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Expansion of higher-dimensional cubical complexes with application to quantum locally testable codes (2402.07476v2)

Published 12 Feb 2024 in quant-ph, cs.CC, cs.IT, and math.IT

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.

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Citations (9)
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Summary

  • The paper presents a novel construction of high-dimensional cubical complexes extending expander code theory to create quantum locally testable codes.
  • It establishes a framework for quantum LTCs with non-trivial dimension, distance, and robustness, addressing long-standing challenges in quantum error correction.
  • The research offers comprehensive theoretical insights that could improve fault tolerance in quantum processors and advance high-performance quantum code designs.

Expansion of Higher-Dimensional Cubical Complexes with Application to Quantum Locally Testable Codes

The paper presents a significant advancement in the paper of high-dimensional chain complexes for constructing quantum locally testable codes (qLTCs). It builds on prior work in the area of quantum low-density parity-check (LDPC) codes and high-dimensional expander graphs, proposing a novel method for constructing qLTCs by generalizing cubical chain complexes to arbitrary dimensions.

Key Contributions

  1. Construction of High-Dimensional Cubical Complexes: The authors introduce a family of high-dimensional cubical chain complexes that extend the Sipser-Spielman construction of expander codes to dimensions greater than two. For dimension four, the construction leads to a family of quantum locally testable codes, contingent upon a conjecture related to the robustness of random linear maps. The results encompass both classical and quantum settings, highlighting the flexibility of the proposed complex.
  2. Framework for Quantum LTCs: The paper establishes a framework for quantum locally testable codes with non-trivial dimension, distance, and soundness, a long-standing open problem in quantum coding theory. The proposed construction has applications in quantum computing for error correction, providing a template for qLTC with substantial soundness and relative distance.
  3. Robustness and Two-Way Robustness: Central to the paper is the notion of two-way robustness for a family of codes. The authors conjecture that random tuples of codes exhibit this property, generalizing known results for pairs of codes. This leads to, under certain assumptions, the construction of qLTCs with linear scaling properties.
  4. Theoretical Analysis and Results: The authors provide comprehensive theoretical analysis, including cycle and co-cycle expansion properties, numerical bounds on the systolic and co-systolic distances, and implications for quantum codes. The work effectively bridges theoretical advancements in high-dimensional expanders and practical quantum code constructions.
  5. Future Directions: The paper speculates on possible future developments, including further construction optimization and possibly proving the existence of two-way robust code families. The robustness conjecture, if resolved positively, could have profound implications for the feasibility of constructing high-performance qLTCs.

Implications and Future Developments

The implications of this research are substantial for both theoretical computer science and quantum computing. Practically, the construction of qLTCs with improved parameters could enhance fault-tolerance in quantum processors, impacting quantum algorithm implementation and complexity theory profoundly. The paper's conjecture about two-way robustness underscores the need for further work to verify and potentially construct robust code families, a task essential for realizing good quantum LTCs.

Furthermore, the work opens new pathways in the paper of high-dimensional expanders, with potential cross-applications in areas such as topological methods in data analysis and advanced cryptographic protocols. If the robustness conjecture is proved, it might unravel new types of high-dimensional expander graphs with broad applicability in complexity, combinatorial optimization, and network theory.

In conclusion, the paper presents a methodologically complex yet intellectually stimulating framework for conceiving quantum codes that incorporate higher-dimensional algebraic structures, paving the way for advancements in computational theory and quantum information science alike. This research contributes to a deeper understanding of the expanding frontier of quantum algorithm development and high-dimensional expander theories.

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