Overview
This research explores the construction of quantum locally testable codes (qLTCs) using a higher-dimensional lossless approach, specifically focusing on cubical complexes. The work develops a theoretical framework without constructing the required high-dimensional lossless cubical complexes directly, but rather investigates the implications of their existence. A central contribution is a local-to-global theorem established for a novel structure: a level chain complex associated with a high-dimensional cubical complex.
Research Context
Prior work in code construction has leveraged specific mathematical structures. Sipser and Spielman, for instance, constructed Low-Density Parity-Check (LDPC) codes using either bipartite spectral expanders or one-sided lossless expanders. In higher dimensions, spectral expansion has been foundational for asymptotically good classical Locally Testable Codes (LTCs) and qLDPC codes, as shown by Dinur, Evra, Livne, Lubotzky, and Mozes, and by Panteleev and Kalachev. An alternative approach by Lin and Hsieh utilized two-dimensional lossless cubical complexes to construct both classical LTCs and qLDPC codes.
Approach
The current study extends the lossless approach to higher dimensions. Instead of constructing high-dimensional lossless cubical complexes, the methodology centers on exploring the consequences of their hypothetical existence. A key step involves associating a level chain complex with a high-dimensional cubical complex. The chain groups of this level chain complex are defined as being supported on the level sets of the Boolean cube, distinguishing them from traditional chain groups supported on cells.
Findings
A primary technical outcome is the development of a local-to-global theorem for the level chain complex structure. This theorem states that suitable one-dimensional lossless expansion observed in the directional graphs implies small-set coboundary expansion of the global level complex. As a direct consequence of this theorem, the research indicates that sufficiently imbalanced, two-sided lossless four-dimensional cubical complexes can give rise to asymptotically good quantum locally testable codes.
Why This Matters
The development of this local-to-global principle is expected to have further applications. The insights gained regarding the properties of lossless cubical complexes and their associated level chain complexes contribute to the theoretical understanding of constructing advanced quantum codes.
Potential Applications
The researchers anticipate that the local-to-global principle developed in this work will have further applications beyond the immediate context of qLTCs.