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1. Unitary Property Testing Lower Bounds by Polynomials

   https://doi.org/10.4230/LIPIcs.ITCS.2023.96  

   She, Adrian; Yuen, Henry (January 2023, Leibniz International Proceedings in Informatics (LIPIcs):14th Innovations in Theoretical Computer Science Conference (ITCS 2023))

   Tauman Kalai, Yael (Ed.)

   We study unitary property testing, where a quantum algorithm is given query access to a black-box unitary and has to decide whether it satisfies some property. In addition to containing the standard quantum query complexity model (where the unitary encodes a binary string) as a special case, this model contains "inherently quantum"; problems that have no classical analogue. Characterizing the query complexity of these problems requires new algorithmic techniques and lower bound methods. Our main contribution is a generalized polynomial method for unitary property testing problems. By leveraging connections with invariant theory, we apply this method to obtain lower bounds on problems such as determining recurrence times of unitaries, approximating the dimension of a marked subspace, and approximating the entanglement entropy of a marked state. We also present a unitary property testing-based approach towards an oracle separation between QMA and QMA(2), a long standing question in quantum complexity theory. 

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2. Efficient Interactive Proofs for Non-Deterministic Bounded Space

   https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.47  

   Cook, Joshua; Rothblum, Ron D. (September 2023, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023))

   Megow, Nicole; Smith, Adam (Ed.)

   The celebrated IP = PSPACE Theorem gives an efficient interactive proof for any bounded-space algorithm. In this work we study interactive proofs for non-deterministic bounded space computations. While Savitch’s Theorem shows that nondeterministic bounded-space algorithms can be simulated by deterministic bounded-space algorithms, this simulation has a quadratic overhead. We give interactive protocols for nondeterministic algorithms directly to get faster verifiers. More specifically, for any non-deterministic space S algorithm, we construct an interactive proof in which the verifier runs in time Õ(n+S²). This improves on the best previous bound of Õ(n+S³) and matches the result for deterministic space bounded algorithms, up to polylog(S) factors. We further generalize to alternating bounded space algorithms. For any language L decided by a time T, space S algorithm that uses d alternations, we construct an interactive proof in which the verifier runs in time Õ(n + S log(T) + S d) and the prover runs in time 2^O(S). For d = O(log(T)), this matches the best known interactive proofs for deterministic algorithms, up to polylog(S) factors, and improves on the previous best verifier time for nondeterministic algorithms by a factor of log(T). We also improve the best prior verifier time for unbounded alternations by a factor of S. Using known connections of bounded alternation algorithms to bounded depth circuits, we also obtain faster verifiers for bounded depth circuits with unbounded fan-in. 

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3. A Quadratic Speedup in Finding Nash Equilibria of Quantum Zero-Sum Games

   Vasconcelos, F; Vlatakis-Gkaragkounis, E-V; Mertikopoulos, P; Piliouras, G; Jordan, M I (November 2023, 7th International Conference on Quantum Techniques in Machine Learning)

   Recent developments in domains such as non-local games, quantum interactive proofs, and quantum generative adversarial networks have renewed interest in quantum game theory and, specifically, quantum zero-sum games. Central to classical game theory is the efficient algorithmic computation of Nash equilibria, which represent optimal strategies for both players. In 2008, Jain and Watrous proposed the first classical algorithm for computing equilibria in quantum zerosum games using the Matrix Multiplicative Weight Updates (MMWU) method to achieve a convergence rate of O(d/ε2) iterations to ε-Nash equilibria in the 4d-dimensional spectraplex. In this work, we propose a hierarchy of quantum optimization algorithms that generalize MMWU via an extra-gradient mechanism. Notably, within this proposed hierarchy, we introduce the Optimistic Matrix Multiplicative Weights Update (OMMWU) algorithm and establish its average-iterate convergence complexity as O(d/ε) iterations to ε-Nash equilibria. This quadratic speed-up relative to Jain and Watrous’ original algorithm sets a new benchmark for computing ε-Nash equilibria in quantum zero-sum games. 

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4. Nearly Optimal Pseudorandomness from Hardness

   https://doi.org/10.1145/3555307  

   Doron, Dean; Moshkovitz, Dana; Oh, Justin; Zuckerman, David (December 2022, Journal of the ACM)

   Existing proofs that deduce BPP = P from circuit lower bounds convert randomized algorithms into deterministic algorithms with a large polynomial slowdown. We convert randomized algorithms into deterministic ones with little slowdown . Specifically, assuming exponential lower bounds against randomized NP ∩ coNP circuits, formally known as randomized SVN circuits, we convert any randomized algorithm over inputs of length n running in time t ≥ n into a deterministic one running in time t 2+α for an arbitrarily small constant α > 0. Such a slowdown is nearly optimal for t close to n , since under standard complexity-theoretic assumptions, there are problems with an inherent quadratic derandomization slowdown. We also convert any randomized algorithm that errs rarely into a deterministic algorithm having a similar running time (with pre-processing). The latter derandomization result holds under weaker assumptions, of exponential lower bounds against deterministic SVN circuits. Our results follow from a new, nearly optimal, explicit pseudorandom generator fooling circuits of size s with seed length (1+α)log s , under the assumption that there exists a function f ∈ E that requires randomized SVN circuits of size at least 2 (1-α′) n , where α = O (α)′. The construction uses, among other ideas, a new connection between pseudoentropy generators and locally list recoverable codes. 

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5. Testing Calibration in Nearly-Linear Time

   Hu, L; Jambulapati, A; Tian, K; Yang, C (June 2024, https://doi.org/10.48550/arXiv.2402.13187)

   In the recent literature on machine learning and decision making, calibration has emerged as a desirable and widely-studied statistical property of the outputs of binary prediction models. However, the algorithmic aspects of measuring model calibration have remained relatively less well-explored. Motivated by [BGHN23], which proposed a rigorous framework for measuring distances to calibration, we initiate the algorithmic study of calibration through the lens of property testing. We define the problem of calibration testing from samples where given n draws from a distribution  on (predictions,binaryoutcomes), our goal is to distinguish between the case where  is perfectly calibrated, and the case where  is ε-far from calibration. We make the simple observation that the empirical smooth calibration linear program can be reformulated as an instance of minimum-cost flow on a highly-structured graph, and design an exact dynamic programming-based solver for it which runs in time O(nlog2(n)), and solves the calibration testing problem information-theoretically optimally in the same time. This improves upon state-of-the-art black-box linear program solvers requiring Ω(nω) time, where ω>2 is the exponent of matrix multiplication. We also develop algorithms for tolerant variants of our testing problem improving upon black-box linear program solvers, and give sample complexity lower bounds for alternative calibration measures to the one considered in this work. Finally, we present experiments showing the testing problem we define faithfully captures standard notions of calibration, and that our algorithms scale efficiently to accommodate large sample sizes. 

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