Classical product code constructions for quantum Calderbank-Shor-Steane codes

Abstract

Several notions of code products are known in quantum error correction, such as hyper-graph products, homological products, lifted products, balanced products, to name a few. In this paper we introduce a new product code construction which is a natural generalisation of classical product codes to quantum codes: starting from a set of component Calderbank-Shor-Steane (CSS) codes, a larger CSS code is obtained where both $X$ parity checks and $Z$ parity checks are associated to classical product codes. We deduce several properties of product CSS codes from the properties of the component codes, including bounds to the code distance, and show that built-in redundancies in the parity checks result in so-called meta-checks which can be exploited to correct syndrome read-out errors. We then specialise to the case of single-parity-check (SPC) product codes which in the classical domain are a common choice for constructing product codes. Logical error rate simulations of a SPC $3$-fold product CSS code having parameters $[[512,174,8]]$ are shown under both a maximum likelihood decoder for the erasure channel and belief propagation decoding for depolarising noise. We compare the results with other codes of comparable length and dimension, including a code from the family of asymptotically good Tanner codes. We observe that our reference product CSS code outperforms all the other examined codes.

References in corpus (28)

• Supplementary information for "Quantum supremacy using a programmable superconducting processor"
• Universal Quantum Computation with ideal Clifford gates and noisy ancillas
• Strong quantum computational advantage using a superconducting quantum processor
• Logical quantum processor based on reconfigurable atom arrays
• A Theory of Fault-Tolerant Quantum Computation
• An atom-by-atom assembler of defect-free arbitrary 2d atomic arrays
• High-threshold and low-overhead fault-tolerant quantum memory
• Stim: a fast stabilizer circuit simulator
• Sparse Graph Codes for Quantum Error-Correction
• Quantum Low-Density Parity-Check Codes
• Building a fault-tolerant quantum computer using concatenated cat codes
• Quantum LDPC codes with positive rate and minimum distance proportional to n^{1/2}
• Degenerate Quantum LDPC Codes With Good Finite Length Performance
• Magic State Distillation: Not as Costly as You Think
• Fault-tolerant quantum computation against biased noise
• Decoding Across the Quantum LDPC Code Landscape
• Repetition Cat Qubits for Fault-Tolerant Quantum Computation
• High-threshold fault-tolerant quantum computation with analog quantum error correction
• Balanced Product Quantum Codes
• Linear-Time Maximum Likelihood Decoding of Surface Codes over the Quantum Erasure Channel
• Single-shot error correction of three-dimensional homological product codes
• Improved single-shot decoding of higher dimensional hypergraph product codes
• Quantum Lego: Building Quantum Error Correction Codes from Tensor Networks
• Finite Rate QLDPC-GKP Coding Scheme that Surpasses the CSS Hamming Bound
• Two-dimensional linear trap array for quantum information processing
• Synthesis of Logical Clifford Operators via Symplectic Geometry
• Concatenation Schemes for Topological Fault-tolerant Quantum Error Correction
• Improved quantum error correction using soft information