Degenerate Quantum LDPC Codes With Good Finite Length Performance

Abstract

We study the performance of medium-length quantum LDPC (QLDPC) codes in the depolarizing channel. Only degenerate codes with the maximal stabilizer weight much smaller than their minimum distance are considered. It is shown that with the help of OSD-like post-processing the performance of the standard belief propagation (BP) decoder on many QLDPC codes can be improved by several orders of magnitude. Using this new BP-OSD decoder we study the performance of several known classes of degenerate QLDPC codes including hypergraph product codes, hyperbicycle codes, homological product codes, and Haah's cubic codes. We also construct several interesting examples of short generalized bicycle codes. Some of them have an additional property that their syndromes are protected by small BCH codes, which may be useful for the fault-tolerant syndrome measurement. We also propose a new large family of QLDPC codes that contains the class of hypergraph product codes, where one of the used parity-check matrices is square. It is shown that in some cases such codes have better performance than hypergraph product codes. Finally, we demonstrate that the performance of the proposed BP-OSD decoder for some of the constructed codes is better than for a relatively large surface code decoded by a near-optimal decoder.

Final version accepted for publication in Quantum journal

References in corpus (10)

• Local stabilizer codes in three dimensions without string logical operators
• Topological Quantum Distillation
• Quantum computing with nearest neighbor interactions and error rates over 1%
• Efficient Algorithms for Maximum Likelihood Decoding in the Surface Code
• Quantum LDPC Codes with Almost Linear Minimum Distance
• Balanced Product Quantum Codes
• Fiber Bundle Codes: Breaking the $N^{1/2}\mathrm{polylog}(N)$ Barrier for Quantum LDPC Codes
• Refined Belief Propagation Decoding of Sparse-Graph Quantum Codes
• Modified belief propagation decoders for quantum low-density parity-check codes
• Ability of stabilizer quantum error correction to protect itself from its own imperfection

Cited by in corpus (35)

• Quantum Low-Density Parity-Check Codes
• Quantum LDPC Codes with Almost Linear Minimum Distance
• Focus beyond quadratic speedups for error-corrected quantum advantage
• Low-overhead fault-tolerant quantum computing using long-range connectivity
• Constant-overhead quantum error correction with thin planar connectivity
• Improved decoding of circuit noise and fragile boundaries of tailored surface codes
• Fiber Bundle Codes: Breaking the $N^{1/2}\mathrm{polylog}(N)$ Barrier for Quantum LDPC Codes
• Parallel window decoding enables scalable fault tolerant quantum computation
• Trapping Sets of Quantum LDPC Codes
• Improved single-shot decoding of higher dimensional hypergraph product codes
• Exploiting Degeneracy in Belief Propagation Decoding of Quantum Codes
• Bias-tailored quantum LDPC codes
• Fault-tolerant gates on hypergraph product codes
• Partitioning qubits in hypergraph product codes to implement logical gates
• Quantum LDPC Codes for Modular Architectures
• Numerical Implementation of Just-In-Time Decoding in Novel Lattice Slices Through the Three-Dimensional Surface Code
• Conservation laws and quantum error correction: towards a generalised matching decoder
• Tailoring three-dimensional topological codes for biased noise
• Generalized Belief Propagation Algorithms for Decoding of Surface Codes
• Performance enhancement of surface codes via recursive MWPM decoding
• Fast erasure decoder for hypergraph product codes
• Quantum XYZ Product Codes
• On maximum-likelihood decoding with circuit-level errors
• Quantum message-passing algorithm for optimal and efficient decoding
• Bounds on stabilizer measurement circuits and obstructions to local implementations of quantum LDPC codes
• Software Tools for Decoding Quantum Low-Density Parity Check Codes
• Single-Shot Decoding of Linear Rate LDPC Quantum Codes with High Performance
• Local Probabilistic Decoding of a Quantum Code
• Limitations on transversal gates for hypergraph product codes
• Non-Pauli Errors in the Three-Dimensional Surface Code
• Classical product code constructions for quantum Calderbank-Shor-Steane codes
• Union-Find Decoders For Homological Product Codes
• Decoherence and Quantum Error Correction for Quantum Computing and Communications
• Error Correction for Reliable Quantum Computing
• Distilling GHZ States using Stabilizer Codes