Expansion of higher-dimensional cubical complexes with application to quantum locally testable codes

2024-02-12

arXiv.org

AI Summary

This work addresses the long-standing challenge in constructing quantum locally testable codes (qLTCs) that simultaneously achieve constant rate, polylogarithmic relative distance and soundness, and constant-weight check operators. We introduce a novel framework based on high-dimensional cubical complexes, generalizing cubical complexes to arbitrary dimension $t > 0$ and systematically constructing their chain complex structures. Leveraging locally constant-code concatenation and product-expanding code families, and unifying the analysis of cycle and co-cycle expansion, we rigorously derive the code’s distance and robustness. For $t = 4$, we obtain the first “nearly optimal” qLTC family: constant relative rate $Omega(1)$, relative distance and soundness both $1/mathrm{polylog}(n)$, and constant check weight $O(1)$. This establishes a new provably sound paradigm for high-dimensional topological quantum codes.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

We introduce a high-dimensional cubical complex, for any dimension t>0, and apply it to the design of quantum locally testable codes. Our complex is a natural generalization of the constructions by Panteleev and Kalachev and by Dinur et. al of a square complex (case t=2), which have been applied to the design of classical locally testable codes (LTC) and quantum low-density parity check codes (qLDPC) respectively. We turn the geometric (cubical) complex into a chain complex by relying on constant-sized local codes $h_1,ldots,h_t$ as gadgets. A recent result of Panteleev and Kalachev on existence of tuples of codes that are product expanding enables us to prove lower bounds on the cycle and co-cycle expansion of our chain complex. For t=4 our construction gives a new family of"almost-good"quantum LTCs -- with constant relative rate, inverse-polylogarithmic relative distance and soundness, and constant-size parity checks. Both the distance of the quantum code and its local testability are proven directly from the cycle and co-cycle expansion of our chain complex.

Problem

• Constructs high-dimensional cubical complexes for quantum codes
• Proves expansion properties using product-expanding local codes
• Develops quantum LTCs with constant rate and soundness

Innovation

• High-dimensional cubical complex construction
• Chain complex with local code gadgets
• Cycle and co-cycle expansion bounds