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SIAM Journal on Computing

An Optimal Separation of Randomized and Quantum Query Complexity

Alexander A. Sherstov, Andrey A. Storozhenko, Pei Wu

Published Apr 26, 2023• 4 citations
Publisher

Abstract

Abstract. We prove that for every decision tree, the absolute values of the Fourier coefficients of a given order [Formula: see text] sum to at most [Formula: see text], where [Formula: see text] is the number of variables, [Formula: see text] is the tree depth, and [Formula: see text] is an absolute constant. This bound is essentially tight and settles a conjecture due to Tal [ Towards optimal separations between quantum and randomized query complexities, in Proceedings of the Sixty-First Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2020, pp. 228–239]. The bounds prior to our work degraded rapidly with [Formula: see text], becoming trivial already at [Formula: see text]. As an application, we obtain, for every integer [Formula: see text], a partial Boolean function on [Formula: see text] bits that has bounded-error quantum query complexity at most [Formula: see text] and randomized query complexity [Formula: see text]. This separation of bounded-error quantum versus randomized query complexity is best possible, by the results of Aaronson and Ambainis [ SIAM J. Comput., 47 (2018), pp. 982–1038] and Bravyi et al. [ Classical Algorithms for Forrelation, arXiv preprint, 2021]. Prior to our work, the best known separation was polynomially weaker: [Formula: see text] versus [Formula: see text] for any [Formula: see text] [A. Tal, Towards optimal separations between quantum and randomized query complexities, in Proceedings of the Sixty-First Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2020, pp. 228–239]. As another application, we obtain an essentially optimal separation of [Formula: see text] versus [Formula: see text] for bounded-error quantum versus randomized communication complexity for any [Formula: see text]. The best previous separation was polynomially weaker: [Formula: see text] versus [Formula: see text] (this is implicit in [A. Tal, Towards optimal separations between quantum and randomized query complexities, in Proceedings of the Sixty-First Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2020, pp. 228–239]).

DOI
10.1137/22m1468943

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We prove that for every parity decision tree of depth d on n variables, the sum of absolute values of Fourier coefficients at level 𝓁 is at most d^{𝓁/2} ⋅ O(𝓁 ⋅ log(n))^𝓁. Our result is nearly tight for small values of 𝓁 and extends a previous Fourier bound for standard decision trees by Sherstov, Storozhenko, and Wu (STOC, 2021). As an application of our Fourier bounds, using the results of Bansal and Sinha (STOC, 2021), we show that the k-fold Forrelation problem has (randomized) parity decision tree complexity Ω̃(n^{1-1/k}), while having quantum query complexity ⌈ k/2⌉. Our proof follows a random-walk approach, analyzing the contribution of a random path in the decision tree to the level-𝓁 Fourier expression. To carry the argument, we apply a careful cleanup procedure to the parity decision tree, ensuring that the value of the random walk is bounded with high probability. We observe that step sizes for the level-𝓁 walks can be computed by the intermediate values of level ≤ 𝓁-1 walks, which calls for an inductive argument. Our approach differs from previous proofs of Tal (FOCS, 2020) and Sherstov, Storozhenko, and Wu (STOC, 2021) that relied on decompositions of the tree. In particular, for the special case of standard decision trees we view our proof as slightly simpler and more intuitive. In addition, we prove a similar bound for noisy decision trees of cost at most d - a model that was recently introduced by Ben-David and Blais (FOCS, 2020).

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Abstract. We prove that for every decision tree, the absolute values of the Fourier coefficients of a given order [Formula: see text] sum to at most [Formula: see text], where [Formula: see text] is the number of variables, [Formula: see text] is the tree depth, and [Formula: see text] is an absolute constant. This bound is essentially tight and settles a conjecture due to Tal [ Towards optimal separations between quantum and randomized query complexities, in Proceedings of the Sixty-First Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2020, pp. 228–239]. The bounds prior to our work degraded rapidly with [Formula: see text], becoming trivial already at [Formula: see text]. As an application, we obtain, for every integer [Formula: see text], a partial Boolean function on [Formula: see text] bits that has bounded-error quantum query complexity at most [Formula: see text] and randomized query complexity [Formula: see text]. This separation of bounded-error quantum versus randomized query complexity is best possible, by the results of Aaronson and Ambainis [ SIAM J. Comput., 47 (2018), pp. 982–1038] and Bravyi et al. [ Classical Algorithms for Forrelation, arXiv preprint, 2021]. Prior to our work, the best known separation was polynomially weaker: [Formula: see text] versus [Formula: see text] for any [Formula: see text] [A. Tal, Towards optimal separations between quantum and randomized query complexities, in Proceedings of the Sixty-First Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2020, pp. 228–239]. As another application, we obtain an essentially optimal separation of [Formula: see text] versus [Formula: see text] for bounded-error quantum versus randomized communication complexity for any [Formula: see text]. The best previous separation was polynomially weaker: [Formula: see text] versus [Formula: see text] (this is implicit in [A. Tal, Towards optimal separations between quantum and randomized query complexities, in Proceedings of the Sixty-First Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2020, pp. 228–239]).
We prove that for every decision tree, the absolute values of the Fourier coefficients of a given order ℓ≥1\ell\geq1ℓ≥1 sum to at most cℓ(dℓ)(1+log⁡n)ℓ−1,c^{\ell}\sqrt{\binom{d}{\ell}(1+\log n)^{\ell-1}},cℓ(ℓd​)(1+logn)ℓ−1​, where nnn is the number of variables, ddd is the tree depth, and c>0c>0c>0 is an absolute constant. This bound is essentially tight and settles a conjecture due to Tal (arxiv 2019; FOCS 2020). The bounds prior to our work degraded rapidly with ℓ,\ell,ℓ, becoming trivial already at ℓ=d.\ell=\sqrt{d}.ℓ=d​. As an application, we obtain, for every integer k≥1,k\geq1,k≥1, a partial Boolean function on nnn bits that has bounded-error quantum query complexity at most kkk and randomized query complexity Ω~(n1−12k).\tildeΩ(n^{1-\frac{1}{2k}}).Ω~(n1−2k1​). This separation of bounded-error quantum versus randomized query complexity is best possible, by the results of Aaronson and Ambainis (STOC 2015) and Bravyi, Gosset, Grier, and Schaeffer (2021). Prior to our work, the best known separation was polynomially weaker: O(1)O(1)O(1) versus Ω(n2/3−ε)Ω(n^{2/3-ε})Ω(n2/3−ε) for any ε>0ε>0ε>0 (Tal, FOCS 2020). As another application, we obtain an essentially optimal separation of O(log⁡n)O(\log n)O(logn) versus Ω(n1−ε)Ω(n^{1-ε})Ω(n1−ε) for bounded-error quantum versus randomized communication complexity, for any ε>0.ε>0.ε>0. The best previous separation was polynomially weaker: O(log⁡n)O(\log n)O(logn) versus Ω(n2/3−ε)Ω(n^{2/3-ε})Ω(n2/3−ε) (implicit in Tal, FOCS 2020).
The query model offers a concrete setting where quantum algorithms are provably superior to randomized algorithms. Beautiful results by Bernstein-Vazirani, Simon, Aaronson, and others presented partial Boolean functions that can be computed by quantum algorithms making much fewer queries compared to their randomized analogs. To date, separations of O(1) vs. √N between quantum and randomized query complexities remain the state-of-the-art (where N is the input length), leaving open the question of whether O(1) vs. N1/2+Ω(1)separations are possible? We answer this question in the affirmative. Our separating problem is a variant of the Aaronson-Ambainis k-fold Forrelation problem. We show that our variant: 1)Can be solved by a quantum algorithm making 2O(k)queries to the inputs. 2)Requires at least ~Ω(N2(k-1)/(3k-1)) queries for any randomized algorithm. For any constant , this gives a O(1) vs. N1/2-εseparation between the quantum and randomized query complexities of partial Boolean functions. Our proof is Fourier analytical and uses new bounds on the Fourier spectrum of classical decision trees, which could be of independent interest. Looking forward, we conjecture that the Fourier bounds could be further improved in a precise manner, and show that such conjectured bounds imply optimal O(1) vs. N1-εseparations between the quantum and randomized query complexities of partial Boolean functions.
We prove that for every parity decision tree of depth ddd on nnn variables, the sum of absolute values of Fourier coefficients at level ℓ\ellℓ is at most dℓ/2⋅O(ℓ⋅log⁡(n))ℓd^{\ell/2} \cdot O(\ell \cdot \log(n))^\elldℓ/2⋅O(ℓ⋅log(n))ℓ. Our result is nearly tight for small values of ℓ\ellℓ and extends a previous Fourier bound for standard decision trees by Sherstov, Storozhenko, and Wu (STOC, 2021). As an application of our Fourier bounds, using the results of Bansal and Sinha (STOC, 2021), we show that the kkk-fold Forrelation problem has (randomized) parity decision tree complexity Ω~(n1−1/k)\tilde{\Omega}\left(n^{1-1/k}\right)Ω~(n1−1/k), while having quantum query complexity ⌈k/2⌉\lceil k/2\rceil⌈k/2⌉. Our proof follows a random-walk approach, analyzing the contribution of a random path in the decision tree to the level-ℓ\ellℓ Fourier expression. To carry the argument, we apply a careful cleanup procedure to the parity decision tree, ensuring that the value of the random walk is bounded with high probability. We observe that step sizes for the level-ℓ\ellℓ walks can be computed by the intermediate values of level ≤ℓ−1\le \ell-1≤ℓ−1 walks, which calls for an inductive argument. Our approach differs from previous proofs of Tal (FOCS, 2020) and Sherstov, Storozhenko, and Wu (STOC, 2021) that relied on decompositions of the tree. In particular, for the special case of standard decision trees we view our proof as slightly simpler and more intuitive. In addition, we prove a similar bound for noisy decision trees of cost at most ddd -- a model that was recently introduced by Ben-David and Blais (FOCS, 2020).
We achieve essentially the largest possible separation between quantum and classical query complexities. We do so using a property-testing problem called Forrelation, where one needs to decide whether one Boolean function is highly correlated with the Fourier transform of a second function. This problem can be solved using 1 quantum query, yet we show that any randomized algorithm needs Ω(N/log⁡N)\Omega(\sqrt{N}/\log N)Ω(N​/logN) queries (improving an Ω(N1/4)\Omega(N^{1/4})Ω(N1/4) lower bound of Aaronson). Conversely, we show that this 1 versus Ω~(N)\widetilde{\Omega}(\sqrt{N})Ω(N​) separation is optimal: indeed, any ttt-query quantum algorithm whatsoever can be simulated by an O(N1−1/2t)O(N^{1-1/2t})O(N1−1/2t)-query randomized algorithm. Thus, resolving an open question of Buhrman et al. [ SIAM J. Comput., 37 (2008), pp. 1387--1400] from 2002, there is no partial Boolean function whose quantum query complexity is constant and whose randomized query complexity is linear. We conjecture that a natural generalization of Forrelation achieves the optimal ttt versus Ω(N1−1/2t)\Omega(N^{1-1/2t})Ω(N1−1/2t) separation for all ttt. As a bonus, we show that this generalization is BQP{BQP}BQP-complete. This yields what is arguably the simplest BQP{BQP}BQP-complete problem yet known and gives a second sense in which Forrelation “captures the maximum power of quantum computation.”
We prove that for every parity decision tree of depth d on n variables, the sum of absolute values of Fourier coefficients at level 𝓁 is at most d^{𝓁/2} ⋅ O(𝓁 ⋅ log(n))^𝓁. Our result is nearly tight for small values of 𝓁 and extends a previous Fourier bound for standard decision trees by Sherstov, Storozhenko, and Wu (STOC, 2021). As an application of our Fourier bounds, using the results of Bansal and Sinha (STOC, 2021), we show that the k-fold Forrelation problem has (randomized) parity decision tree complexity Ω̃(n^{1-1/k}), while having quantum query complexity ⌈ k/2⌉. Our proof follows a random-walk approach, analyzing the contribution of a random path in the decision tree to the level-𝓁 Fourier expression. To carry the argument, we apply a careful cleanup procedure to the parity decision tree, ensuring that the value of the random walk is bounded with high probability. We observe that step sizes for the level-𝓁 walks can be computed by the intermediate values of level ≤ 𝓁-1 walks, which calls for an inductive argument. Our approach differs from previous proofs of Tal (FOCS, 2020) and Sherstov, Storozhenko, and Wu (STOC, 2021) that relied on decompositions of the tree. In particular, for the special case of standard decision trees we view our proof as slightly simpler and more intuitive. In addition, we prove a similar bound for noisy decision trees of cost at most d - a model that was recently introduced by Ben-David and Blais (FOCS, 2020).