Explicit Construction of High-Dimensional Cubical Complexes for Quantum Locally Testable Codes

Core Concepts

The authors introduce a high-dimensional cubical complex construction that can be used to design new families of quantum locally testable codes with improved parameters compared to previous constructions.

Abstract

The authors introduce a high-dimensional cubical complex construction that generalizes previous work on 2-dimensional complexes used for classical and quantum error-correcting codes. The complex is constructed from a set G and subsets of permutations A1, ..., At acting on G.
The complex is endowed with a system of local coefficients, defined using a family of constant-sized local codes h1, ..., ht. The authors prove lower bounds on the cycle and co-cycle expansion of the resulting chain complex, which depend on the spectral expansion of the associated Cayley graphs and a "two-way robustness" property of the local codes.
By instantiating the construction using an abelian lift and leveraging recent results on product expansion of random tuples of codes, the authors obtain an explicit family of quantum locally testable codes with constant relative rate, inverse-polylogarithmic relative distance and soundness, and constant-size parity checks. This significantly improves upon previous constructions of quantum locally testable codes.
The authors also introduce new techniques, including a reduction from cycle expansion to co-cycle expansion via an auxiliary "dual" chain complex, that may be of independent interest in the study of high-dimensional expanders and their applications.

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Key Insights Distilled From

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

by Irit Dinur,T... at arxiv.org 04-12-2024

https://arxiv.org/pdf/2402.07476.pdf

Deeper Inquiries

How can the dependence on the dimension t in the expansion bounds be improved

To improve the dependence on the dimension t in the expansion bounds, one approach could be to explore alternative constructions of the high-dimensional cubical complex. By investigating different choices for the sets G and {Ai} that form the complex, it may be possible to find configurations that lead to better expansion properties. Additionally, refining the system of local coefficients and exploring different types of local codes could also potentially reduce the dependence on the dimension t. By optimizing the underlying structures and parameters of the complex, it may be feasible to achieve expansion bounds that are less sensitive to the dimension t.

Can the techniques developed here be applied to construct high-dimensional expanders with stronger expansion properties

The techniques developed in this work could potentially be applied to construct high-dimensional expanders with stronger expansion properties. By adapting the concept of two-way robustness and product expansion to different types of complexes, such as simplicial complexes or other geometric structures, it may be possible to create expanders with enhanced expansion characteristics. By leveraging the insights and methodologies from this study, researchers could explore the construction of high-dimensional expanders that exhibit robust expansion in multiple dimensions, leading to more powerful tools for various applications in quantum computing, complexity theory, and beyond.

What are the potential applications of the quantum locally testable codes constructed here to quantum complexity theory

The quantum locally testable codes constructed in this study have the potential to have significant applications in quantum complexity theory. These codes could be utilized in the development of quantum algorithms, quantum error correction schemes, and quantum communication protocols. By providing a framework for verifying the correctness of quantum states with high soundness and locality properties, these codes could enhance the efficiency and reliability of quantum computations. Additionally, the construction of good quantum LTCs could have implications for quantum complexity classes, quantum PCP theorems, and other fundamental questions in quantum complexity theory. The ability to efficiently test quantum states against a quantum code could lead to advancements in quantum verification and validation processes, ultimately contributing to the advancement of quantum computing technologies.