BiBiEQ: Bivariate Bicycle Codes on Erasure Qubits

TL;DR

The paper tackles the high overhead of fault-tolerant quantum memory by leveraging erasure qubits to reveal fault locations. It introduces BiBiEQ, an end-to-end framework that builds BB-code memory circuits, compiles them under an erasure-biased noise model, and maps the erasure-annotated circuits to stabilizer circuits using two engines—BiBiEQ-Exact (posterior-faithful) and BiBiEQ-Approx (throughput-focused). Through simulations with 2EC and 4EC schedules and BP+OSD decoding, the study demonstrates substantial subthreshold improvements and distance-driven gains, notably a major improvement when increasing distance from $d=6$ to $d=10$ and strong agreement between engines under 4EC. The results provide practical guidance on scheduling and engine choice, showing that erasure-aware BB codes can achieve low logical error rates with favorable thresholds, thereby reducing the qubit and circuit overhead in fault-tolerant quantum memories. The work lays groundwork for integrating erasure-native decoders and extending BiBiEQ to broader QLDPC families, with clear implications for scalable quantum architectures.

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 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.

Figures (8)

• Figure 1: BiBiEQ framework overview: (1) Circuit builder from code parameters [[$n,k,d$]]. (2) Compilation to apply the noise law and collect schedule-aware segments. (3) Convert $C_{E}$ into a stabilizer circuit $C$. (4) Evaluation to get LER.
• Figure 2: Ideal vs. simulated operations under the erasure model. Each simulated block appends an erasure/Pauli channel with probabilities $p$ or $q$, and EC triggers a reset $R$.
• Figure 3: Tanner graph of the $[[72,12,6]]$ BB code. Two interleaved data lattices (blue/yellow) connect to sparse $Z$ (green) and $X$ (red) checks. Arrows show toroidal wraparound. Monomials in $A$ generate dotted "A" edges, and monomials in $B$ generate solid "B" edges.
• Figure 4: Segment abstraction. (a) A segment $s$ over qubit $q$ in an erasure circuit $C_{E}$. (b) The associated spacetime locations are realized as fault channels $\mathcal{F}_i$ in the stabilizer circuit.
• Figure 5: BB memory circuit for syndrome measurement with checkpoints $A$–$D$ (EC + reset sites).