CSS quantum codes approach capacity with sparse decoding graphs
Achievability of CSS codes performance limits over Pauli channels under separate ML decoding with bounded graphical complexity
Information Theory
Summary
Quantum computers need special error-correcting codes to keep their information safe, but the best codes are usually complicated and hard to decode efficiently. The authors show a way to build CSS codes that get very close to the best possible performance while keeping the decoding process manageable and simple. They combine different coding techniques to keep the structure sparse, which helps separate and correct bit and phase errors effectively. This work opens the door for practical quantum error correction that balances power and efficiency.
What this means in practice
- •For quantum hardware engineers: Design quantum memory systems that use CSS codes with simpler, bounded-degree decoding graphs to improve error correction performance near capacity limits.
- •For classical error correction developers: Incorporate combined LDPC and LDGM code structures to optimize decoding efficiency for classical systems facing Pauli-like noise models.
Authors
Gulnihal Mercanoglu, Achilleas Anastasopoulos, S. Sandeep Pradhan
Abstract
Capacity-approaching quantum error correction faces a fundamental tension: the required parity-check matrices need to become increasingly dense, while practical decoding and syndrome extraction favor sparse, bounded-degree structures. Reconciling these competing requirements remains an important open problem in quantum code design. We construct Calderbank-Shor-Steane (CSS) code ensembles that approach the CSS rate over memoryless Pauli channels under maximum-likelihood decoding while retaining sparse, bounded-degree graphical representations. The construction separates parity-check density from graphical complexity: although capacity-approaching components may require dense parity-check matrices, they are represented by sparse concatenated factor graphs. We first study a construction combining low-density parity-check (LDPC) and low-density generator-matrix (LDGM) component codes and numerically analyze the average weight distributions of the resulting codes. Motivated by these results, we introduce a second construction based on concatenated LDPC and LDGM codes and generalized irregular repeat-accumulate (GIRA) codes. We derive bounds on the average weight distribution of the resulting sum ensemble and use the Miller-Burshtein framework to prove simultaneous reliability of the two CSS decoding components. Consequently, the proposed ensemble achieves rates arbitrarily close to the CSS hashing bound under separate bit- and phase-error correction, while maintaining bounded graphical complexity, thus providing a foundation for future iterative decoders and indirect syndrome-extraction architectures.