2025/07/22 by Xiaojun Fu, Han Zheng, Fu, Esther Xiaozhen +5 · 1 citation
Computer Science · #FOS: Physical sciences #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum Physics (quant-ph)
paper · pdf · doi:10.48550/arxiv.2507.16797
openalex publication_date 2025/07/22 · openalex created_date 2025/09/23 · openalex updated_date 2026/07/31
Quantum error-correcting codes are essential to the implementation of fault-tolerant quantum computation. Homological products of classical codes offer a versatile framework for constructing quantum error-correcting codes with desirable properties, especially quantum low-density parity check (qLDPC) codes. Based on extensions of the Bravyi--König theorem that encompass codes without geometric locality, we establish a series of general no-go theorems for fault-tolerant logical gates supported by hypergraph product codes. Specifically, we show that non-Clifford logical gates cannot be implemented transversally on hypergraph product codes of all product dimensions, and that the dimensions impose various limitations on the accessible level of the Clifford hierarchy gates by constant-depth local circuits. We also discuss examples both with and without geometric locality which attain the Clifford hierarchy bounds. Our results reveal fundamental restrictions on logical gates originating from highly general algebraic structures, extending beyond existing knowledge only in geometrically local, finite logical qubits, transversal, or 2-dimensional product cases, and may guide the vital study of fault-tolerant quantum computation with qLDPC codes.