Efficient learning of logical noise from syndrome data

TL;DR

This work develops a theory and protocol for learning the logical error channel of fault-tolerant quantum circuits from syndrome data. By unifying circuit-level faults with spacetime stabilizer codes and applying Fourier analysis plus compressed sensing, it derives necessary and sufficient learnability conditions, provides provable sample-cost guarantees, and delivers an end-to-end framework to estimate logical error probabilities from syndrome measurements. The approach yields substantial sample-efficiency improvements over direct logical benchmarking and demonstrates practical viability on syndrome-extraction circuits. The results pave the way for scalable, syndrome-based characterization of logical noise in realistic fault-tolerant quantum devices, enabling more accurate calibration and benchmarking of quantum memories and processors.

Abstract

Characterizing errors in quantum circuits is essential for device calibration, yet detecting rare error events requires a large number of samples. This challenge is particularly severe in calibrating fault-tolerant, error-corrected circuits, where logical error probabilities are suppressed to higher order relative to physical noise and are therefore difficult to calibrate through direct logical measurements. Recently, Wagner et al. [PRL 130, 200601 (2023)] showed that, for phenomenological Pauli noise models, the logical channel can instead be inferred from syndrome measurement data generated during error correction. Here, we extend this framework to realistic circuit-level noise models. From a unified code-theoretic perspective and spacetime code formalism, we derive necessary and sufficient conditions for learning the logical channel from syndrome data alone and explicitly characterize the learnable degrees of freedom of circuit-level Pauli faults. Using Fourier analysis and compressed sensing, we develop efficient estimators with provable guarantees on sample complexity and computational cost. We further present an end-to-end protocol and demonstrate its performance on several syndrome-extraction circuits, achieving orders-of-magnitude sample-complexity savings over direct logical benchmarking. Our results establish syndrome-based learning as a practical approach to characterizing the logical channel in fault-tolerant quantum devices.

Figures (5)

• Figure 1: Illustration of the learnability problem studied in this work. On the left, we give a schematic illustration of utilizing the syndrome data to infer the effective error probabilities. We provide the necessary and sufficient learnability condition in Theorem \ref{['thm: main-physical-learnability']} and Theorem \ref{['thm: main-logical-learnability']}. The efficient sampling and robust prediction procedure are given respectively from Theorem \ref{['thm: main-row-subsampled-rip-A']} and Theorem \ref{['thm: background-free-estimation-prior']}. Finally, the logical error probability conditioned on a given circuit-level decoder is computed through Eqs. \ref{['eq: main-logical-error-probability']} & \ref{['eq: main-prior-distribution-logical-error']}. The entire learning procedure is summarized in Algorithm \ref{['algo: end-2-end-framework-from-syndrome']}. On the right, we show that, under a circuit-level fault model with memory experiments, the required sample complexity in estimating the logical error probability within $10\%$ of relative precision. For the square-octagon $(4.8.8)$ color code bombin2007exact and rotated surface code with distance $7$, we demonstrate orders-of-magnitude savings in estimating the logical error probability to $10\%$ of relative precision.
• Figure 2: (a) Sample complexity (number of shots) for learning priors (here, for the $d=5$ surface code) across several physical error probabilities, where measurement errors and data errors are modeled by the single-qubit depolarizing channel of fixed error rate $p$. For fixed $p$, the sample size ($N$) required to reduce the maximum relative deviation between the true and predicted priors below a tolerance $\tau$ exhibits the expected scaling $N \propto \tau^{-2}$. (b) Fixing a target accuracy (here $\tau = 0.8$), the sample size scales approximately inversely with the error rate, $N \propto p^{-1}$ (fit shown in blue). (c) Sample complexity comparison at $p = 5\times 10^{-4}$ for achieving $10\%$ relative error: the predicted logical error probability requires $2$–$3$ orders of magnitude smaller sample size than the (naively) sampled logical error probability estimation for the rotated surface code at distances $d=3,5,7$.
• Figure 3: A schematic illustration of the repetition code with three rounds of syndrome extraction. As in Eq. \ref{['eq: rep-circuit-stabilizer-no-ancilla']}, there are 8 independent stabilizer generators given in backward propagation of $6$ mid-circuit stabilizers and $2$ output stabilizers. A schematic plot featuring ancillas for the stabilizer measurements is given in Figure \ref{['fig: rep-circuit-ancilla']}.
• Figure 4: The Bacon-Shor code defined on the $2 \times 2$ and $3 \times 3$ lattices. The stabilizers are given as weight-$6$ operators.
• Figure 5: The syndrome extraction circuit that implements the $3$-qubit repetition code. Note that the stabilizer measurement fault at $\eta_2(X_4)$ can be gauge-related to $\eta_3(X_1) \eta_0(X_1)$, which explains the modeling of measurement faults of the spacetime circuit without the ancilla.

Theorems & Definitions (179)

• Theorem 1: Learnability
• Theorem 2: Learnability up to logical equivalence
• Theorem 3: Informal
• Theorem 4: Informal
• Definition 3.1: Syndrome extraction circuit
• Definition 3.2: Forward and backward propagation
• Proposition 3.3: Corollary 1 and Proposition 3 in Ref. delfosse2023spacetimecodescliffordcircuits
• Theorem 5: Circuit-to-code isomorphism
• Remark
• Example : 3 rounds of syndrome extraction with repetition codes