Quantum error correction plays a prominent role in the realization of quantum computation, and quantum low-density parity-check (qLDPC) codes are believed to be practically useful stabilizer codes. While qLDPC codes are defined to have constant weight parity-checks, the weight of these parity checks could be large constants that make implementing these codes challenging. Large constants can also result in long syndrome extraction times and bad error propagation that can impact error correction performance. Hastings recently introduced weight reduction techniques for qLDPC codes that reduce the weight of the parity checks as well as the maximum number of checks that acts on any data qubit. However, the fault tolerance of these techniques remains an open question. In this paper, we analyze the effective distance of the weight-reduced code when single-ancilla syndrome extraction circuits are considered for error correction. We prove that there exists single-ancilla syndrome extraction circuits that largely preserve the effective distance of the weight-reduced qLDPC codes. In addition, we also show that the distance balancing technique introduced by Evra et al. \cite{evra2022decodable} preserves effective distance. As a corollary, our result shows that higher-dimensional hypergraph product (HGP) codes, also known as homological product codes corresponding to the product of 1-complexes, have no troublesome hook errors when using any single-ancilla syndrome extraction circuit.
Quantum error correction plays a prominent role in the realization of quantum computation, and quantum low-density parity-check (qLDPC) codes are believed to be practically useful stabilizer codes. While qLDPC codes are defined to have constant weight parity-checks, the weight of these parity checks could be large constants that make implementing these codes challenging. Large constants can also result in long syndrome extraction times and bad error propagation that can impact error correction performance. Hastings recently introduced weight reduction techniques for qLDPC codes that reduce the weight of the parity checks as well as the maximum number of checks that acts on any data qubit. However, the fault tolerance of these techniques remains an open question. In this paper, we analyze the effective distance of the weight-reduced code when single-ancilla syndrome extraction circuits are considered for error correction. We prove that there exists single-ancilla syndrome extraction circuits that largely preserve the effective distance of the weight-reduced qLDPC codes. In addition, we also show that the distance balancing technique introduced by Evra et al. \cite{evra2022decodable} preserves effective distance. As a corollary, our result shows that higher-dimensional hypergraph product (HGP) codes, also known as homological product codes corresponding to the product of 1-complexes, have no troublesome hook errors when using any single-ancilla syndrome extraction circuit.
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.
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.
Unlike the surface code, quantum low-density parity-check (QLDPC) codes can have a finite encoding rate, potentially lowering the error correction overhead. However, finite-rate QLDPC codes have nonlocal stabilizers, making it difficult to design stabilizer measurement circuits that are low-depth and do not decrease the effective distance. Here, we demonstrate that a popular family of finite-rate QLDPC codes, hypergraph product codes, has the convenient property of distance-robustness: any stabilizer measurement circuit preserves the effective distance. In particular, we prove the depth-optimal circuit in [Tremblay et al, PRL 129, 050504 (2022)] is also optimal in terms of effective distance.
We design low-complexity, fault-tolerant decoders for quantum low-density parity-check (QLDPC) codes with the goal of reducing decoding latency. We target two major bottlenecks of decoding under the \emph{circuit-level} noise model: (i) post-processing via order-statistics decoding (OSD), and (ii) the large number of auxiliary variable nodes commonly introduced to represent CNOT-induced correlations during syndrome extraction. Our key observation is that propagating CNOT faults (\emph{hook errors}) create \emph{stabilizer-induced} trapping sets (TSs) that are intrinsic to hypergraph-product (HGP) and lifted-product (LP) constructions. Therefore, instead of modeling each such fault with an explicit correlation node and relying on OSD to clean up the resulting failures, we design message-passing decoders that resolve the corresponding \emph{stabilizer-induced} TSs directly. We obtain these decoders by deriving QLDPC decoders from decoders for the parent classical LDPC codes and using them collectively to correct broad families of \emph{stabilizer-induced} TSs. For CNOT faults that manifest primarily as syndrome errors, we show that their effect is equivalent to a data error together with syndrome-bit measurement errors. Consequently, given repeated measurements and a decoding graph that already includes nodes representing syndrome-bit errors, no distinct variable node is needed for each CNOT fault. Using a \emph{phenomenological} Tanner graph with nodes representing only data errors and syndrome-bit errors, simulations on LP codes show a significant improvement in logical error rate relative to BP+OSD. Moreover, the improved error correction relies on low-complexity finite-state iterative decoders, targeted for fast and low-latency FPGA implementation.
Quantum low-density parity-check (QLDPC) codes with asymptotically non-zero rates are prominent candidates for achieving fault-tolerant quantum computation, primarily due to the low operational depth of their syndrome-measurement circuits. Numerous studies advocate the necessity of fast decoders to fully harness the capabilities of QLDPC codes, thus driving the focus towards designing low-complexity iterative decoders. However, empirical investigations indicate that such iterative decoders are susceptible to having a high error floor when decoding QLDPC codes. The main objective of this paper is to analyze the decoding failures of thehypergraph-product(HGP) andlifted-product(LP) codes and to design decoders that mitigate these failures, thus achieving a reduced error floor. The suboptimal performance of these codes can predominantly be ascribed to two structural phenomena: (1) stabilizer-induced trapping sets (TS), which correspond to stabilizer-induced subgraphs in the Tanner graphs, and (2) classical trapping sets (TS), which originate from the classical codes used in the construction of HGP and LP codes. The dynamics of stabilizer-induced TSs are examined, and a straightforward modification of iterative decoders is proposed to circumvent these TSs. Moreover, this work proposes a systematic methodology for designing decoders that can circumvent the classical TSs in both HGP and LP codes by deriving them from decoders capable of avoiding the TSs in the parent classical LDPC codes. When decoders that can avoid stabilizer-induced TSs are run in parallel with those that can mitigate the effect of classical TSs, the logical error rate improves significantly in the error-floor region.
Quantum error correction is necessary to perform large-scale quantum computation but requires extremely large overheads in both space and time. High-rate quantum low-density-parity-check (qLDPC) codes promise a route to reduce qubit numbers, but performing computation while maintaining low space cost has required serialization of operations and extra time costs. In this work, we design fast and parallelizable logical gates for qLDPC codes and demonstrate their utility for key algorithmic subroutines such as the quantum adder. Our gate gadgets utilize transversal logical s between a data qLDPC code and a suitably constructed ancilla code to perform parallel Pauli product measurements (PPMs) on the data logical qubits. For hypergraph product codes, we show that the ancilla can be constructed by simply modifying the base classical codes of the data code, achieving parallel PPMs on a subgrid of the logical qubits with a lower space-time cost than existing schemes for an important class of circuits. Generalizations to 3D and 4D homological product codes further feature fast PPMs in constant depth. While prior work on qLDPC codes has focused on individual logical gates, we initiate the study of fault-tolerant compilation with our expanded set of native qLDPC code operations, constructing algorithmic primitives for preparing k -qubit Greenberger-Horne-Zeilinger states and distilling or teleporting k magic states with O ( 1 ) space overhead in O ( 1 ) and O ( k log k ) logical cycles, respectively. We further generalize this to key algorithmic subroutines, demonstrating the efficient implementation of quantum adders using parallel operations. Our constructions are naturally compatible with reconfigurable architectures such as neutral atom arrays, paving the way to large-scale quantum computation with low space and time overheads.