Toward Low-latency Iterative Decoding of QLDPC Codes Under Circuit-Level Noise

TL;DR

The paper tackles the challenge of decoding quantum LDPC codes under circuit-level noise with low latency. It introduces a sliding-window BP-based decoder whose inner loop uses guided decimation guessing (GDG) to accelerate convergence, particularly for BB codes. GDG achieves comparable logical error rates to BP+OSD while delivering millisecond-scale per-window latency on multi-threaded CPUs, enabling real-time decoding in streaming syndrome scenarios. The work demonstrates the practicality of BDG-based window decoding for sub-threshold circuit-noise regimes and outlines directions for hardware-oriented optimizations and extensions.

Abstract

We introduce a sliding window decoder based on belief propagation (BP) with guided decimation for the purposes of decoding quantum low-density parity-check codes in the presence of circuit-level noise. Windowed decoding keeps the decoding complexity reasonable when, as is typically the case, repeated rounds of syndrome extraction are required to decode. Within each window, we employ several rounds of BP with decimation of the variable node that we expect to be the most likely to flip in each round, Furthermore, we employ ensemble decoding to keep both decimation options (guesses) open in a small number of chosen rounds. We term the resulting decoder BP with guided decimation guessing (GDG). Applied to bivariate bicycle codes, GDG achieves a similar logical error rate as BP with an additional OSD post-processing stage (BP+OSD) and combination-sweep of order 10. For a window size of three syndrome cycles, a multi-threaded CPU implementation of GDG achieves a worst-case decoding latency of 3ms per window for the [[144,12,12]] code.

Figures (7)

• Figure 1: (3,1) sliding window decoding of the circuit code PCM $\mathbf{H}_{circ}$. For the BB code family with circuit in bravyi2023highthreshold, $\mathbf{H}_0$ has shape $w\times 3w$, $\mathbf{H}_1, \mathbf{H}_2$ both have shape $w\times 9w$, where $w=N/2$ is the number of detectors used in one round for a block-length $N$ code.
• Figure 2: The decision tree for the BPGDG algorithm. VN selection and decimation are made after each step, where one step is defined to be six BP iterations. The red path is the main branch. The solid dots (red or black) are the only places where guessing is allowed. No splitting from the main branch into side branches (blue) after reaching depth 10. No guessing for the tree branches (green) after depth 4. The main branch terminates after 25 steps ($D_{max}=25$), regardless of convergence. The side branches are allowed to run 10 more steps after their split-off at depth $D_{splitt}$, i.e., $D_{max}=D_{splitt}+10$. The tree branches are also allowed to run 10 more steps after splitting from their neighbors at depth $4$, which means $D_{max}=4+10$ for all tree branches.
• Figure 3: Logical error rate per round. The syndrome measurement is repeated $d$ rounds for a distance $d$ code. Dotted lines are global decoding over $d$ rounds using BP (1000 iterations) + OSD-CS10. Dashed lines are (3,1)-sliding window decoding where each window uses BP (200 iterations) + OSD-CS10 as the inner decoder. Solid lines use GDG as the inner decoder in (3,1)-sliding window. For GDG, low error mode (no aggressive decimation) is used for $p\leq 0.002$ for all codes.
• Figure 4: Data qubit noise, X-noise only. Solid lines are GDG in low error mode with main branch maximum depth $40$, side or tree branches do not split after depth $20$ or $5$ and are allowed to proceed for $30$ more steps. Scaling factor $0.625$. Dashed and dotted lines are BP+OSD-CS10 and BP+OSD-0 respectively. BP preprocessing for both OSD methods is $100$ iterations. Scaling factor $0.5$ is used for $N\leq 144$ and $0.625$ is used for $N=288$.
• Figure 5: Logical error rate for the $\llbracket 288,12,18\rrbracket$ code at data qubit X-noise $p_d=0.03 \sim 0.06$ and i.i.d. syndrome bit-flip with probability $p_s\in [10^{-5}, 10^{-3}]$. The dotted lines are the lower bound $p_L(p_d)+864\cdot p_s^2+2592\cdot p_d\cdot p_s^2$.