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arXiv:2305.03004 · New Journal of Physics

Calibration of syndrome measurements in a single experiment

Christian L. Wimmer, Jochen Szangolies, Michael Epping

Published Nov 28, 2024• 1 citation
arXivHTMLPDF

Abstract

Abstract Quantum error correction can reduce the effects of noise in quantum systems, e.g. in metrology or most notably in quantum computing. Typically, this requires making measurements that provide information about the errors that have occurred in the system. However, these syndrome measurements themselves introduce noise into the system, for example by using noisy gates. A complete characterization of the measurements is very costly. Here we describe a calibration method to obtain the syndrome statistics taking into account the additional noise sources. All calibration data are extracted from a single experiment in which the syndrome measurement is performed twice in a row. Thus, our method allows an accurate evaluation of syndrome measurements with significantly less effort than existing methods. We give examples of the application of this method to noise estimation and error correction. Finally, we discuss the results of experiments performed on an IBM quantum computer.

DOI
10.1088/1367-2630/ad9828

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Abstract Quantum error correction can reduce the effects of noise in quantum systems, e.g. in metrology or most notably in quantum computing. Typically, this requires making measurements that provide information about the errors that have occurred in the system. However, these syndrome measurements themselves introduce noise into the system, for example by using noisy gates. A complete characterization of the measurements is very costly. Here we describe a calibration method to obtain the syndrome statistics taking into account the additional noise sources. All calibration data are extracted from a single experiment in which the syndrome measurement is performed twice in a row. Thus, our method allows an accurate evaluation of syndrome measurements with significantly less effort than existing methods. We give examples of the application of this method to noise estimation and error correction. Finally, we discuss the results of experiments performed on an IBM quantum computer.
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.
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.
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.
Syndrome measurements made in quantum error correction contain more information than is typically used. We show that the statistics of data from syndrome measurements can be used to do the following: (i) estimation of parameters of an error channel, including the ability correct away the invertible part of the error channel, once it is estimated; (ii) hypothesis testing (or model selection) to distinguish error channels, e.g., to determine if the errors are correlated. The unifying theme is to make use of all of the information in the statistics of the data collected from syndrome measurements using machine learning and control algorithms.
Quantum error correction (QEC) enables fault-tolerant quantum computation but requires operating quantum hardware at physical error rates below code-dependent thresholds, which remains challenging for current devices. We introduce syndrome resampling, a general method that increases QEC thresholds of any decoder and suppresses logical errors without additional hardware, decoding modifications, or code-specific assumptions beyond syndrome statistics. The method exploits the fact that syndromes with low probability are likely to lead to logical failure, therefore biasing syndrome averages towards most likely syndromes effectively increases logical fidelities. We establish a direct connection between the Rényi coherent information (RCI) and powers of the syndrome probability distribution, showing that resampling syndromes according to these powers combined with maximum likelihood decoding (MLD) realizes a family of optimal thresholds associated with phase transitions in the RCI. Numerical simulations of surface codes demonstrate that syndrome resampling substantially increases thresholds for both optimal and suboptimal decoders and reduces logical error rates by up to four orders of magnitude in experimentally relevant regimes. We further show that syndrome resampling can be effectively implemented from finite data and combined with decoding-based post-selection to achieve additional gains. Finally, applying the method to existing experimental QEC data yields up to two orders of magnitude reduction in logical error rates without requiring additional measurements. Our results provide a practical and decoder-agnostic route to improved logical fidelities in near-term QEC experiments.
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.