Quantum XYZ Product Codes
Anthony Leverrier, Simon Apers, Christophe Vuillot

TL;DR
This paper explores a novel three-fold hypergraph product code construction called XYZ product, which produces non-CSS quantum LDPC codes with potentially large minimum distances and unique logical operator structures, advancing quantum error correction theory.
Contribution
Introduces and analyzes the XYZ product code construction, highlighting its algebraic complexity, logical operator geometry, and potential for high-distance quantum codes with local 3D embeddings.
Findings
XYZ product codes depend on algebraic properties of classical codes' parity-check matrices.
Certain XYZ codes may achieve a minimum distance of Θ(N^{2/3}).
Some codes could serve as self-correcting quantum memories with a logarithmic energy barrier.
Abstract
We study a three-fold variant of the hypergraph product code construction, differing from the standard homological product of three classical codes. When instantiated with 3 classical LDPC codes, this "XYZ product" yields a non CSS quantum LDPC code which might display a large minimum distance. The simplest instance of this construction, corresponding to the product of 3 repetition codes, is a non CSS variant of the 3-dimensional toric code known as the Chamon code. The general construction was introduced in Denise Maurice's PhD thesis, but has remained poorly understood so far. The reason is that while hypergraph product codes can be analyzed with combinatorial tools, the XYZ product codes also depend crucially on the algebraic properties of the parity-check matrices of the three classical codes, making their analysis much more involved. Our main motivation for studying XYZ product…
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Taxonomy
TopicsQuantum Computing Algorithms and Architecture · Semiconductor Quantum Structures and Devices
