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International Conference on Quantum Communications, Networking, and Computing (QCNC)

BiBiEQ: Bivariate Bicycle Codes on Erasure Qubits

Ameya Bhave, Navnil Choudhury, Andrew Nemec, Kanad Basu

Published Apr 6, 2026• 0 citations
Publisher

Abstract

Erasure qubits reduce overhead in fault-tolerant quantum error correction (QEC) by converting dominant faults into detectable errors known as erasures. They have demonstrated notable improvements in thresholds and scaling in surface and Floquet code memories. In this work, we use erasure qubits on Bivariate Bicycle (BB) codes from the quantum low-density parity-check (QLDPC) regime. Owing to their sparse structure and favorable rate-distance trade-offs, BB codes are practical candidates for QEC. We introduce BiBiEQ, a novel framework that compiles a given BB code into an erasure-aware memory circuit$C_{E}$. This erasure circuit$C_{E}$comprises erasure checks (ECs), resets, and erasures spread over a user-specified erasure check schedule (2EC, 4EC). BiBiEQ converts this erasure circuit$C_{E}$into the stabilizer circuit$C$for general-purpose decoding. BiBiEQ provides two engines for this conversion, BiBiEQ-Exact and BiBiEQ-Approx. BiBiEQ-Exact preserves the joint-erasure correlations and serves as our accuracy benchmark, while BiBiEQ-Approx uses an independence approximation to accelerate large sweeps and expose accuracy-throughput tradeoffs. Using BiBiEQ, we decode the stabilizer circuits to get a per-round logical error rate (LER) for the BB codes and quantify the effect of the EC schedules on the correctable operating region below the pseudo-threshold. The$4 E C$schedule keeps the accuracy of both engines close to one another, making BiBiEQ-Approx a reliable proxy for BiBiEQ-Exact for faster sweeps. Below the pseudo-threshold, the code distance ($d$) hop from$d: 6 \rightarrow 10$yields a drop in LER by$\approx 10-17 \times$larger than$d: 10 \rightarrow 12$, showing that most gains are realized by$d=10$.

DOI
10.1109/qcnc69040.2026.00101

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quant-pharXiv:2609.06572•Sep 8, 2026•0 citations
Erasure qubits reduce overhead in fault-tolerant quantum error correction (QEC) by converting dominant faults into detectable errors known as erasures. They have demonstrated notable improvements in thresholds and scaling in surface and Floquet code memories. In this work, we use erasure qubits on Bivariate Bicycle (BB) codes from the quantum low-density parity-check (QLDPC) regime. Owing to their sparse structure and favorable rate-distance trade-offs, BB codes are practical candidates for QEC. We introduce BiBiEQ, a novel framework that compiles a given BB code into an erasure-aware memory circuitCEC_{E}CE​. This erasure circuitCEC_{E}CE​comprises erasure checks (ECs), resets, and erasures spread over a user-specified erasure check schedule (2EC, 4EC). BiBiEQ converts this erasure circuitCEC_{E}CE​into the stabilizer circuitCCCfor general-purpose decoding. BiBiEQ provides two engines for this conversion, BiBiEQ-Exact and BiBiEQ-Approx. BiBiEQ-Exact preserves the joint-erasure correlations and serves as our accuracy benchmark, while BiBiEQ-Approx uses an independence approximation to accelerate large sweeps and expose accuracy-throughput tradeoffs. Using BiBiEQ, we decode the stabilizer circuits to get a per-round logical error rate (LER) for the BB codes and quantify the effect of the EC schedules on the correctable operating region below the pseudo-threshold. The4EC4 E C4ECschedule keeps the accuracy of both engines close to one another, making BiBiEQ-Approx a reliable proxy for BiBiEQ-Exact for faster sweeps. Below the pseudo-threshold, the code distance (ddd) hop fromd:6→10d: 6 \rightarrow 10d:6→10yields a drop in LER by≈10−17×\approx 10-17 \times≈10−17×larger thand:10→12d: 10 \rightarrow 12d:10→12, showing that most gains are realized byd=10d=10d=10.
Erasure qubits reduce overhead in fault-tolerant quantum error correction (QEC) by converting dominant faults into detectable errors known as erasures. They have demonstrated notable improvements in thresholds and scaling in surface and Floquet code memories. In this work, we use erasure qubits on Bivariate Bicycle (BB) codes from the quantum low-density parity-check (QLDPC) regime. Owing to their sparse structure and favorable rate-distance trade-offs, BB codes are practical candidates for QEC. We introduce BiBiEQ, a novel framework that compiles a given BB code into an erasure-aware memory circuit C_E. This erasure circuit C_E comprises erasure checks (ECs), resets, and erasures spread over a user-specified erasure check schedule (2EC, 4EC). BiBiEQ converts this erasure circuit C_E into the stabilizer circuit C for general-purpose decoding. BiBiEQ provides two engines for this conversion, BiBiEQ-Exact and BiBiEQ-Approx. BiBiEQ-Exact preserves the joint-erasure correlations and serves as our accuracy benchmark, while BiBiEQ-Approx uses an independence approximation to accelerate large sweeps and expose accuracy-throughput trade-offs. Using BiBiEQ, we decode the stabilizer circuits to get a per-round logical error rate (LER) for the BB codes and quantify the effect of the EC schedules on the correctable operating region below the pseudo-threshold. The 4EC schedule keeps the accuracy of both engines close to one another, making BiBiEQ-Approx a reliable proxy for BiBiEQ-Exact for faster sweeps. Below the pseudo-threshold, the code distance (d) hop from distance (d) 6 to 10 yields a drop in LER by 10-17x larger than distance (d) 10 to 12, showing that most gains are realized by d=10.
Fair threshold estimation for bivariate bicycle (BB) codes on the quantum erasure channel runs into two recurring problems: decoder-baseline unfairness and the conflation of finite-size pseudo-thresholds with true asymptotic thresholds. We run both uninformed and \emph{erasure-aware} minimum-weight perfect matching (MWPM) toric code baselines alongside BP-OSD decoding of BB codes. With standard depolarizing-weight MWPM and no erasure information, performance matches random guessing on the erasure channel in our tested regime -- so prior work that compares against this baseline is really comparing decoders, not codes. Using 200{,}000 shots per point and bootstrap confidence intervals, we sweep five BB code sizes from N=144N=144N=144 to N=1296N=1296N=1296. Pseudo-thresholds (WER = 0.10) run from p∗=0.370p^* = 0.370p∗=0.370 to 0.4710.4710.471; finite-size scaling (FSS) gives an asymptotic threshold p∞∗≈0.488p^*_\infty \approx 0.488p∞∗​≈0.488, within 2.4\% of the zero-rate limit and without maximum-likelihood decoding. On the fair baseline, BB at N=1296N=1296N=1296 has a modest edge in threshold over the toric code at twice the qubit count, and a 12×\times× lower normalized overhead -- the latter is where the practical advantage sits. All runs are reproducible from recorded seeds and package versions.
Quantum low density parity check (qLDPC) codes, particularly bivariate bicycle (BB) codes, achieve competitive fault tolerance thresholds while offering substantially higher encoding rates than planar surface codes. However, their intrinsically long-range stabilizer structure makes them difficult to implement on monolithic devices with nearest neighbor connectivity and limited qubit capacity. In this work, we study the realization of a BB code in a modular multiprocessor architecture, where quantum processors are interconnected through shared Bell pairs. We consider processors with all to all internal connectivity, which is feasible on trapped ion and neutral atom platforms, enabling flexible local gate execution while inter-processor (nonlocal) gates are mediated by shared entanglement. We describe a star network architecture that can realize this distributed setting. We partition the qubits of the [[144,12,12]] BB code across 4, 6, and 12 quantum processors and analyze the resulting logical error rates and pseudo-threshold performance under circuit level noise by varying the number of processors and a scaling factor that captures the additional noise associated with nonlocal operations. We use Monte Carlo simulations with BP+OSD decoding and extend the previously known BB code ansatz to the distributed setting. Our results provide architectural insight and design considerations for distributed BB codes in modular quantum computing architectures.
Identifying the best families of quantum error correction (QEC) codes for near-term experiments is key to enabling fault-tolerant quantum computing. Ideally, such codes should have low overhead in qubit number, high physical error thresholds, and moderate requirements on qubit connectivity to simplify experiments, while allowing for high logical error suppression. Quantum low-density parity-check (LDPC) codes have been recently shown to provide a path towards QEC with low qubit overhead and small logical error probabilities. Here, we demonstrate that when the dominant errors are erasures, as can be engineered in different quantum computing architectures, quantum LDPC codes additionally provide high thresholds and even stronger logical error suppression in parameter regimes that are accessible to current experiments. Using large-scale QEC numerical simulations, we benchmark the performance of two families of high-rate quantum LDPC codes, namely the so-called Clifford-deformed La-cross codes and the bivariate bicycle codes, under a noise model strongly biased towards erasure errors. Both codes outperform the surface code by offering up to orders of magnitude lower logical error probabilities. Interestingly, we find that this decrease in the logical error probability may not be accompanied by an increase in the code threshold, as different QEC codes benefit differently from large erasure fractions. While here we focus on neutral atom qubits, the results also hold for other quantum platforms, such as trapped ions and superconducting qubits, for which erasure conversion has been demonstrated.
Quantum low-density parity-check (qLDPC) codes are promising candidates for fault-tolerant quantum computation due to their high encoding rates and distances. However, implementing logical operations using qLDPC codes presents significant challenges. Previous research has demonstrated that self-dual qLDPC codes facilitate the implementation of transversal Clifford gates. Here we introduce a method for constructing self-dual qLDPC codes by stacking non-self-dual qLDPC codes. Leveraging this methodology, we develop double-chain bicycle codes, double-layer bivariate bicycle (BB) codes, double-layer twisted BB codes, and double-layer reflection codes, many of which exhibit favorable code parameters. Additionally, we conduct numerical calculations to assess the performance of these codes as quantum memory under the circuit-level noise model, revealing that the logical failure rate can be significantly reduced with high pseudo-thresholds.
Bivariate bicycle (BB) codes of Bravyi \emph{et al.}~\cite{Bravyi2024} are quantum low-density parity-check codes with weight-666 checks, exemplified by [[144,12,12]][[144,12,12]][[144,12,12]] with kd2/n=12kd^2/n=12kd2/n=12. We develop the algebraic structure theory of BB-type codes with weight-888 checks (weight-444 generator polynomials) and use it, together with an exactly validated search pipeline, to construct and certify new codes. We prove an exact dimension formula k=2dim⁡R/(A,B)k=2\dim R/(A,B)k=2dimR/(A,B) (forcing even kkk), a 4ℓm4\ell m4ℓm-element symmetry group on generator pairs, an X/ZX/ZX/Z distance equality dX=dZd_X=d_ZdX​=dZ​, and a family of subgroup-coset kernel vectors giving rigorous distance upper bounds and a design rule for high-distance constructions; all distances are computed exhaustively by a cross-validated bit-mask verifier. At n=144n=144n=144 the pipeline returns a census of 535353 codes whose strongest members surpass the BB benchmark: [[144,6,d≥15]][[144,6,d\ge 15]][[144,6,d≥15]] exceeds the benchmark distance 121212 (certified d≥15d\ge 15d≥15), [[144,10,12]][[144,10,12]][[144,10,12]] reaches it with weight-888 checks, and [[144,16,10]][[144,16,10]][[144,16,10]] encodes a third more logical qubits at kd2/n=11.11kd^2/n=11.11kd2/n=11.11 (7.4%7.4\%7.4% below benchmark) while decoding no worse. At n=72n=72n=72, [[72,14,8]][[72,14,8]][[72,14,8]] attains kd2/n=12.44kd^2/n=12.44kd2/n=12.44---more than twice the same-length BB code---and decodes better; a circuit-level memory experiment places our weight-888 codes at ≈0.1%\approx 0.1\%≈0.1% pseudo-threshold versus ≈0.4%\approx 0.4\%≈0.4% for the BB reference under an identical model, quantifying the threshold cost of the heavier checks. All structural statements are verified numerically on the whole census.