Derandomized Non-Abelian Homomorphism Testing in Low Soundness Regime (2405.18998v2)

Abstract: We give a randomness-efficient homomorphism test in the low soundness regime for functions, $f: G\to \mathbb{U}_t$, from an arbitrary finite group $G$ to $t\times t$ unitary matrices. We show that if such a function passes a derandomized Blum--Luby--Rubinfeld (BLR) test (using small-bias sets), then (i) it correlates with a function arising from a genuine homomorphism, and (ii) it has a non-trivial Fourier mass on a low-dimensional irreducible representation. In the full randomness regime, such a test for matrix-valued functions on finite groups implicitly appears in the works of Gowers and Hatami [Sbornik: Mathematics '17], and Moore and Russell [SIAM Journal on Discrete Mathematics '15]. Thus, our work can be seen as a near-optimal derandomization of their results. Our key technical contribution is a "degree-2 expander mixing lemma'' that shows that Gowers' $\mathrm{U}^2$ norm can be efficiently estimated by restricting it to a small-bias subset. Another corollary is a "derandomized'' version of a useful lemma due to Babai, Nikolov, and Pyber [SODA'08] and Gowers [Comb. Probab. Comput.'08].

• The paper demonstrates that if a matrix-valued function passes a derandomized BLR test, it either closely approximates a true homomorphism or exhibits significant low-dimensional Fourier mass.
• The authors introduce a degree-2 expander mixing lemma that efficiently estimates the Gowers U-norm while reducing the required random bits to near-optimal levels.
• These insights have practical implications for probabilistically checkable proofs, quantum complexity, and locally testable codes, paving the way for more resource-efficient algorithms.

Derandomized Non-Abelian Homomorphism Testing in Low Soundness Regime

The paper, "Derandomized Non-Abelian Homomorphism Testing in Low Soundness Regime" by Mittal and Roy, addresses the complexity of analyzing functions from a finite group $G$ to $t \times t$ unitary matrices. It develops a randomness-efficient homomorphism test in low soundness regimes, leveraging small-bias sets.

Overview of Contributions

Key Results

1. Correlation with Homomorphisms:
   • The authors prove that if a function $f: G \rightarrow U(t)$ passes the derandomized three-query Blum--Luby--Rubinfeld (BLR) test with some probability, it either closely correlates with a function derived from a genuine homomorphism or demonstrates significant Fourier mass on a low-dimensional irreducible representation.
   • This test uses a reduced number of random bits—approaching the optimal $(1+o(1))\log |G|$ bits required for non-Abelian groups, which is a noteworthy improvement over prior work by Gowers and Hatami (2017), and Moore and Russell (2015).
2. Technical Contributions:
   • The central technical advancement introduced is a "degree-2 expander mixing lemma," which enables efficient estimation of Gowers' $U$-norm when restricted to small-bias subsets.
   • A corollary from this is a "derandomized" version of a lemma from Babai, Nikolov, and Pyber (2008), utilized in analyzing mixing in progressions and hardness of approximation.

Theoretical Implications

The insights provided by Mittal and Roy extend beyond mere randomness-efficient testing. They lay the groundwork for further explorations in non-Abelian homomorphism testing. Specifically, the Fourier analysis and the segmental construct of irreducible representations reveal deeper connections between structure and randomness within groups. The degree-2 expander mixing lemma, in particular, might find broader applications in spectral graph theory and communication complexity.

Practical Implications

This research carries significant implications for probabilistically checkable proofs (PCPs), quantum complexity, and locally testable codes, among other areas. By reducing randomness in testing, one can construct more resource-efficient algorithms. Such developments are crucial in settings involving large-scale computations or limited random sources, demonstrating practical relevance across multiple domains in theoretical computer science and quantum information science.

Future Directions

The implications of these results suggest several avenues for future explorations:

• Approximation Algorithms: The connection with small-bias sets and Fourier analysis can be exploited to create approximation algorithms with enhanced performance guarantees.
• Quantum Homomorphism Testing: The constraints of quantum measurements and the specific structure of groups like the Pauli group call for a refined quantum testing framework, leveraging the some techniques developed in this work.
• Algorithmic Expansions: Extending the methods and findings to broader classes of groups, including infinite groups or more complex structures beyond matrix-valued functions, can yield deeper insights.

The research by Mittal and Roy delineates a sophisticated approach to derandomizing homomorphism tests, pushing the boundaries in low-soundness regimes and providing crucial technical tools for future advancements in both theoretical and practical facets of computational group theory and complexity theory.