Quantum bootstrap product codes

TL;DR

This paper introduces the quantum bootstrap product (QBP), a new framework for constructing quantum CSS codes that unifies various code families, including hypergraph product and fracton codes, and enables the design of self-correcting quantum memories.

Contribution

The paper develops the QBP paradigm, extending beyond homological methods, and introduces fork complexes, broadening the scope of quantum product codes and fault-tolerant quantum memory design.

Findings

QBP unifies hypergraph product and fracton codes.

Solutions to the bootstrap equation produce fork complexes with multiple components.

QBP can generate self-correcting codes with high energy barriers.

Abstract

Product constructions constitute a powerful method for generating quantum CSS codes, yielding celebrated examples such as toric codes and asymptotically good low-density parity check (LDPC) codes. Since a CSS code is fully described by a chain complex, existing product formalisms are predominantly homological, defined via the tensor product of the underlying chain complexes of input codes, thereby establishing a natural connection between quantum codes and topology. In this Letter, we introduce the \textit{quantum bootstrap product} (QBP), an approach that extends beyond this standard homological paradigm. Specifically, a QBP code is determined by solving a consistency condition termed the ``bootstrap equation''. We find that the QBP paradigm unifies a wide range of important codes, including general hypergraph product (HGP) codes of arbitrary dimensions and fracton codes typically represented by the X-cube code. Crucially, the solutions to the bootstrap equation yield chain complexes where the chain groups and associated boundary maps consist of multiple components. We term such structures \textit{fork complexes}. This structure elucidates the underlying topological structures of fracton codes, akin to foliated fracton order theories. Beyond conceptual insights, we demonstrate that the QBP paradigm can generate self-correcting quantum codes from input codes with constant energy barriers and surpass the code-rate upper bounds inherent to HGP codes. Our work thus substantially extends the scope of quantum product codes and provides a versatile framework for designing fault-tolerant quantum memories.