Effective Distance of Higher Dimensional HGPs and Weight-Reduced Quantum LDPC Codes

Abstract

Quantum error correction plays a prominent role in the realization of quantum computation, and quantum low-density parity-check (qLDPC) codes are believed to be practically useful stabilizer codes. While qLDPC codes are defined to have constant weight parity-checks, the weight of these parity checks could be large constants that make implementing these codes challenging. Large constants can also result in long syndrome extraction times and bad error propagation that can impact error correction performance. Hastings recently introduced weight reduction techniques for qLDPC codes that reduce the weight of the parity checks as well as the maximum number of checks that acts on any data qubit. However, the fault tolerance of these techniques remains an open question. In this paper, we analyze the effective distance of the weight-reduced code when single-ancilla syndrome extraction circuits are considered for error correction. We prove that there exists single-ancilla syndrome extraction circuits that largely preserve the effective distance of the weight-reduced qLDPC codes. In addition, we also show that the distance balancing technique introduced by Evra et al. preserves effective distance. As a corollary, our result shows that higher-dimensional hypergraph product (HGP) codes, also known as homological product codes corresponding to the product of 1-complexes, have no troublesome hook errors when using any single-ancilla syndrome extraction circuit.

arXiv commentUpdated the manuscript to include distance preservation for coning for completeness

References in corpus (23)

• Fault-tolerant quantum computation by anyons
• Surface codes: Towards practical large-scale quantum computation
• Good Quantum Error-Correcting Codes Exist
• Topological quantum memory
• Multiple Particle Interference and Quantum Error Correction
• Operator Quantum Error Correcting Subsystems for Self-Correcting Quantum Memories
• Quantum LDPC codes with positive rate and minimum distance proportional to n^{1/2}
• Subsystem fault tolerance with the Bacon-Shor code
• Dynamically Generated Logical Qubits
• String and Membrane condensation on 3D lattices
• A theory of single-shot error correction for adversarial noise
• Single-shot error correction of three-dimensional homological product codes
• Logical blocks for fault-tolerant topological quantum computation
• Local Hamiltonians Whose Ground States are Hard to Approximate
• Subsystem codes with high thresholds by gauge fixing and reduced qubit overhead
• Higher-dimensional quantum hypergraph-product codes
• Fault-tolerant gates on hypergraph product codes
• A four-dimensional toric code with non-Clifford transversal gates
• A Pair Measurement Surface Code on Pentagons
• Distance-preserving stabilizer measurements in hypergraph product codes
• Improved Pairwise Measurement-Based Surface Code
• Weight Reduced Stabilizer Codes with Lower Overhead
• Dynamical weight reduction of Pauli measurements