Bias-tailoring allows quantum error correction codes to exploit qubit noise asymmetry. Recently, it was shown that a modified form of the surface code, the XZZX code, exhibits considerably improved performance under biased noise. In this work, we demonstrate that quantum low density parity check codes can be similarly bias-tailored. We introduce a bias-tailored lifted product code construction that provides the framework to expand bias-tailoring methods beyond the family of 2D topological codes. We present examples of bias-tailored lifted product codes based on classical quasi-cyclic codes and numerically assess their performance using a belief propagation plus ordered statistics decoder. Our Monte Carlo simulations, performed under asymmetric noise, show that bias-tailored codes achieve several orders of magnitude improvement in their error suppression relative to depolarising noise.
Bias-tailoring allows quantum error correction codes to exploit qubit noise asymmetry. Recently, it was shown that a modified form of the surface code, the XZZX code, exhibits considerably improved performance under biased noise. In this work, we demonstrate that quantum low density parity check codes can be similarly bias-tailored. We introduce a bias-tailored lifted product code construction that provides the framework to expand bias-tailoring methods beyond the family of 2D topological codes. We present examples of bias-tailored lifted product codes based on classical quasi-cyclic codes and numerically assess their performance using a belief propagation plus ordered statistics decoder. Our Monte Carlo simulations, performed under asymmetric noise, show that bias-tailored codes achieve several orders of magnitude improvement in their error suppression relative to depolarising noise.
Abstract To improve the decoding performance of quantum error-correcting codes in asymmetric noise channels, a neural network-based decoding algorithm for bias-tailored quantum codes is proposed. The algorithm consists of a biased noise model, a neural belief propagation decoder, a convolutional optimization layer, and a multi-objective loss function. The biased noise model simulates asymmetric error generation, providing a training dataset for decoding. The neural network, leveraging dynamic weight learning and a multi-objective loss function, mitigates error degeneracy. Additionally, the convolutional optimization layer enhances early-stage convergence efficiency. Numerical results show that for bias-tailored quantum codes, our decoder performs much better than the belief propagation (BP) with ordered statistics decoding (BP + OSD). Our decoder achieves an order of magnitude improvement in the error suppression compared to higher-order BP + OSD. Furthermore, the decoding threshold of our decoder for surface codes reaches a high threshold of 20%.
Quantum error correction (QEC) for generic errors is challenging due to the demanding threshold and resource requirements. Interestingly, when physical noise is biased, we can tailor our QEC schemes to the noise to improve performance. Here we study a family of codes having XZZX-type stabilizer generators, including a set of cyclic codes generalized from the five-qubit code and a set of topological codes that we call generalized toric codes (GTCs). We show that these XZZX codes are highly qubit efficient if tailored to biased noise. To characterize the code performance, we use the notion of effective distance, which generalizes code distance to the case of biased noise and constitutes a proxy for the logical failure rate. We find that the XZZX codes can achieve a favorable resource scaling by this metric under biased noise. We also show that the XZZX codes have remarkably high thresholds that reach what is achievable by random codes, and furthermore they can be efficiently decoded using matching decoders. Finally, by adding only one flag qubit, the XZZX codes can realize fault-tolerant QEC while preserving their large effective distance. In combination, our results show that tailored XZZX codes give a resource-efficient scheme for fault-tolerant QEC against biased noise.
We find that the substantial advantages predicted for bias-tailored quantum error correction (QEC) under code-capacity noise are strongly reduced once realistic syndrome extraction and circuit-level noise models are considered. We start by comparing XZZX codes to rectangular surface codes with a bias-dependent optimised anisotropy. Although code-capacity simulations predict an advantage of rectangular surface codes in the limit of high noise bias, this actually disappears under circuit-level noise, making the XZZX codes the preferred and simplest choice even for platforms that allow for a flexible variation of the code layout adapted to changes in noise calibration. Our results identify bias degradation during syndrome extraction under circuit-level noise as the central limitation of biased-tailored QEC. To partially mitigate this effect, we introduce a bias-filtering CNOT gadget that temporarily encodes the ancillary target qubit during syndrome extraction in a repetition code and, upon measurement and feed forward, manages to reduce the bias degradation. In a regime of high-bias and low-idle errors, this bias-filtering gadget yields a few-percent relative improvement of the XZZX code error threshold, demonstrating that lightweight bias-filtering strategies can recover part of the lost bias-tailoring advantage for realistic circuit-level noise.
Quantum computing is one of the most promising technological revolutions of the last few decades. Despite its potential, one problem it suffers from is that quantum bits (or qubits), its basic units of information, are intrinsically susceptible to noise. To counter the effects of such noise, we need fault-tolerant quantum computing, and for that, we need a systematic method to protect quantum information. This is the role of quantum error correction (QEC). However, most QEC codes are built on the assumption that X- and Z-type errors occur with equal frequency, which is not always the case for some realistic quantum hardware platforms. Among the several QEC protocols currently developed, surface codes are the leading implementation due to their high thresholds and local connectivity. Previous results have shown that modified variants of these codes, such as the XZZX surface code, can perform well under biased noise. Despite being realizable in two-dimensional real architectures, surface codes have poor encoding rates, leading to large overheads that prevent them from being scalable in fault-tolerant quantum architectures. Quantum low-density parity-check (qLDPC) codes provide a promising alternative, since they can achieve higher encoding rates. In this thesis, we study the code-capacity performance of bias-tailored qLDPC codes under both X- and Z-bias Pauli noise. We construct bias-tailored qLDPC codes, including bivariate bicycle codes, generalized toric codes, and clustered-cyclic codes, by applying Hadamard deformations that transform the Calderbank-Shor-Steane (CSS) codes into non-CSS codes. Using Monte Carlo simulations with minimum-weight perfect matching and belief propagation with localized-statistics decoding, we evaluate the word error rate of these bias-tailored qLDPC codes across different bias regimes. We show that Hadamard-deformed qLDPC codes can yield substantial performance improvements under biased noise and can match or even surpass those performances obtained from XZZX surface code implementations. In particular, we find that generalized toric codes perform consistently the best under bias noise in comparison to the bivariate and clustered-cyclic codes. These results indicate that bias tailoring can be extended to high-rate qLDPC constructions, leading to future work to study their performance in more realistic settings, such as circuit-level noise.
Quantum low-density parity-check (qLDPC) codes are an important component in the quest for quantum fault tolerance. Dramatic recent progress on qLDPC codes has led to constructions which are asymptotically good, and which admit linear-time decoders to correct errors affecting a constant fraction of codeword qubits. These constructions, while theoretically explicit, rely on inner codes with strong properties only shown to exist by probabilistic arguments, resulting in lengths that are too large to be practically relevant. In practice, the surface/toric codes, which are the product of two repetition codes, are still often the qLDPC codes of choice. A previous construction of qLDPC codes based on the lifted product of an expander-based classical LDPC code with a repetition code (Panteleev and Kalachev, 2020) achieved a near-linear distance, and avoids the need for such intractable inner codes. Our main result is an efficient decoding algorithm for these codes that corrects a near-linear number of adversarial errors. En route, we give a similar algorithm for the hypergraph product version these codes, which are simpler but have distance growing only as the square root of the block length. Our decoding algorithms leverage the fact that the codes we consider are quasi-cyclic, meaning that they respect a cyclic group symmetry. Since the repetition code is not based on expanders, previous approaches to decoding expander-based qLDPC codes, which typically worked by greedily flipping code bits to reduce some potential function, do not apply in our setting. Instead, we reduce our decoding problem (in a black-box manner) to that of decoding classical expander-based LDPC codes under noisy parity-check syndromes. For completeness, we also include a treatment of such classical noisy-syndrome decoding that is sufficient for our application to the quantum setting.
Remarkable technical progress has made high-rate, high-distance, quantum low-density parity-check codes (QLDPC) promising candidates for scalable quantum computing. However, it is hard to design low-depth syndrome extraction circuits that do not spread errors from ancilla qubits to multiple data qubits, also known as hook errors, for general QLDPC codes. Additionally, widely used decoders for these codes based on belief propagation are impaired due to short loops in the Tanner graph. Here, we investigate a hardware-aware approach to avoid these hooks and loops using biased noise ancillas. Using examples of bicycle bivariate codes and a cyclic hypergraph product code, which have been widely considered for practical application, we show that the effective fault-distance of the conventional syndrome extraction circuit can be significantly higher and the number of short loops can be significantly lower when the ancillas are subject to phase-flip errors only, compared to when they are also subject to bit-flip errors. This can result in almost an order of magnitude improvement in the logical error rate at circuit noise of 2×10−3 and when bit-flip errors in the ancilla are 50 times less likely than phase-flip errors. Our work demonstrates a significant and practical quantum error correction advantage with biased noise qubits in which full-bias cannot be maintained.