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Accelerating A*-Based Algorithms for Decoding Quantum Low-Density Parity-Check Codes
Lamia Yous, Francisco Garcia Herrero, Mark F. Flanagan
TL;DR
QLDPC decoding must be accurate and efficient, while Tesseract’s search can be costly on large graphs. This paper combines BP preprocessing and reliability gating with Tesseract for residual decoding; the hybrid reduces node expansions while keeping LER close to stand-alone Tesseract.
Problem
QLDPC codes need accurate, computationally efficient decoders that can process syndromes within the quantum state’s strict decoherence time.
Source: §I · p. 1Method
The hybrid uses BP for an initial error estimate and reliability information, then gates soft information and applies Tesseract to the residual syndrome when BP does not converge.
Source: §I · pp. 1–2, A. Hybrid Decoding Pipeline · p. 3Results
The hybrid reduces Tesseract node expansions by nearly 15× at peak while keeping LER extremely close to stand-alone Tesseract for both tested codes.
Source: §V · p. 6, §IV · p. 6Takeaways & Limitations
The approach substantially reduces Tesseract decoding complexity while maintaining LER values comparable to stand-alone Tesseract.
Source: §V · p. 6
Abstract
from arXiv · showhide
Quantum low-density parity-check (QLDPC) codes represent a promising approach for error correction in quantum computing. The recently proposed Tesseract decoder uses the A* search algorithm that guarantees finding the most likely error pattern. However, practical implementations of Tesseract often involve an extremely large graph, and the inherently sequential nature of the search results in high computational overhead and long runtime. To improve decoding efficiency, we propose a two-stage decoding framework. First, a belief propagation (BP) decoder efficiently processes the syndrome. This step generates hard decisions (a binary error vector) and soft information (per-qubit confidence levels). In non-convergent BP cases, a gating mechanism examines the BP decoder's output to identify and filter out qubits with oscillating confidence values. In these cases, the refined output serves as input to Tesseract, which then attempts to identify and correct any residual errors. This hybrid approach leverages the high speed of BP decoding and delegates the more challenging decoding instances to Tesseract. Numerical results demonstrate a substantial reduction in overall decoding complexity while maintaining the logical error rate (LER) of the stand-alone Tesseract. Across all tested physical error rates, the proposed method achieves at least 5x reduction in the number of expanded nodes in the Tesseract, with a peak reduction of nearly 15x at a physical error rate of p = 0.05 for the [[126, 12, d < 11]] T1 code, and approximately 8.8x at p = 0.05 for the [[72, 12, 6]] bicycle bivariate (BB) code.
I. INTRODUCTION
QLDPC codes offer higher code rates and lower qubit overhead than surface codes, but practical use requires accurate decoding fast enough to keep pace with quantum-state decoherence. This section motivates combining BP and Tesseract, with reliability gating, to balance decoding speed and performance.
I. INTRODUCTION
- Practical QLDPC decoding must be accurate and fast enough to operate within qubit decoherence time and keep pace with syndrome generation.Otherwise, syndrome-processing backlog can cause exponential slowdown during quantum computation.Source: §I · p. 1
- BP is fast, but Tanner-graph cycles can correlate messages, prevent convergence, and produce high error floors in some cases.Tesseract [1] guarantees the most likely error pattern, but its sequential A* search scales poorly with graph size.Source: §I · p. 1
- The proposed hybrid uses BP to form an initial error estimate and residual syndrome, reducing the problem Tesseract must decode while adding per-qubit reliability to guide its search.The framework also gates BP confidence values before passing soft information to Tesseract.Source: §I · pp. 1–2
- The framework is validated for X-type error correction on the T-family T1 code and the [[72, 12, 6]] bicycle bivariate code.The T1 code reference is, and the bicycle bivariate code reference is.Source: §I · p. 2
A. QLDPC codes
QLDPC codes extend sparse-graph coding to quantum error correction, with bounded check and variable-node degrees. Their Tanner-graph representation connects parity-check constraints to candidate error patterns consistent with measured syndromes.
A. QLDPC codes
- QLDPC codes bound the number of bits in each check and the number of checks acting on each bit, keeping the parity-check matrix sparse [12].Many use the CSS construction from two classical binary linear codes, yielding sparse matrices H_X and H_Z with an orthogonality condition,.Source: A. QLDPC codes · p. 2
- Under the CSS structure, X- and Z-type errors can be decoded independently, and this paper focuses on X-type errors without loss of generality.The two error types commute because H_X H_Z^T = 0.Source: A. QLDPC codes · p. 2
- A Tanner graph represents parity checks as check nodes and matrix columns as variable nodes, with edges encoding the check constraints.Decoding seeks an error pattern satisfying the measured syndrome; e is the binary error vector and s the measured syndrome, where n and m count variable and check nodes.Source: A. QLDPC codes · p. 2
- The error pattern must satisfy s = He over the binary field, linking the candidate error vector to the measured syndrome.The multiplication is over the binary field.Source: A. QLDPC codes · p. 2
B. Tesseract Decoder [1]
Tesseract casts syndrome-constrained minimum-cost decoding as a shortest-path search over candidate error vectors. Its admissible heuristic and pruning reduce search, but large graphs can still make decoding prohibitively expensive.
B. Tesseract Decoder [1]
- Tesseract seeks a minimum-cost error vector consistent with the measured syndrome under an independent noise model, using A* with an admissible heuristic and pruning.The task minimizes w(e) subject to the syndrome constraint.Source: B. Tesseract Decoder [1] · p. 2
- In the search graph, each node is a candidate error vector, each edge adds one qubit error, and path cost is the associated error-vector cost.The initial node is the zero vector.Source: B. Tesseract Decoder [1] · p. 2
- A* expands nodes by estimated total cost, while its lower-bound heuristic preserves optimality and pruning avoids redundant configurations and restricts candidate expansions.The paper uses total node expansions as a measure of decoding complexity.Source: B. Tesseract Decoder [1] · p. 2
- Despite its optimality guarantee and pruning, Tesseract can incur prohibitive time and complexity costs when the graph is extremely large.The search graph grows with the Tanner graph, or number of qubits n.Source: B. Tesseract Decoder [1] · p. 2
C. Belief-Propagation Decoder [4]
BP iteratively exchanges messages over the Tanner graph to estimate an error vector matching the measured syndrome. Short cycles can correlate messages and cause oscillation, but BP remains fast through parallel processing and edge-linear per-iteration complexity.
C. Belief-Propagation Decoder [4]
- BP estimates an error vector matching the measured syndrome through iterative message passing along Tanner-graph edges.Messages are exchanged between variable and check nodes, then updated across iterations.Source: C. Belief-Propagation Decoder [4] · pp. 2–3
- Short cycles in QLDPC Tanner graphs can correlate messages and cause oscillatory behavior when BP does not converge.Decoding is considered successful if the estimated error satisfies the syndrome equation or the maximum iteration count is reached.Source: C. Belief-Propagation Decoder [4] · p. 3
- BP has per-iteration complexity linear in Tanner-graph edges and can process nodes in parallel.The paper uses BP soft output to construct reliability-informed inputs that guide Tesseract in non-convergent cases.Source: C. Belief-Propagation Decoder [4] · p. 3
A. Hybrid Decoding Pipeline
BP first attempts to decode the measured syndrome, and its hard decisions and soft confidence estimates feed a residual-syndrome correction when BP does not converge. Tesseract then finds a lowest-cost correction to the residual and combines it with the BP estimate.
A. Hybrid Decoding Pipeline
- BP is the preliminary decoder, and successful convergence makes its output the final result; otherwise, decoding proceeds to Tesseract.The described setting assumes independent, identically distributed symmetric depolarizing errors.Source: A. Hybrid Decoding Pipeline · p. 3
- The BP output includes a hard error decision and per-qubit soft information, with pflip measuring uncertainty in each hard decision.Values of pBP,i above 0.5 indicate a likely error, while values below 0.5 indicate likely no error.Source: A. Hybrid Decoding Pipeline · p. 3
- The framework computes the residual syndrome as the measured syndrome XOR the syndrome of the BP hard decisions, then uses Tesseract to find a lowest-cost correction satisfying it.The final decoded vector combines the BP estimate with that correction.Source: A. Hybrid Decoding Pipeline · p. 3
B. Preprocessing Decoder
The preprocessing stage assesses BP confidence and stability before passing information to Tesseract. Its gating mechanism discounts unreliable estimates, including those whose confidence appears high despite persistent oscillation.
B. Preprocessing Decoder
- Qubit confidence estimates can oscillate substantially during non-convergent BP, while convergent estimates stabilize near zero or one.The examples include qubit 32 switching from above 80% estimated error at iteration 45 to about 25% at iteration 48, and qubit 33 fluctuating despite an unchanged hard decision.Source: B. Preprocessing Decoder · pp. 3–4
- Correctly decoded qubits tend to have low pflip, motivating a threshold that filters qubits with large pflip from influencing Tesseract.Figure 2 compares mean pflip for correct and incorrect qubits across physical error rates.Source: B. Preprocessing Decoder · p. 4, Figure 2 · p. 4
- Because pflip alone can miss oscillations, the method measures average probability step changes over the final 20% of BP iterations.Early fluctuations are not the target; the metric is intended to detect persistent oscillations.Source: B. Preprocessing Decoder · p. 4
- The reliability score αBP,i combines temporal variation and pflip, and threshold τ classifies each qubit as reliable or unreliable.A low pflip can still be unreliable when the estimate oscillates, while a stable estimate can be unreliable when pflip is high.Source: B. Preprocessing Decoder · pp. 4–5
- Unreliable qubits receive fixed prior pgated in Tesseract, while reliable qubits retain their original pflip prior.The paper defines speed-up as the reduction in Tesseract expanded nodes relative to Tesseract.Source: B. Preprocessing Decoder · p. 5
C. Parameter Optimization
The section selects τ and p_gated to maximize node-expansion reduction while keeping gated-qubit costs within constraints. At p = 0.05, the maximum reduction factor is 14.79 at τ = 0.0025 and p_gated = 0.1.
C. Parameter Optimization
- For the T1 code, sum-product BP with serial scheduling and at most 50 iterations achieved a maximum node-expansion reduction factor of 14.79 at p = 0.05, pgated = 0.1, and τ = 0.0025.The reduction factor compares node expansions by stand-alone Tesseract with those by the hybrid BP-Tesseract decoder.Source: C. Parameter Optimization · p. 5
- p_gated must lie in (pflip,max, 1/2] so gated qubits cost more to modify than reliable qubits without creating negative Tesseract weights.Values above 0.5 yield negative weights; values at or below pflip,max could make gated qubits cheaper than reliable ones.Source: C. Parameter Optimization · p. 5
- 14.79 is the maximum node-expansion reduction factor, reached at p_gated = 0.1 and τ = 0.0025.Similar peak values occur across different τ settings, suggesting the optimal p_gated is independent of τ.Source: C. Parameter Optimization · p. 5
- Too-high τ leaves too many qubits ungated, while too-low τ gates too many; p_gated must balance under-penalizing unreliable qubits against over-penalizing potentially correct ones.The comparison varies p_gated across τ values at fixed p = 0.05.Source: Figure 3 · p. 5
IV. RESULTS
The gated BP-Tesseract decoder reduces mean node expansions relative to both stand-alone Tesseract and ungated BP-Tesseract, while retaining LER extremely close to stand-alone Tesseract for both tested codes.
IV. RESULTS
- The evaluation optimizes τ and p_gated over physical error rates from p = 0.03 to p = 0.1 for the T1 and bicycle bivariate codes.Figure 4 compares average node expansions for stand-alone Tesseract, BP-Tesseract, and gated BP-Tesseract.Source: §IV · p. 5, Figure 4 · p. 6
- Nearly 15× fewer node expansions are reached for T1, with approximately 5× reduction at p = 0.03 and approximately 8.8× for the bicycle bivariate code at p = 0.05.The gated decoder has fewer average node expansions than both stand-alone Tesseract and BP-Tesseract without probability gating.Source: §IV · p. 5, Figure 4 · p. 6
- The gated decoder's LER is extremely close to stand-alone Tesseract for both the T1 and bicycle bivariate codes.The comparison includes stand-alone Tesseract, BP-Tesseract, and gated BP-Tesseract across physical error rates.Source: §IV · p. 6, Figure 5 · p. 6
V. CONCLUSION
The conclusion presents a hybrid BP-Tesseract decoder for QLDPC codes that uses BP estimates to simplify Tesseract's residual-error task and gates unreliable soft information. Simulations report fewer node expansions while keeping LER comparable to stand-alone Tesseract.
V. CONCLUSION
- The framework uses BP estimates to turn the original decoding task into correcting BP's errors, then uses Tesseract to address the residual errors.BP provides soft information and initial probabilistic estimates.Source: §V · p. 6
- Reliability-based gating filters oscillatory and uncertain BP soft information before it reaches Tesseract.The method combines this filtering with the hybrid decoding framework.Source: §V · p. 6
- The method achieves nearly 15× fewer node expansions while maintaining LER comparable to stand-alone Tesseract.The conclusion describes reliability-guided preprocessing as improving efficiency without substantially compromising accuracy.Source: §V · p. 6
- Future work includes testing larger QLDPC codes and incorporating detector error models to assess scalability more fully.These are stated directions for further research.Source: §V · p. 6