The leading exact tensor-network contraction requires time exponential in treewidth. A new decoding algorithm based on rank-decomposition dynamic programming (Rank DP) is introduced. Decoding partition functions with independent local fault factors are expressed as quadratic sums of powers and Rank DP is applied. The resulting exact evaluator has arithmetic complexity polynomial in input size and exponential in the Tanner graph’s rank-width concerning the fault partition, including decomposition construction. Following Gaussian elimination, it gives polynomial-time decoding for punctured quantum Reed-Muller codes and a Steane-concatenated family with growing distance under indepen.

## Decoding Quantum Error Correction Codes via Rank Decomposition Dynamic Programming Scientists at National University of Singapore, alongside collaborators at Singapore University of Technology and Design, have fully evaluated quantum Reed-Muller codes containing up to 1,023 qubits using maximum likelihood; this represents a leap beyond previous capabilities limited by computational demands scaling exponentially with problem size.

Instead of relying on existing tensor network contraction techniques which struggle with computations increasing rapidly in graph treewidth, the team focused on the rank-width of a Tanner graph, a measure of its structure, to achieve arithmetic complexity that scales polynomially. The new algorithm, called Rank-Decomposition Dynamic Programming (Rank DP), expresses decoding probabilities as sums of powers enabling exact computation while offering substantial runtime advantages over standard methods tested on selected code instances.

It was also successfully applied to Steane-concatenated codes, achieving polynomial time decoding under specific noise conditions where conventional tensor network methods would require considerably longer computation times. Furthermore, this method accurately learns parameters describing circuit noise from observed syndromes, allowing for better characterisation of errors within quantum circuits and quantifying how close decoder results are to an ideal solution. This opens up possibilities for more efficient error correction in complex quantum systems with a greater number of qubits.

Standard contraction of tensor networks requires superpolynomial time for single-qubit Pauli noise; however, a nonnegative realization gives relative floating-point error bounds under explicit arithmetic assumptions. Numerical experiments demonstrate runtime advantages over tested tensor-network implementations on selected code-capacity and circuit-level instances including full likelihood evaluation for Reed-Muller codes reaching 1,023 qubits. These findings suggest that Rank DP offers significant improvements in computational efficiency compared to traditional methods when dealing with large qubit numbers.

Likelihoods were further used to learn circuit noise parameters from syndromes, evaluate rare postselection probabilities, and quantify decoder optimality gaps. This work opens new avenues for exploiting algebraic structure in quantum decoding and noise characterisation; it provides a pathway towards more robust and reliable quantum computations by improving our ability to understand and correct errors. Quantum error correction protects logical quantum information by encoding it across physical qubits and repeatedly measuring error syndromes, its effectiveness depends on a classical decoder using these observations to determine a recovery operation.

Maximum-likelihood decoding minimizes the probability of selecting an incorrect logical class under a correctly specified stochastic Pauli noise model providing both an optimal decoding rule and a benchmark for practical decoders. The computational cost is central as maximum-likelihood decoding requires accounting for degeneracy where errors within the same logical class differ only by a stabilizer, thus being corrected by identical recovery operators. Consequently, this approach selects the class with the largest total error probability which reduces, under local stochastic noise, to evaluating and comparing partition functions motivating algorithms exploiting additional structure.

Tensor-network methods evaluate or approximate these decoding partition functions developed for surface codes and general two-dimensional Pauli codes extended to higher-dimensional codes and circuit-level noise; recent work uses exact tensor network likelihoods to estimate circuit-level noise parameters directly from syndrome data enabling high accuracy inference but requiring time exponential in treewidth of underlying graph while approximate contraction using truncation does not generally preserve exact likelihoods.

Decoding problems contain algebraic structure as syndrome and logical constraints are linear equations over F2 representing Pauli faults suggesting independent parity conditions rather than dense tensor boundary size may provide a more succinct description of computation. This raises an important question: Can exact maximum-likelihood decoding surpass the limitations of tensor-network contraction.

Scientists answer this by expressing decoding likelihoods as weighted quadratic sums of powers applying rank-decomposition dynamic programming, obtaining an exact evaluator whose complexity is exponential in the rank-width of the underlying Tanner graph. This represents the first application of this framework to quantum maximum-likelihood decoding including its formulation, resulting complexity guarantees and tractable code families alongside numerical evidence identifying instances where it proves useful. Gaussian elimination further reduces the graph on which this evaluator operates giving polynomial-time maximum-likelihood decoding for two families based on punctured quantum Reed-Muller codes whereas dense pairwise contraction requires superpolynomial time.

A nonnegative realization preserves table sizes and arithmetic complexity using only nonnegative sums and products avoiding cancellation admitting relative roundoff error bounds under explicit floating point assumptions detailed in supplemental material. Complementary theoretical results are supported by numerical experiments assessing practical performance with full likelihood evaluation up to 1,023 qubits demonstrating a gap between decomposition widths after reduction and treewidth lower bounds for original graphs. Rank DP achieves shorter runtimes than tested tensor network implementations allowing additional algebraic optimizations without code specific optimization; it computes both logical observable sector probabilities with certified accuracy determining maximum likelihood labels.

Maximum-likelihood decoding offers an optimal strategy for quantum error correction under stochastic Pauli noise; however, calculating logical-class probabilities presents challenges as leading tensor-network contraction requires time exponential in treewidth. Standard tensor network contraction demands superpolynomial time to achieve similar results and a nonnegative realisation yields relative floating-point error bounds assuming explicit arithmetic assumptions.

Numerical tests demonstrate runtime benefits over tested tensor-network implementations on specific code instances allowing full likelihood evaluation for Reed-Muller codes up to 1,023 qubits, these likelihoods also support learning circuit noise parameters from syndromes and evaluating postselection probabilities. Leading tensor-network contraction requires time exponential in treewidth.

A new algorithm utilising rank-decomposition dynamic programming expresses partition functions with independent local faults as quadratic sums of powers applying Rank DP. Gaussian elimination then provides polynomial-time decoding for punctured Reed-Muller codes and a Steane family exhibiting growing distance under single-qubit Pauli noise, a significant improvement over existing methods. Maximum-likelihood decoding provides an optimal strategy for quantum error correction under stochastic Pauli noise; however, computing logical-class probabilities presents challenges as leading tensor-network contraction requires time exponential in treewidth.

To address this issue, a new decoding algorithm based on rank-decomposition dynamic programming (Rank DP) has been introduced. Following Gaussian elimination, it enables polynomial-time decoding for punctured quantum Reed-Muller codes and a Steane-concatenated family experiencing growing distance under independent single-qubit Pauli noise; standard tensor network contraction of corresponding representations demands superpolynomial computation time.

A nonnegative realization provides relative floating-point error bounds assuming explicit arithmetic assumptions.

The researchers developed a new algorithm called Rank DP that allows for faster computation of logical-class probabilities during quantum error correction under stochastic Pauli noise. This method offers polynomial time decoding for punctured quantum Reed-Muller codes and certain Steane code families, representing an improvement over standard tensor network contraction which requires significantly more computational time. Authors demonstrated runtime advantages compared with existing methods on selected instances.

👉 More information
🗞 Exact Maximum Likelihood Decoding beyond Treewidth via Rank-Decomposition Dynamic Programming
✍️ Bin Cheng (National University of Singapore); Feng Pan (Singapore University of Technology and Design)
🧠 ArXiv: https://arxiv.org/abs/2609.39556

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