrm{A}}[m]\cdot M_{\mathrm{A}}[m-1]\cdot M_{\mathrm{A}}[m-2]\cdot M_{\mathrm{A}}[m-3]$, as we need to ‘undo’ the interleaving of the $\mathrm{ZZ}$ and $\mathrm{XX}$ checks to detect errors. For an input state $|\texttt{+}0\rangle=(|0\rangle+|1\rangle)/\sqrt{2}\otimes|0\rangle$, a first pair of checks ideally projects the data qubits to one of the four Bell states with equal probability. Expanding the PFU to $X$ and/or $Z$ on $Q_{\mathrm{DH}}$ we find $F_{\mathrm{|\Phi^{+}\rangle}}=83.8\pm 0.8\%$ (Fig. S6). For subsequent rounds, the ‘final’ strategy now relies on the final three $M_{\mathrm{A}}$. We observe a decay towards a steady state $F_{\mathrm{|\Phi^{+}\rangle}}=73.7\pm 0.9\%$ (Fig. 4), consistent with previously observed leakage. We battle this decay by adapting the HMMs (detailed in Secs. IV.5 and IV.6). We find an improved ROC for $Q_{\mathrm{A}}$ leakage (Fig. S7). For data-qubit leakage however, the ROC is degraded. This is to be expected — when one data qubit is leaked in this experiment, the ancilla effectively performs interleaved $Z$ and $X$ measurements on the unleaked qubit. This leads to a signal of random noise $P(s_{\mathrm{D}}[m]=-1)=0.5$, which is less distinguishable from unleaked experiments $P(s_{\mathrm{D}}[m]=-1)\sim 0$ than the signal of a leaked data-qubit during the \langle Z\texttt{\otimes}Z\rangle-only experiment $P(s_{\mathrm{D}}[m]=-1)\sim 1$. Most importantly, thresholding to $\mathrm{TPR}=0.7$ restores \langle X\texttt{\otimes}X\rangle and \langle Z\texttt{\otimes}Z\rangle, leading to an almost constant $F_{\mathrm{|\Phi^{+}\rangle}}=82.8\pm 0.2\%$ with $f_{\mathrm{post}}=0.81$ (averaged over $M$), as expected from the no-leakage model results in Fig. S8, E to H. In this experiment, the simple two-state HMMs performs almost identically compared to the complex HMM, achieving Bell-state fidelities within $2\%$ whilst retaining the same amount of data (Fig. S10N).

III Discussion

This HMM demonstration provides exciting prospects for leakage detection and correction. In larger systems, independent HMMs can be dedicated to each qubit because leakage produces local error signals Ghosh and Fowler (2015). An HMM for an ancilla only needs its measurement outcomes while a data-qubit HMM only needs the outcomes of the nearest-neighbor ancillas [details in Sup ]. Therefore, the computational power grows linearly with the number of qubits, making the HMMs a small overhead when running parallel to MWPM. HMM outputs could be used as inputs to MWPM, allowing MWPM to dynamically adjust its weights. The outputs could also be used to trigger leakage reduction units Aliferis and Terhal (2007); Fowler (2013); Ghosh and Fowler (2015); Suchara et al. (2015) or qubit resets Magnard et al. (2018).

In summary, we have performed the first experimental investigation of leakage detection during repetitive parity checking, successfully protecting an entangled state from qubit errors and leakage in a circuit quantum electrodynamics processor. Future work will extend this protection to logical qubits, e.g., the 17-qubit surface code Tomita and Svore (2014); O’Brien et al. (2017). The low technical overhead and scalability of HMMs is attractive for performing leakage detection and correction in real time using the same parity outcomes as traditionally used to correct qubit errors only.

IV Materials and methods

IV.1 Device

Our quantum processor (Fig. S1) follows a three-qubit-frequency extensible layout with nearest-neighbor interactions that is designed for the surface code Versluis et al. (2017). Our chip contains low- and high-frequency data qubits ($Q_{\mathrm{DL}}$ and $Q_{\mathrm{DH}}$), and an intermediate-frequency ancilla ($Q_{\mathrm{A}}$). Single-qubit gates around axes in the equatorial plane of the Bloch sphere are performed via a dedicated microwave drive line for each qubit. Two-qubit interactions between nearest neighbors are mediated by a dedicated bus resonator (extensible to four per qubit) and controlled by individual tuning of qubit transition frequencies via dedicated flux-bias lines DiCarlo et al. (2009). For measurement, each qubit is dispersively coupled to a dedicated readout resonator (RR) which is itself connected to a common feedline via a dedicated Purcell resonator (PR). The RR-PR pairs allow frequency-multiplexed readout of selected qubits with negligible backaction on untargeted qubits Heinsoo et al. (2018).

IV.2 Hidden Markov models

HMMs provide an efficient tool for indirect inference of the state of a system given a set of output data Baum and Petrie (1966). A hidden Markov model describes a time-dependent system as evolving between a set of $N_{h}$ hidden states $\{h\}$ and returning one of $N_{o}$ outputs $\{o\}$ at each timestep $m$. The evolution is stochastic: the system state $H[m]$ of the system at timestep $m$ depends probabilistically on the state $H[m-1]$ at the previous timestep, with probabilities determined by a $N_{h}\times N_{h}$ transition matrix $A$

[TABLE]

The user cannot directly observe the system state, and must infer it from the outputs $O[m]\in\{o\}$ at each timestep $m$. This output is also stochastic: $O[m]$ depends on $H[m]$ as determined by a $N_{o}\times N_{h}$ output matrix $B$

[TABLE]

If the $A$ and $B$ matrices are known, along with the expected distribution $\vec{\pi}^{(\mathrm{prior})}[0]$ of the system state over the $N_{h}$ possibilities,

[TABLE]

one may simulate the experiment by generating data according to the above rules. Moreover, given a vector $\vec{o}$ of observations, we may calculate the distribution $\vec{\pi}[m]$ over the possible states at a later time $m$,

[TABLE]

by interleaving rounds of Markovian evolution,

[TABLE]

and Bayesian update,

[TABLE]

IV.3 Hidden Markov models for QEC experiments

To maximize the discrimination ability of HMMs in the various settings studied in this work, we choose different quantities to use for our output vectors $\vec{o}$. In all experiments in this work, the signature of a leaked ancilla is repeated $M_{\mathrm{A}}[m]=-1$, and so we choose $\vec{o}=\vec{M_{\mathrm{A}}}$. By contrast, the signature of leaked data qubits in both experiments may be seen as an increased error rate in their corresponding syndromes $\vec{s_{\mathrm{D}}}$, and we choose $\vec{o}=\vec{s_{\mathrm{D}}}$ for the corresponding HMMs.

One may predict the computational likelihood for data-qubit (D) leakage at timestep $M$ in the $\mathrm{ZZ}$-check experiment given $\vec{\pi}(M)$. In particular, once we have declared which states $h$ correspond to leakage, we may write

[TABLE]

However, in the repeated $\mathrm{ZZ}$-check experiment, the ancilla (A) needs to be within the computational subspace for two rounds to perform a correct parity measurement. Therefore, the computational likelihood is slightly more complicated to calculate,

[TABLE]

In the interleaved $\mathrm{ZZ}$- and $\mathrm{XX}$-check experiment, the situation is more complicated as we require data from the final two parity checks to fully characterize the quantum state. This implies that we need unleaked data qubits for the last two rounds and unleaked ancillas for the last three. The likelihood of the latter may be calculated by similar means to the above.

IV.4 Simplest models for leakage discrimination

One need not capture the full dynamics of the quantum system in a HMM to infer whether a qubit is leaked. This is of critical importance if we wish to extend this method for the purposes of leakage mitigation in a large QEC code [as we discuss in Sup ]. The simplest possible HMM (Fig. 3A) has two hidden states: $H[m]=1$ if the qubit(s) in question are within the computational subspace, and $H[m]=2$ if $Q_{\mathrm{A}}$ (or either data qubit) is leaked. (The labels $1$ and $2$ are arbitrary here, and explicitly have no correlation with the qubit states $|1\rangle$ and $|2\rangle$.) Then, the $2\times 2$ transition matrix simply captures the leakage and seepage rates of the system in question:

[TABLE]

The $2\times 2$ output matrices then capture the different probabilities of seeing output $O[m]=0$ or $O[m]=1$ when the qubit(s) are leaked or unleaked:

[TABLE]

When studying data-qubit leakage, $p_{0,1}$ simply captures the rate of errors within the computational subspace. Then, in the repeared $\mathrm{ZZ}$-check experiment, $p_{1,0}$ captures events such as ancilla or measurement errors that cancel the error signal of a leakage event. However, in the interleaved $\mathrm{ZZ}$ and $\mathrm{XX}$ experiment, a leaked qubit causes the syndrome to be random, so we expect $p_{1,0}\sim 0.5$. When studying ancilla leakage, $p_{1,0}$ is simply the probability of $|2\rangle$ state being read out as $|0\rangle$, and is also expected to be close to [math]. However, $p_{0,1}\sim 0.5$, as we do not reset $Q_{\mathrm{A}}$ or the logical state between rounds of measurement, and thus any measurement in isolation is roughly equally-likely to be [math] or $1$. In all situations, we assume that the system begins in the computational subspace — $\pi_{n}(0)=\delta_{n,0}$. With this fixed, we may choose the parameters $p_{\mathrm{leak}}$, $p_{\mathrm{seep}}$, $p_{0,1}$ and $p_{1,0}$ to maximize the likelihood $L(\{\vec{o}\})$ of observing the recorded experimental data $\{o\}$. (Note that $L(\{\vec{o}\})$ is not the computational likelihood $L_{\mathrm{comp,Q}}$.)

IV.5 Modeling additional noise

The simple model described above does not completely capture all of the details of the stabilizer measurements $\vec{M}_{A}$. For example, the data-qubit HMM will overestimate the leakage likelihood when an ancilla error occurs, as this gives a signal with a time correlation that is unaccounted for. As the signature of a leakage event in a fully fault-tolerant code will be large Sup , we expect these details to not significantly hinder the simple HMM in a large-scale QEC simulation. However, this lack of accuracy makes evaluating HMM performance somewhat difficult, as internal metrics may not be so trustworthy. We also risk overestimating the HMM performance in our experiment, as our only external metrics for success (e.g., fidelity) do just as poorly when errors occur near the end of the experiment as they do when leakage occurs. Therefore, we extend the set of hidden states in the HMMs to account for ancilla and measurement errors, and to allow the ancilla HMM to keep track of the stabilizer state. To attach physical relevance to the states in our Markovian model, and to limit ourselves to the noise processes that we expect to be present in the system, we generalize Eqs. 19 and 27 to a linearly-parametrized model,

[TABLE]

Here, we choose the matrices $D^{(\mathrm{A})}_{i}$ and $D^{(\mathrm{B})}_{i}$ such that the error rates $p^{(\mathrm{A})}_{i},p^{(\mathrm{B})}_{i}$ correspond to known physical processes. (We add the superscripts $(A)$ and $(B)$ here to the $D$ matrices to emphasize that each error channel only appears in one of the two above equations.)

The error generators $D^{(\mathrm{A})}$, $D^{(\mathrm{B})}$ may be identified as derivatives of $A$ with respect to these error rates:

[TABLE]

This may be extended to calculate derivatives of the likelihood $L(\{\vec{o}\})$ (or more practically, the log-likelihood) with respect to the various parameters $p_{i}$. This allows us to obtain the maximum likelihood model within our parametrization via gradient descent methods (in particular the Newton-CG method), instead of resorting to more complicated optimization algorithms such as the Baum-Welch algorithm Baum and Petrie (1966). All models were averaged over between $10$ and $20$ optimizations using the Newton-CG method in scipy Jones et al. (2001), calculating likelihoods, gradients and Hessians over $10,000$-$20,000$ experiments per iteration, and rejecting any failed optimizations. As the signal of ancilla leakage is identical to the signal for even $\mathrm{ZZ}$ and $\mathrm{XX}$ parities with ancilla in $|1\rangle$ and no errors, we find that the optimization is unable to accurately estimate the ancilla leakage rate, and so we fix this in accordance with independent calibration to $0.0040$/round using averaged homodyne detection of $|2\rangle$ (making use of a slightly different homodyne voltage for $|1\rangle$ and $|2\rangle$).

IV.6 Hidden Markov models used in Figs. 2 and 4

Different Markov models (with independently optimized parameters) were used to optimize ancilla and data-qubit leakage estimation for both the $\mathrm{ZZ}$ experiment and the experiment interleaving $\mathrm{ZZ}$ and $\mathrm{XX}$ checks. This lead to a total of four HMMs, which we label $H_{\mathrm{ZZ}}$-D, $H_{\mathrm{ZZ}}$-A, $H_{\mathrm{ZZ},\mathrm{XX}}$-D and $H_{\mathrm{ZZ},\mathrm{XX}}$-A. A complete list of parameter values used in each HMM is given in Table 1. We now describe the features captured by each HMM. As we show in Sup , these additional features are not needed to increase the error mitigation performance of the HMMs, but rather to ensure their closeness to the experiment and increase trust in their internal metrics.

To go beyond the simple HMM in the $\mathrm{ZZ}$-check experiment when modeling data-qubit leakage ($H_{\mathrm{ZZ}}$-D), we need to include additional states to account for the correlated signals of ancilla and readout error. If we assume data-qubit errors (that remain within the logical subspace) are uncorrelated in time, they are already well-captured in the simple model. This is because any single error on a data qubit may be decomposed into a combination of $Z$ errors (which commute with the measurement and thus are not detected) and $X$ errors (which anti-commute with the measurement and thus produce a single error signal $s_{\mathrm{D}}[m]=1$), and is thus captured by the $p_{0,1}$ parameter. When one of the data qubits is leaked, uncorrelated $X$ errors on the other data qubit cancel the constant $s_{\mathrm{D}}[m]=-1$ signal for a single round, and are thus captured by the $p_{1,0}$ parameter. However, errors on the ancilla, and readout errors, give error signals that are correlated in time (separated by $1$ or $2$ timesteps, respectively). This may be accounted for by including extra ‘ancilla error states’. These may be most easily labeled by making the $h$ labels a tuple $h=(h_{0},h_{1})$, where $h_{0}$ keeps track of whether or not the qubit is leaked, and $h_{1}=1,2,3$ keeps track of whether or not a correlated error has occurred. In particular, we encode the future syndrome for 2 cycles in the absence of error on $h_{1}$, allowing us to account for any correlations up to 2 rounds in the future. This extends the model to a total of $4\times 2=8$ states. The transition and output matrices in the absence of error for the unleaked $h_{0}=0$ states may then be written in a compact form (noting that leakage errors cancel out with correlated ancilla and readout errors to give $s_{\mathrm{D}}[m]=+1$),

[TABLE]

where the double slash $//$ refers to integer division.

Let us briefly demonstrate how the above works for ancilla error in the system. Suppose the system was in the state $h=(0,3)$ at time $m$. It would output $M_{\mathrm{A}}[m]=-1$, and then evolve to $h=(0,3//2)=(0,1)$ at time $m+1$ (in the absence of additional error). Then, it would output a second error signal [$M_{\mathrm{A}}[m+1]=-1$] and finally decay back to the $h=(0,1//2)=(0,0)$ state. This gives the HMM the ability to model ancilla error as an evolution from $h=(0,0)$ to $h(0,3)$. Formally, we assign the matrix $D^{(A)}_{\mathrm{ancilla}}$ to this error process, and following this argument we have

[TABLE]

The corresponding error rate $p_{\mathrm{ancilla}}$ is then an additional free parameter to be optimized to maximize the likelihood. To finish the characterization of this error channel, we need to consider the effect of ancilla error in states other than $h=(0,0)$. Two ancilla errors in the same timestep cancel, but two ancilla errors in subsequent timesteps will cause the signature $s_{\mathrm{D}}=\ldots,-1,+1,-1,\ldots$. This may be captured by an evolution from $h=(0,2)$ to $h=(0,3)$ [instead of $h=(0,1)$], which implies we should set

[TABLE]

(Note that $A_{0}$ already captures a decay from $h=(0,2)\rightarrow(0,1)\rightarrow(0,0)$, which will give the desired signal.) We note that this also matches the signature of readout error, which can then be captured by a separate error channel $D^{(A)}_{\mathrm{readout}}$ which increases this correlation

[TABLE]

One can then check that ancilla errors in $h=(0,3)$ should cause the system to remain in $h=(0,3)$, and that ancilla or readout errors in $h=(0,1)$ should evolve the system to $h=(0,2)$. We note that this model cannot account for the $s_{\mathrm{D}}=\ldots-1,+1,+1,+1,-1$ signature of readout error at time $m$ and $m+2$, but adjusting the model to include this has negligible effect.

Ancilla error in the $\mathrm{ZZ}$-check experiment when the data qubits are leaked has the same correlated behavior as when they are not, but may occur at a different rate. This requires that we define a new matrix $D^{(A)}_{\mathrm{ancilla,leaked}}$ by

[TABLE]

with a separate error rate $p_{\mathrm{ancilla,leaked}}$. As we do not expect the readout of the ancilla to be significantly affected by whether the data qubit is leaked, we do not add an extra parameter to account for this behavior, and instead simply set

[TABLE]

We also assume that leakage $p_{\mathrm{leak}}$ and seepage $p_{\mathrm{seep}}$ rates are independent of these correlated errors (i.e., $[D^{(A)}_{\mathrm{leak}}]_{(0,j),(0,k)},[D^{(A)}_{\mathrm{seep}}]_{(1,j),(1,k)}]\in\{0,-1\}$). We then assume that the first measurement made following a leakage/seepage event is just as likely to have an additional error (corresponding to an evolution to $(h_{0},1)$) or not (corresponding to an evolution to $(h_{0},0)$). We finally account for data-qubit error in the output matrices in the same way as in the simple model, but with different error rates $p_{\mathrm{data,leaked}}$ for the leaked states $(1,h_{1})$ and $p_{\mathrm{data}}$ for the unleaked states $(0,h_{1})$.

There are a few key differences between the interleaved $\mathrm{ZZ}$—$\mathrm{XX}$ and $\mathrm{ZZ}$ experiments that need to be captured in the data-qubit HMM $H_{\mathrm{ZZ},\mathrm{XX}}-D$. Firstly, as the syndrome is now given by $s_{\mathrm{D}}[m]=M_{\mathrm{A}}[m]\cdot M_{\mathrm{A}}[m-1]\cdot M_{\mathrm{A}}[m-2]\cdot M_{\mathrm{A}}[m-3]$, ancilla and classical readout error can then generate a signal stretching up to $4$ steps in time. This implies that we require $2^{4}$ possibilities for $h_{1}$ to keep track of all correlations. However, as a leaked data qubit makes ancilla output random in principle, we no longer need to keep track of the ancilla output upon leakage. This implies that we can accurately model the experiment with $16+1=17$ states, which we can label by $h\in\{2,(1,h_{1})\}$. The $A_{0}$ and $B_{0}$ matrices in the unleaked states $(1,h_{1})$ follow Eq. 30, and we fix $[A_{0}]_{2,2}=1$ (as in the absence of $p_{\mathrm{seep}}$ a leaked state stays leaked). However, we allow for some bias in the leaked state error rate - $B_{-1,2}=p_{\mathrm{data,leaked}}$ is not fixed to $0.5$. (For example, this accounts for a measurement bias towards a single state, which will reduce the error rate below $0.5$.) The non-zero elements in the matrices $D^{(A)}_{\mathrm{ancilla}}$ and $D^{(A)}_{\mathrm{readout}}$ may be written:

[TABLE]

Here, $a\oplus b$ refers to addition of each binary digit of $a$ and $b$ modulo $2$. We may use this formalism to additionally keep track of $Y$ data-qubit errors, which show up as correlated errors on subsequent $\mathrm{XX}$ and $\mathrm{ZZ}$ stabilizer checks, by introducing a new error channel

[TABLE]

with a corresponding error rate $p_{\mathrm{data,Y}}$. As before, we assume that leakage occurs at a rate $p_{\mathrm{leak}}$ independently of $h_{1}$, and that seepage takes the system either to the state with either no error signal $h=(1,0)$ or one error signal $h=(0,1)$ with a rate $p_{\mathrm{seep}}$.

As the output used for the $H_{\mathrm{ZZ}}-A$ HMM is the pure measurement outcomes $M_{\mathrm{A}}$, the dominant signal that must be accounted for is that of the stabilizer $\mathrm{ZZ}$ itself. This either causes a constant signal $M_{\mathrm{A}}[m]=M_{\mathrm{A}}[m-1]$ or a constant flipping signal $M_{\mathrm{A}}[m]=-M_{\mathrm{A}}[m-1]$. This cannot be accounted for in the simple HMM, as it cannot contain any history in a single unleaked state. To deal with this, we extend the set of states in the $H_{\mathrm{ZZ}}-A$ HMM to include both an estimate of the ancilla state $a\in\{0,1,2\}$ at the point of measurement, and the stabilizer state $s\in\{0,1\}$, and label the states by the tuple $(a,s)$. The ancilla state then immediately defines the device output in the absence of any error:

[TABLE]

while the stabilizer state defines the transitions in the absence of any error or leakage:

[TABLE]

The only thing that affects the output matrices is readout error:

[TABLE]

Data-qubit errors flip the stabilizer with probability $p_{\mathrm{data}}$:

[TABLE]

Ancilla errors flip the ancilla with probability $p_{\mathrm{ancilla}}$, but these are dominated by $T_{1}$ decay, and so are highly asymmetric. To account for this, we used different error rates $p_{\mathrm{anc},a,a^{\prime}}$ for the four possible combinations of ancilla measurement at time $m-1$ and expected ancilla measurement at time $m$:

[TABLE]

(Note that this asymmetry could not be accounted for in the data-qubit HMM as the state of the ancilla was not contained within the output vector.) As with the data-qubit HMMs, we assume that ancilla leakage is HMM-state independent, as it is dominated by CZ gates during the time that the ancilla is either in $|+\rangle$ or $|-\rangle$. We also assume that leakage (with rate $p_{\mathrm{leak}}$) and seepage (with rate $p_{\mathrm{seep}}$) have equal chances to flip the stabilizer state, as ancilla leakage has a good chance to cause additional error on the data qubits.

The ancilla-qubit HMMs need little adjustment between the $\mathrm{ZZ}$-check experiment and the experiment interleaving $\mathrm{ZZ}$ and $\mathrm{XX}$ checks. The $H_{\mathrm{ZZ},\mathrm{XX}}$-A HMM behaves almost identically to the $H_{\mathrm{ZZ}}$-A HMM, but we include in the state information on the $\mathrm{XX}$ stabilizer as well as the $\mathrm{ZZ}$ stabilizer. This leaves the states indexed as $(a,s_{1},s_{2})$. The HMM needs to also keep track of which stabilizer is being measured. This may be achieved by shuffling the stabilizer labels at each timestep: for $a=0,1$, we set

[TABLE]

Other than this, the HMM follows the same equations as above (with the additional index added as expected.)

IV.7 Uncertainty calculations

All quoted uncertainties are an estimation of standard error of the mean (SEM). SEMs for the independent device characterizations (Sec. II.2, Table 1) are either obtained from at least three individually fitted repeated experiments ($T_{2}^{\mathrm{echo}}$, $T_{1}$, $T_{2}^{*}$, $\eta$, $e_{\mathrm{a}}$, $e_{\mathrm{a,ZZ}}$) or in the case that the quantitiy is only measured once ($e_{\mathrm{SQ}}$, $e_{\mathrm{CZ}}$, $L_{1}$), the SEM is estimated from least-squares fitting by the LmFit fitting module using the covariance matrix Newville et al. (2014).

SEMs in the first-round Bell-state fidelities (Figs. 1 and S6, Secs. II.2 and II.5) are obtained through bootstrapping. For bootstrapping, a data-set (in total 4096 runs with each 36 tomographic elements and 28 calibration points) is subdivided into four subsets and tomography is performed on each of these subsets individually. As verification, subdivision was performed with eight subsets leading to similar SEMs.

SEMs in the multi-round experiment parameters (steady-state fidelities, decay constants) are also estimated from least-squares fitting by the LmFit fitting module using the covariance matrix Newville et al. (2014) (Secs. II.3, II.4 and II.5).

Acknowledgements.

We thank W. Oliver and G. Calusine for providing the parametric amplifier, J. van Oven and J. de Sterke for experimental assistance and F. Battistel, C. Beenakker, C. Eichler, F. Luthi, B. Terhal, and A. Wallraff for discussions. Funding: This research is 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-16-1-0071, and by Intel Corporation. T.E.O. is funded by Shell Global Solutions BV. 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. X.F. was funded by China Scholarship Council (CSC). Author contributions: C.C.B. performed the experiment. R.V. and M.W.B. designed the device with input from C.C.B. and B.T. N.M. and A.B. fabricated the device. T.E.O. devised the HMMs with input from B.T., B.V., and V.O. X.F. and M.A.R. contributed to the experimental setup and tune-up. C.C.B., T.E.O., and L.D.C. co-wrote the manuscript with feedback from all authors. L.D.C. supervised the project. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.

Supplementary Materials for: Protecting quantum entanglement from leakage and qubit errors via repetitive parity measurements

V Materials and methods

V.1 Setup

A full wiring diagram of the setup is provided in (Fig. S2). All operations are controlled by a fully digital device, the central controller (CC7), which takes as input a binary in an executable quantum instruction set architecture [eQASM Fu et al. (2019)], and outputs digital codeword triggers based on the execution result of these instructions. These digital codeword triggers are issued every 20 ns to arbitrary waveform generators (AWGs) for single-qubit gates and two-qubit gates, a vector switch matrix (VSM) for single-qubit gate routing and a readout module (AWG and acquisition) for frequency-multiplexed readout. Single-qubit gate generation, readout pulse generation and readout signal integration are performed by single-sideband mixing. The measurement signal is amplified with a JTWPA Macklin et al. (2015) at the front end of the amplification chain. Following Ref. Bultink et al. (2018), we extract an overall measurement efficiency $\eta=48\pm 1.0\%$ by comparing the integrated signal-to-noise ratio of single-shot readout to the integrated measurement-induced dephasing.

V.2 Cross-measurement-induced dephasing of data qubits

During ancilla measurement, data-qubit coherence is susceptible to intrinsic decoherence, phase shifts via residual ZZ interactions and cross-measurement-induced dephasing Saira et al. (2014); Heinsoo et al. (2018). For the single-data-qubit subspace we investigate the different contributions experimentally and assess the benefit of an echo pulse on the data qubits halfway through the ancilla measurement. We study this by including the ancilla measurement (with amplitude $\varepsilon$) in a Ramsey-type sequence (Fig. S3A). By varying the azimuthal phase of the second $\pi/2$ pulse, we obtain Ramsey fringes from which we extract the coherence $|\rho_{\mathrm{01}}|$ and phase $arg\left(\rho_{\mathrm{01}}\right)$. Several features of these curves explain the need for the echo pulse on the data qubits. Firstly, at $\varepsilon=0$, the echo pulse improves data-qubit coherence (for both ancilla states) by reducing the effect of low-frequency noise (Fig. S3, B and C). This is confirmed by individual Ramsey and echo experiments. Secondly, the echo pulse almost perfectly cancels ancilla-state dependent phase shifts due to residual ZZ interactions (Fig. S3, D and E). When gradually turning on the ancilla measurement towards the nominal value $\varepsilon=1$, we furthermore observe that: thirdly, the echo pulse almost perfectly cancels the measurement-induced Stark shift (Fig. S3, D and E). When increasing the measurement amplitude beyond the operation amplitude (indicated by the vertical dashed lines), we see rapid non-Gaussian decay of data-qubit coherence. We attribute this to measurement-induced relaxation of the ancilla: via the ZZ interaction, this can lead to probabilistic phase shifts on the data qubit. This effect is stronger for $Q_{\mathrm{DL}}$ than for $Q_{\mathrm{DH}}$ due to its higher residual interaction with $Q_{\mathrm{A}}$ (Table S1).

VI Supplementary Text

VI.1 Performance of the simple hidden Markov model

In this section we detail the performance of the simple HMMs, as described in Fig. 3A and Sec. IV.4 of the main text. In Fig. S10, A and B, we plot a histogram of the computational likelihoods $L_{\mathrm{comp,Q}}$ of $10^{5}$ simulated and actual $\mathrm{ZZ}$ experiments as calculated with the simple HMMs $H_{\mathrm{ZZ}}-D^{(\mathrm{simple})}$ and $H_{\mathrm{ZZ}}-A^{(\mathrm{simple})}$. This can be compared with Fig. 3, B and C, of the main text. We plot similar histograms for the interleaved $\mathrm{ZZ}$—$\mathrm{XX}$ experiment in Fig. S10, H and I. We see reasonable agreement, but noticeably worse agreement than that in the detailed model. This is underscored by the Akaike information criterion (Eq. 1 of the main text), which is significantly reduced compared to the more detailed HMMs:

[TABLE]

Indeed, in all cases the Akaike information criterion for the simple HMM is lower than that for the detailed HMM without leakage. This makes complete sense, as even though the simple HMMs might capture leakage fairly well, the additional effects captured in the detailed HMMs are far more dominant in the measurement signals than that of leakage. As such, the internal metrics, such as the ROC curves (Fig. S11) for the simplified model are significantly less trustworthy than those of the detailed model. This exemplifies the need for external HMM verification, as achieved in the main text by testing the HMM in a leakage mitigation scheme. We now repeat this verification procedure for the simple model. We see that in the $\mathrm{ZZ}$ experiment the performance is significantly degraded; although the flat line in the \langle Z\texttt{\otimes}Z\rangle curve is restored after about $8$ parity checks, it requires rejecting $47\%$ of the data, and is restored to a point $\sim 8\%$ below the performance of the detailed HMM. By contrast, the simple HMM performs almost identically to the complex HMM in the interleaved $\mathrm{ZZ}$—$\mathrm{XX}$ experiment, achieving Bell-state fidelities within $2\%$ whilst retaining the same amount of data. As the signal from a large-scale QEC code is more similar to the latter experiment than the former (See Sec. VI.2), this strongly suggests that the detailed modeling used in this text will not be needed in such experiments.

VI.2 Hidden Markov models for large-scale QEC

The hidden Markov models used in this text provide an exciting prospect for the indirect detection of leakage on both data qubits and ancillas in a QEC code. This is essential for accurate decoding of stabilizer measurements made during QEC. Furthermore, this idea can be combined with proposals for leakage reduction Aliferis and Terhal (2007); Fowler (2013); Ghosh and Fowler (2015); Suchara et al. (2015) to target such efforts, reducing unnecessary overhead. As leakage does not spread in superconducting qubits (to lowest order), and gives only local error signals Ghosh and Fowler (2015), such a scheme would require a single HMM per (data and ancilla) qubit. Each individual HMM needs only to process the local error syndrome, and as demonstrated in this work, completely independent HMMs may be used for the detection of nearby data-qubit and ancilla leakage. This implies that the computational overhead of leakage detection via HMMs in a larger QEC code will grow only linearly with the system size. Previous leakage reduction units are designed to act as the identity on the computational subspace (up to additional noise), so we do not require perfect discrimination between leaked and computational states. However, optimizing this discrimination (and investigating threshold levels for the application of targeted leakage reduction) will boost the code performance. Also, near-perfect discrimination could allow for the direct resetting of leaked data qubits Magnard et al. (2018), which would completely destroy an error correcting code if not targeted.

On the other hand, for implementation on classical hardware within the sub-$1~{}\mu s$ QEC cycle time on superconducting qubits O’Brien et al. (2017), one may wish to strip back some of the optimization used in this work. The minimal HMM that could be used in QEC for detection has only two states, leaked and unleaked (Fig. 3A), and $2^{n_{\mathrm{A}}}$ outputs, where $n_{\mathrm{A}}$ is the number of neighboring ancilla on which a signature of leakage is detected. (For the surface code, $n_{\mathrm{A}}\leq 4$ in all situations.) Such a simple model cannot perfectly deal with correlated errors, such as ancilla errors (which give multiple error signals separated in time). However, this should only cause a slight reduction in the discrimination capability whenever such correlations remain local. If the loss in accuracy is acceptable, one may store only $\pi_{0}^{(\mathrm{post})}$, and update it following a measurement $M_{\mathrm{A}}[m]$ as

[TABLE]

which is trivial compared to the overhead for most QEC decoders.

A key question about the use of HMMs for leakage detection in future QEC experiments is whether leakage in larger codes is reliably detectable. In previous theoretical work Ghosh et al. (2013), data-qubit leakage in repetition codes has been sometimes hidden, a phenomenon known as ‘leakage paralysis’ or ‘silent stabilizer’ Waintal (2019). This effect occurs when the relative phase $\varphi$ accumulated between the $|20\rangle$ and $|21\rangle$ states during a CZ gate is a multiple of $\pi$. In the absence of additional error, an indirect measurement of the data qubit via an ancilla would return a result $\frac{\varphi}{\pi}$ mod $2$. (By comparison, if $\varphi=\pi/2$, the ancilla would return measurements of [math] or $1$ at random.) This is then identical to the measurement of a data qubit in the $|\frac{\varphi}{\pi}\;\mathrm{mod}\;2\rangle$ state, and no discrimination between the two may be achieved. However, in an $N$-qubit parity check $S$, the ancilla continues to accumulate phase from the other qubits, reducing this to an $N-1$-qubit effective parity check $S^{\prime}$ (plus a well-defined, constant phase). Such a parity check may no longer commute with other effective parity checks $R^{\prime}$ that share the leaked qubit, even though we would require $[S,R]=0$ in stabilizer QEC. This is demonstrated in our second experiment measuring both $\mathrm{ZZ}$ and $\mathrm{XX}$ parity checks; though these commute when no data qubit is leaked, leakage reduces the checks to non-commuting $Z$ and $X$ measurements (of the unleaked data qubit). (In the $\mathrm{ZZ}$ experiment, the leakage paralysis was broken by the echo pulse on the data qubits, which flips the effective stabilizer of a leaked qubit at each round.) The repeated measurement of these non-commuting operators generates random results, similar to the case when $\varphi=\pi/2$. To the best of our knowledge, in all fully fault-tolerant stabilizer QEC codes, the removal of a single data qubit breaks the commutativity of at least two neighboring stabilizers. As such, data-qubit leakage will always be detectable in QEC experiments with superconducting circuits.

Beyond the proof-of-principle argument above, one might question whether the signal of leakage is improved or reduced when going from our prototype experiment to a larger QEC code, and when the underlying physical-qubit error rate is reduced. Fortunately, we can expect an improvement in the HMM discrimination capability in both situations. To see this, consider the example of a data qubit which is either leaked at round $1$ with probability $p_{\mathrm{leak}}$ or never leaks. Let us further assume that in the absence of leakage, a number of neighboring ancillas $n_{\mathrm{A}}$ incur errors (where the parity check reports a flip) at a rate $p$, whereas in the presence of leakage these ancillas incur errors at a rate $0.5$. (For example, in the bulk of the surface code, $n_{\mathrm{A}}=4$.) The computational likelihood at round $m>0$ after seeing $e$ errors may be calculated as

[TABLE]

If the data qubit was leaked, $e\sim mn_{\mathrm{A}}/2$, and the computational likelihood on average is approximately

[TABLE]

which is of the form

[TABLE]

We see that the signal of leakage ($L_{\mathrm{comp}}[m]\rightarrow 0$) switches on exponentially in time, with a rate proportional to $\log(p^{-n_{\mathrm{A}}/2})$. Any decrease in $p$ (from better qubits) or increases in $n_{\mathrm{A}}$ (from additional ancillas surrounding the leaked qubit in a QEC code) will serve to increase, and not decrease this rate. The exponential decay constant is inversely proportional to the leakage rate (as this corresponds to an initial HMM skepticism towards unlikely leakage events). However, as the likelihood ’switch’ is exponential, a decrease in $p_{\mathrm{leak}}$ by even an order of magnitude should only increase the time before definite detection by a single step or so. The above analysis is complicated in a real scenario, as single physical errors give correlated detection signals, and as leakage may occur at any time, and as leaked qubits may seep. Correlations in the detection signals will serve to renormalize the switching time $\lambda$ (but not remove the generic feature of exponential onset). Seepage causes individual leakage events to be finite (with some average lifetime $T_{\mathrm{seep}}$); an individual leakage event of length $\ll\lambda^{-1}$ will not be detectable by the HMM. However, when the system returns to the computational subspace in such a short period of time, the leakage event may be treated as a ‘regular’ error, and does not need complicated leakage-detection hardware for fault tolerance. For example, a leakage event followed by immediate decay to $|1\rangle$ is indistinguishable from a direct transition to $|1\rangle$ for all practical purposes in QEC.

VII Additional Figures