Quantum error correction allows to actively correct errors occurring in a quantum computation when the noise is weak enough.To make this error correction competitive information about the specific noise is required.Traditionally, this information is obtained by benchmarking the device before operation.We address the question of what can be learned from only the measurements done during decoding.Such estimation of noise models was proposed for surface codes, exploiting their special structure, and in the limit of low error rates, also for other codes.However, so far it has been unclear under what general conditions noise models can be estimated from the syndrome measurements.In this work, we derive a general condition for identifiability of the error rates.For general stabilizer codes, we prove identifiability under the assumption that the rates are small enough.Without this assumption, we prove a result for perfect codes.Finally, we propose a practical estimation method with linear runtime for concatenated codes.We demonstrate that it outperforms other recently proposed methods and that the estimation is optimal in the sense that it reaches the Cramér-Rao bound.Our method paves the way for practical calibration of error corrected quantum devices during operation.
Quantum error correction allows to actively correct errors occurring in a quantum computation when the noise is weak enough.To make this error correction competitive information about the specific noise is required.Traditionally, this information is obtained by benchmarking the device before operation.We address the question of what can be learned from only the measurements done during decoding.Such estimation of noise models was proposed for surface codes, exploiting their special structure, and in the limit of low error rates, also for other codes.However, so far it has been unclear under what general conditions noise models can be estimated from the syndrome measurements.In this work, we derive a general condition for identifiability of the error rates.For general stabilizer codes, we prove identifiability under the assumption that the rates are small enough.Without this assumption, we prove a result for perfect codes.Finally, we propose a practical estimation method with linear runtime for concatenated codes.We demonstrate that it outperforms other recently proposed methods and that the estimation is optimal in the sense that it reaches the Cramér-Rao bound.Our method paves the way for practical calibration of error corrected quantum devices during operation.
The performance of quantum error correction can be significantly improved if detailed information about the noise is available, allowing to optimize both codes and decoders. It has been proposed to estimate error rates from the syndrome measurements done anyway during quantum error correction. While these measurements preserve the encoded quantum state, it is currently not clear how much information about the noise can be extracted in this way. So far, apart from the limit of vanishing error rates, rigorous results have only been established for some specific codes. In this work, we rigorously resolve the question for arbitrary stabilizer codes. The main result is that a stabilizer code can be used to estimate Pauli channels with correlations across a number of qubits given by the pure distance. This result does not rely on the limit of vanishing error rates, and applies even if high weight errors occur frequently. Moreover, it also allows for measurement errors within the framework of quantum data-syndrome codes. Our proof combines Boolean Fourier analysis, combinatorics and elementary algebraic geometry. It is our hope that this work opens up interesting applications, such as the online adaptation of a decoder to time-varying noise.
Quantum computers require error correction to achieve universal quantum computing. However, current decoding of quantum error-correcting codes relies on classical computation, which is slower than quantum operations in superconducting qubits. This discrepancy makes the practical implementation of real-time quantum error correction challenging. In this work, we propose a decoding scheme that leverages the operations of the quantum circuit itself. Given a noisy quantum circuit A, we train a decoding quantum circuit B using syndrome measurements to identify the logical operators needed to correct errors in circuit A. The trained quantum circuit B can be deployed on quantum devices, such as superconducting qubits, to perform real-time decoding and error correction. Our approach is applicable to general quantum codes with multiple logical qubits and operates efficiently under various noise conditions, and the decoding speed matches the speed of the quantum circuits being corrected. We have conducted numerical experiments using surface codes up to distance 7 under circuit-level noise, demonstrating performance on par with the classical minimum-weight perfect matching algorithm. Interestingly, our method reveals that the traditionally classical task of decoding error-correcting codes can be accomplished without classical devices or measurements. This insight paves the way for the development of self-correcting quantum computers.
What must a quantum error-correction decoder know about temporally correlated noise, and what does it cost to learn? We study this question for fixed stabilizer memory experiments with known system-environment interactions. We formulate calibration in terms of the fidelity differences that select a Pauli recovery and relate missing information to the resulting fidelity loss. Our main example uses weak coherent X rotations in a class of CSS codes including Shor, Steane, and odd-distance rotated surface codes. All physical qubits share a sign either fixed throughout a run or independently redrawn each interval. The two models have identical syndrome statistics and the same spatiotemporal Pauli process (SPP), obtained by Pauli-twirling every noise interval. Yet at suitable storage times, they require different logical corrections. In the weak-noise limit, each model is almost perfectly correctable when known. Without model information, even the best final recovery has a worst-case fidelity loss approaching 1/2 relative to the noise-aware optimum, despite access to the full syndrome record. For this pair of models, at fixed code and fixed positive target loss below 1/2, we prove matching calibration bounds. With logical preparation and final readout, complete QEC cycles require a number of noise intervals scaling as the inverse rotation angle raised to the code's X distance. Allowing two uninterrupted intervals before one extraction reduces the optimal cost to inverse-square scaling. Physical Pauli twirling of every noise interval changes both recovery performance and its information requirements. Discarding the record of applied random Pauli operations makes the SPP sufficient for optimal recovery. Keeping this record can improve the attainable fidelity, but reaching that optimum may require additional noise information.
The characterization of quantum devices is crucial for their practical implementation but can be costly in experimental effort and classical postprocessing. Therefore, it is desirable to measure only the information that is relevant for specific applications and develop protocols that require little additional effort. In this Letter, we focus on the characterization of quantum computers in the context of stabilizer quantum error correction. For arbitrary stabilizer codes, subsystem codes, and data syndrome codes, we prove that the logical error channel induced by Pauli noise can be estimated from syndrome data under minimal conditions. More precisely, for any such code, we show that the estimation is possible as long as the code can correct the noise.
Building error-corrected quantum computers relies crucially on measuring and modeling noise on candidate devices.In particular, optimal error correction requires knowing the noise that occurs in the device as it executes the circuits required for error correction.As devices increase in size, we will become more reliant on efficient models of this noise.However, such models must still retain the information required to optimize the algorithms used for error correction.Here, we propose a method of extracting detailed information of the noise in a device running syndrome extraction circuits.We introduce and execute an experiment on a superconducting device using 39 of its qubits in a surface code doing repeated rounds of syndrome extraction but omitting the midcircuit measurement and reset.We show how to extract from the 20 data qubits the information needed to build noise models of various sophistication in the form of graphical models.These models give efficient descriptions of noise in large-scale devices and are designed to illuminate the effectiveness of error correction against correlated noise.Our estimates are furthermore precise: we learn a consistent global distribution where all one-and two-qubit error rates are known to a relative error of 0.1%.By extrapolating our experimentally learned noise models toward lower error rates, we demonstrate that accurate correlated noise models are increasingly important for successfully predicting subthreshold behavior in quantum error-correction experiments.
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.