In-situ benchmarking of fault-tolerant quantum circuits. I. Clifford circuits

TL;DR

This work develops an in-situ framework to benchmark fault-tolerant quantum circuits by learning physical and logical Pauli noise directly from syndrome data. It maps Clifford circuits to spacetime (subsystem) codes, enabling a Walsh-Hadamard-based decomposition of noise into local channel eigenvalues and syndrome classes, with learnability conditions rigorously established. The authors prove constant-sample learnability for syndrome-class physical noise on qLDPC codes and polynomial-sample learnability for circuit-level logical noise, yielding an exponential advantage over direct logical measurements at very low logical error rates. They introduce efficient algorithms to identify a minimal, independent set of stabilizer measurements and validate the approach on synthetic benchmarks and experimental data from fault-tolerant GHZ-state demonstrations, showing accurate prediction of logical fidelities and useful diagnostics for gate calibration and decoding. The framework offers a scalable, in-situ method to characterize, verify, and benchmark fault-tolerant quantum computations, with extensions to non-Clifford settings and Part II addressing magic-state and more general circuits.

Abstract

Benchmarking physical devices and verifying logical algorithms are important tasks for scalable fault-tolerant quantum computing. Numerous protocols exist for benchmarking devices before running actual algorithms. In this work, we show that both physical and logical errors of fault-tolerant circuits can even be characterized in-situ using syndrome data. To achieve this, we map general fault-tolerant Clifford circuits to subsystem codes using the spacetime code formalism and develop a scheme for estimating Pauli noise in Clifford circuits using syndrome data. We give necessary and sufficient conditions for the learnability of physical and logical noise from given syndrome data, and show that we can accurately predict logical fidelities from the same data. Importantly, our approach requires only a polynomial sample size, even when the logical error rate is exponentially suppressed by the code distance, and thus gives an exponential advantage against methods that use only logical data such as direct fidelity estimation. We demonstrate the practical applicability of our methods in various scenarios using synthetic data as well as the experimental data from a recent demonstration of fault-tolerant circuits by Bluvstein et al. [Nature 626, 7997 (2024)]. Our methods provide an efficient, in-situ way of characterizing a fault-tolerant quantum computer to help gate calibration, improve decoding accuracy, and verify logical circuits.

Figures (19)

• Figure 1: An example of the total Pauli error channel $\mathcal{N}_{\Gamma}$ as the composition of local Pauli channels $\mathcal{N}_{\gamma}$ (enclosed in dotted lines) acting on subsets $\gamma\in\Gamma$ of qubits (circles), where $\Gamma=\{\gamma_1,\gamma_2,\gamma_3,\gamma_4\}$. (a) The error rate $P$ is the convolution of the error rates $P_{\gamma}$ of all local error channels. (b) The factor graph of Pauli eigenvalues of the total channel $\mathcal{N}_{\Gamma}$ can be written as the product of eigenvalues $\Lambda_{\gamma}$ of the local channels after performing Walsh-Hadamard transformation of the error rates in (a).
• Figure 2: Simulation result on various codes of learning syndrome class total error rates (true error rates in blue) using exact syndrome expectation values (light blue) and expectation values (red) from 100k samples. Here, single-qubit Pauli error rates are randomly generated around the mean. Codes in (a) seven-qubit Steane code and (c) the $[[72,12,6]]$ bivariate bicycle codes with polynomial $A(x,y)=x^3+y+y^2$, $B(x,y)=y^3+x+x^2$, have $d_{\text{pure}}\geq 3$, therefore all error rates are learnable. (b) The $3 \times 3$ rotated surface code has weight-2 $XX$ (blue) and $ZZ$ (white) generators on the boundary. (d) The $[[8,3,2]]$ 3D color code have $Z$ stabilizers on the faces and a single $X$ stabilizer generator $X^{\otimes 8}$. Therefore, all $Z$ errors, denoted as $\mathcal{Z}s$, are indistinguishable. Both codes in (b) and (d) have $d_{\text{pure}}=2$, and the sum of error rates in each syndrome error class can be learned.
• Figure 3: Numerical results of relative error of learned single-qubit error rates vs sample size. The true error rates are sampled from $p\sim\mathcal{N}(\mu,\sigma^2)$, where $\mu=5/3\times10^{-3},\sigma=1/3\times10^{-3}$.(a) Estimated sample size needed for surface code with various code distances using the optimization \ref{['equation: optimization']} with uniform constraints $B$ within each syndrome class. This does not involve avoiding overfitting, and only the minimal stabilizer subset is used. Sample size is almost constant with respect to code size $n$. (b) Result of optimization while avoiding overfitting. We test the relative error of learning for surface code $d=23$ (blue) and the $[[72,12,6]]$ bivariate bicycle codes (orange). With the same stabilizer subset $\mathcal{M}'$, the optimization (diamonds) outperforms the analytical solution (dotted lines) in \ref{['equation: recursive solving']} at low sample sizes ($N\leq 30\,k$) and is comparable to it at larger sample sizes. Using only a minimal stabilizer subset gives similar performance (crosses) at low sample size. The result is the average of 20 sets of error rates from the normal distribution $\mathcal{N}$.
• Figure 4: Circuit-to-code mapping illustrated using a syndrome extraction circuit for the 3-qubit repetition code. (a) $r$ rounds of 3-layer syndrome extraction circuits that measure $X_1X_2$ for the repetition code with stabilizer group $\mathcal{S}^{(\text{ini})}=\langle X_1X_2,X_2X_3\rangle$. The last qubit is the ancilla qubit for syndrome readout. (b) The spacetime code with $4\times(3r+1)$ qubits mapped from the circuit in (a). Pauli operator in blue is a measured stabilizer generator of the spacetime code due to the parity constraints on the first and third measurement results (blue squares). The stabilizer elements are local within the associated measurements in the temporal direction. One logical representative $\overline{X}^{\text{(ST)}}$ as the back-culument of $\overline{X}$ at the last layer is shown in black. It will be a measured spacetime logical operator if the first qubit is eventually measured. Correlated $ZZ$ error on the CNOT gate shown in (a) is mapped to $ZZ$ errors at $t=3.5$ (in red). Its syndrome and logical effect are given by the commutation with the spacetime stabilizer and logical operator respectively. The gauge group $\mathcal{G}^{(\text{ST})}$ contains errors with trivial syndrome and logical effect. $\mathcal{G}^{(\text{ST})}$ is non-abelian in general (see a pair of errors in light and dark orange).
• Figure 5: (a) Error correction circuit for the $3 \times 3$ rotated surface code and its corresponding spacetime code. The surface code starts from its logical $\ket{\overline 0}$ state and we simulate $r$ rounds of syndrome extractions followed by a transversal $Z$ measurement. The learning algorithm converts the whole circuit into a spacetime code where spacetime qubits (empty circles) are placed before every layer of gates. (b) The relative error $\overline{\eta}_{C}$ of the learned syndrome class total error rate averaged over all classes. We simulated for different rounds of syndrome extraction. (c) relative error $\overline{\eta}_{\text{layer}}$ of the estimated average single-qubit (1Q) and two-qubit (2Q) error rates per layer. The true circuit-level error rates are sampled from a Gaussian distribution ($\mu=0.001/3,\sigma=0.0002/3$ and $\mu=0.01/15,\sigma=0.002/15$ for the Pauli error components of single-qubit and two-qubit locations respectively).

Theorems & Definitions (57)

• Theorem 2: wagner2022pauli
• Theorem 3: Learnability conditions
• Theorem 4: Sample complexity for syndrome class learning
• Definition 5: Circuit-level Pauli error model
• Definition 6: Circuit location
• Definition 7: Standard circuit-level Pauli error model
• Theorem 8: wagner2023learning
• Definition 9: Fault tolerance
• Theorem 10: Learnability of logical errors
• Theorem 11: Sample complexity of logical error estimation