No-go theorems for logical gates on product quantum codes

TL;DR

This paper proves fundamental limitations on implementing non-Clifford logical gates transversally in hypergraph product quantum codes, extending no-go theorems beyond geometrically local cases and influencing fault-tolerant quantum computing.

Contribution

It establishes new no-go theorems for fault-tolerant logical gates on hypergraph product codes, covering non-geometric and higher-dimensional cases, thus broadening understanding of quantum code limitations.

Findings

Non-Clifford gates cannot be implemented transversally on hypergraph product codes.

Constraints on Clifford hierarchy gates imposed by code dimensions.

Examples show bounds are attainable with and without geometric locality.

Abstract

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--K\"{o}nig 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.