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arXiv:2311.16044 · IEEE International Symposium on Information Theory (ISIT)

Robust Syndrome Extraction via BCH Encoding

Eren Guttentag, Andrew Nemec, Kenneth R. Brown

Published Jul 7, 2024• 2 citations
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Abstract

Quantum data-syndrome (QDS) codes are a class of quantum error-correcting codes that protect against errors both on the data qubits and on the syndrome itself via redundant measurement of stabilizer group elements. One way to define a QDS code is to choose a syndrome measurement code, a classical block code that encodes the syndrome of the underlying quantum code by defining additional stabilizer measurements. We propose the use of primitive narrow-sense BCH codes as syndrome measurement codes. We show that these codes asymptotically require$O(t\log \ell)$extra measurements, where$\ell$is the number of stabilizer generators of the quantum code and$t$is the number of syndrome measurement errors corrected by the BCH code. Previously, the best known general method of constructing QDS codes out of quantum codes required$O(t^{3}\log \ell)$extra measurements. As the number of additional syndrome measurements is a reasonable metric for the amount of additional time a general QDS code requires, we conclude that our construction protects against the same number of syndrome errors with significantly less time overhead.

DOI
10.1109/isit57864.2024.10619479

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Quantum data-syndrome (QDS) codes are a class of quantum error-correcting codes that protect against errors both on the data qubits and on the syndrome itself via redundant measurement of stabilizer group elements. One way to define a QDS code is to choose a syndrome measurement code, a classical block code that encodes the syndrome of the underlying quantum code by defining additional stabilizer measurements. We propose the use of primitive narrow-sense BCH codes as syndrome measurement codes. We show that these codes asymptotically requireO(tlog⁡ℓ)O(t\log \ell)O(tlogℓ)extra measurements, whereℓ\ellℓis the number of stabilizer generators of the quantum code andtttis the number of syndrome measurement errors corrected by the BCH code. Previously, the best known general method of constructing QDS codes out of quantum codes requiredO(t3log⁡ℓ)O(t^{3}\log \ell)O(t3logℓ)extra measurements. As the number of additional syndrome measurements is a reasonable metric for the amount of additional time a general QDS code requires, we conclude that our construction protects against the same number of syndrome errors with significantly less time overhead.
Performing active quantum error correction to protect fragile quantum states highly depends on the correctness of measured error syndromes. To obtain reliable error syndromes using imperfect physical circuits, we propose syndrome measurement (SM) and quantum data-syndrome (DS) codes. SM codes protect syndrome with linearly dependent redundant stabilizer measurements. DS codes generalize this idea for simultaneous correction of both data qubits and syndrome bits errors. We study fundamental properties of quantum DS codes, including split weight enumerators, generalized MacWilliams identities, and linear programming bounds. In particular, we derive Singleton and Hamming-type upper bounds on the minimum distance of degenerate quantum DS codes. Then we study random DS codes and show that random DS codes with a relatively small additional syndrome measurements achieve the Gilbert-Varshamov bound of stabilizer codes. Finally, we propose a family of CSS-type quantum DS codes based on classical cyclic codes, which include the Steane code and the quantum Golay code.
Quantum stabilizer codes often struggle with syndrome errors due to measurement imperfections. Typically, multiple rounds of syndrome extraction are employed to ensure reliable error information. In this paper, we consider phenomenological decoding problems, where data qubit errors may occur between extractions, and each measurement can be faulty. We introduce generalized quantum data-syndrome codes along with a generalized check matrix that integrates both quaternary and binary alphabets to represent diverse error sources. This results in a Tanner graph with mixed variable nodes, enabling the design of belief propagation (BP) decoding algorithms that effectively handle phenomenological errors. Importantly, our BP decoders are applicable to general sparse quantum codes. Through simulations, we achieve an error threshold of more than 3% for quantum memory protected by rotated toric codes, using solely BP without post-processing. Our results indicate that d rounds of syndrome extraction are sufficient for a toric code of distance d. We observe that at high error rates, fewer rounds of syndrome extraction tend to perform better, while more rounds improve performance at lower error rates. Additionally, we propose a method to construct effective redundant stabilizer checks for single-shot error correction. Our simulations show that BP decoding remains highly effective even with a high syndrome error rate.
Stabilizer codes are a fundamental class of error-correcting codes for quantum information that allow for syndrome decoding in the quantum domain. One of the substantial challenges in quantum error correction is that the quantum gates that perform error correction themselves are faulty in practice, which makes fault-tolerant implementation a vital component. Recently, a coding theoretic technique is proposed that makes it possible for syndrome decoding to help correct imperfectly extracted syndromes instead of fully relying on an external fault-tolerant mechanism. In particular, it was proved that any single-error-correcting stabilizer code can be made robust against single errors on either a data qubit or a syndrome bit by using at most one more stabilizer operator than necessary for standard syndrome decoding, while analogous overhead for making double-error-correcting stabilizer codes robust against double errors involving data qubits and/or syndrome bits was shown to be at most logarithmic. We generalize this result to t-error-correcting stabilizer codes and show that the overhead for achieving analogously defined global t-error correction for data qubits and/or syndrome bits is also at most logarithmic. The proof exploits combinatorial arrays that may be seen as parity-check matrices that detect errors of even weight but may overlook errors of odd weight.
Quantum error correction is necessary to protect logical quantum states and operations. However, no meaningful data protection can be made when the syndrome extraction is erroneous due to faulty measurement gates. Quantum data-syndrome (DS) codes are designed to protect the data qubits and syndrome bits concurrently. In this paper, we propose an efficient decoding algorithm for quantum DS codes with sparse check matrices. Based on a refined belief propagation (BP) decoding for stabilizer codes, we propose a DS-BP algorithm to handle the quaternary quantum data errors and binary syndrome bit errors. Moreover, a sparse quantum code may inherently be able to handle minor syndrome errors so that fewer redundant syndrome measurements are necessary. We demonstrate this with simulations on a quantum hypergraph-product code.
Quantum error-correcting codes are a vital technology for demonstrating reliable quantum computation. They require data qubits for encoding quantum information and ancillary qubits for taking error syndromes necessary for error correction. The need for a large number of ancillary qubits is an overhead specific to quantum computing, and it prevents the scaling of quantum computers to a useful size. In this work, we propose a framework for generating efficient syndrome measurement circuits with a few ancillary qubits in CSS codes and provide a method to minimize the total number of physical qubits in general settings. We demonstrated our proposal by applying it to surface codes, and we generated syndrome measurement circuits under several constraints of total qubit count. As a result, we find that balanced data and ancillary qubit counts achieve the lower logical error rates under a fixed total number of physical qubits. This result indicates that using fewer ancillary qubits than the number of stabilizers can be effective for reducing logical error rates in a practical noise model.
Code-word-stabilized (CWS) codes are, in general, nonadditive quantum codes that can correct errors by an exhaustive search of different error patterns, similar to the way that we decode classical nonlinear codes. For an nnn-qubit quantum code correcting errors on up to ttt qubits, this brute-force approach consecutively tests different errors of weight ttt or less and employs a separate nnn-qubit measurement in each test. In this article, we suggest an error grouping technique that allows one to simultaneously test large groups of errors in a single measurement. This structured error recovery technique exponentially reduces the number of measurements by about 3t{3}^{t}3t times. While it still leaves exponentially many measurements for a generic CWS code, the technique is equivalent to syndrome-based recovery for the special case of additive CWS codes.