Automorphism gadgets in homological product codes

TL;DR

This paper explores how automorphism symmetries in homological product codes enable logical operations and fault-tolerance, broadening the potential for practical quantum error correction beyond topological codes.

Contribution

It introduces a theoretical framework linking automorphisms of input codes to logical operations and fault-tolerance in homological product codes, including special cases with purely permutation-based operations.

Findings

Logical operations can be derived from automorphisms of input codes.

Automorphism gadgets can preserve code distance under certain conditions.

Classical codes with rich automorphisms fit into the proposed framework.

Abstract

The homological product is a general-purpose recipe that forges new quantum codes from arbitrary classical or quantum input codes, often providing enhanced error-correcting properties. When the input codes are classical linear codes, it is also known as the hypergraph product. We investigate structured homological product codes that admit logical operations arising from permutation symmetries in their input codes. We present a broad theoretical framework that characterizes the logical operations resulting from these underlying automorphisms. In general, these logical operations can be performed by a combination of physical qubit permutations and a subsystem circuit. In special cases related to symmetries of the input Tanner graphs, logical operations can be performed solely through qubit permutations. We further demonstrate that these "automorphism gadgets" can possess inherent fault-tolerant properties such as effective distance preservation, assuming physical permutations are free. Finally, we survey the literature of classical linear codes with rich automorphism structures and show how various classical code families fit into our framework. Complementary to other fault-tolerant gadgets for homological product codes, our results further advance the search for practical fault tolerance beyond topological codes in platforms capable of long-range connectivity.