Quantum bootstrap product codes

• URL: http://arxiv.org/abs/2601.22363v1
• Date: Thu, 29 Jan 2026 22:16:31 GMT
• Title: Quantum bootstrap product codes
• Authors: Meng-Yuan Li,
• Abstract summary: Product constructions constitute a powerful method for generating quantum CSS codes.<n>We introduce the textitquantum bootstrap productfork (QBP)<n>We show that QBP can generate self-correcting quantum codes from input codes with constant energy barriers.
• Score: 8.649956206461296
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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