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1. The Polynomial Method Strikes Back: Tight Quantum Query Bounds via Dual Polynomials

   https://doi.org/10.1145/3188745.3188784  

   Bun, Mark; Kothari, Robin; Thaler, Justin (October 2020, Theory of computing)

   null (Ed.)

   The approximate degree of a Boolean function f is the least degree of a real polynomial that approximates f pointwise to error at most 1/3. The approximate degree of f is known to be a lower bound on the quantum query complexity of f (Beals et al., FOCS 1998 and J. ACM 2001). We find tight or nearly tight bounds on the approximate degree and quantum query complexities of several basic functions. Specifically, we show the following. k-Distinctness: For any constant k, the approximate degree and quantum query complexity of the k-distinctness function is Ω(n3/4−1/(2k)). This is nearly tight for large k, as Belovs (FOCS 2012) has shown that for any constant k, the approximate degree and quantum query complexity of k-distinctness is O(n3/4−1/(2k+2−4)). Image size testing: The approximate degree and quantum query complexity of testing the size of the image of a function [n]→[n] is Ω~(n1/2). This proves a conjecture of Ambainis et al. (SODA 2016), and it implies tight lower bounds on the approximate degree and quantum query complexity of the following natural problems. k-Junta testing: A tight Ω~(k1/2) lower bound for k-junta testing, answering the main open question of Ambainis et al. (SODA 2016). Statistical distance from uniform: A tight Ω~(n1/2) lower bound for approximating the statistical distance of a distribution from uniform, answering the main question left open by Bravyi et al. (STACS 2010 and IEEE Trans. Inf. Theory 2011). Shannon entropy: A tight Ω~(n1/2) lower bound for approximating Shannon entropy up to a certain additive constant, answering a question of Li and Wu (2017). Surjectivity: The approximate degree of the surjectivity function is Ω~(n3/4). The best prior lower bound was Ω(n2/3). Our result matches an upper bound of O~(n3/4) due to Sherstov (STOC 2018), which we reprove using different techniques. The quantum query complexity of this function is known to be Θ(n) (Beame and Machmouchi, Quantum Inf. Comput. 2012 and Sherstov, FOCS 2015). Our upper bound for surjectivity introduces new techniques for approximating Boolean functions by low-degree polynomials. Our lower bounds are proved by significantly refining techniques recently introduced by Bun and Thaler (FOCS 2017). 

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2. Crossing the logarithmic barrier for dynamic Boolean data structure lower bounds

   https://doi.org/10.1145/3188745.3188790  

   Larsen, Kasper Green; Weinstein, Omri; Yu, Huacheng (June 2018, STOC 2018 Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing)

   This paper proves the first super-logarithmic lower bounds on the cell probe complexity of dynamic boolean (a.k.a. decision) data structure problems, a long-standing milestone in data structure lower bounds. We introduce a new approach and use it to prove a Ω(log1.5 n) lower bound on the operational time of a wide range of boolean data structure problems, most notably, on the query time of dynamic range counting over F2. Proving an ω(lgn) lower bound for this problem was explicitly posed as one of five important open problems in the late Mihai Pǎtraşcu’s obituary . This result also implies the first ω(lgn) lower bound for the classical 2D range counting problem, one of the most fundamental data structure problems in computational geometry and spatial databases. We derive similar lower bounds for boolean versions of dynamic polynomial evaluation and 2D rectangle stabbing, and for the (non-boolean) problems of range selection and range median. Our technical centerpiece is a new way of “weakly” simulating dynamic data structures using efficient one-way communication protocols with small advantage over random guessing. This simulation involves a surprising excursion to low-degree (Chebyshev) polynomials which may be of independent interest, and offers an entirely new algorithmic angle on the “cell sampling” method of Panigrahy et al. 

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3. Shuffles and Circuits

   Roughgarden, Tim; Vassilvitskii, Sergei and (November 2018, Journal of the ACM)

   We introduce an abstract and strong model of massively parallel computation, where essentially the only restrictions are that the “fan-in” of each machine is limited to s bits, where s is smaller than the input size n, and that computation proceeds in synchronized rounds, with no communication between different machines within a round. Lower bounds on round complexity in this model apply to every computing platform that shares the most basic design principles of MapReduce-type systems. We apply a variant of the “polynomial method” to capture restrictions obeyed by all such massively parallel computations. This connection allows us to translate a lower bound on the (approximate) polynomial degree of a Boolean function to a lower bound on the round complexity of every (randomized) massively parallel computation of that function. These lower bounds apply even in the “unbounded width” version of our model, where the number of machines can be arbitrarily large. As one example of our general results, computing any non-trivial monotone graph property — such as any of the standard connectivity problems — requires a super-constant number of rounds when every machine can accept only a sub-polynomial (in n) number of input bits s. This lower bound constitutes significant progress on a major open question in the area, 

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4. Quantum Time-Space Tradeoffs for Matrix Problems

   https://doi.org/10.1145/3618260.3649700  

   Beame, Paul; Kornerup, Niels; Whitmeyer, Michael (June 2024, ACM)

   We prove lower bounds on the time and space required for quantum computers to solve a wide variety of problems involving matrices, many of which have only been analyzed classically in prior work. Using a novel way of applying recording query methods we show that for many linear algebra problems—including matrix-vector product, matrix inversion, matrix multiplication and powering—existing classical time-space tradeoffs also apply to quantum algorithms with at most a constant factor loss. For example, for almost all fixed matrices A, including the discrete Fourier transform (DFT) matrix, we prove that quantum circuits with at most T input queries and S qubits of memory require T=Ω(n^2/S) to compute matrix-vector product Ax for x ∈ {0,1}^n. We similarly prove that matrix multiplication for nxn binary matrices requires T=Ω(n^3/√S). Because many of our lower bounds are matched by deterministic algorithms with the same time and space complexity, our results show that quantum computers cannot provide any asymptotic advantage for these problems at any space bound. We also improve the previous quantum time-space tradeoff lower bounds for n× n Boolean (i.e. AND-OR) matrix multiplication from T=Ω(n^2.5/S^0.5) to T=Ω(n^2.5/S^0.25) which has optimal exponents for the powerful query algorithms to which it applies. Our method also yields improved lower bounds for classical algorithms. 

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5. 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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