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arXiv:2408.15570

Direct sum theorems beyond query complexity

Daiki Suruga

Published Aug 28, 2024• 0 citations
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Abstract

A fundamental question in computer science is: Is it harder to solve $n$ instances independently than to solve them simultaneously? This question, known as the direct sum question or direct sum theorem, has been paid much attention in several research fields. Despite its importance, however, little has been discovered in many other research fields. In this paper, we introduce a novel framework that extends to classical/quantum query complexity, PAC-learning for machine learning, statistical estimation theory, and more. Within this framework, we establish several fundamental direct sum theorems. The main contributions of this paper include: (i) establishing a complete characterization of the amortized query/oracle complexities, and (ii) proving tight direct sum theorems when the error is small. Note that in our framework, every oracle access needs to be performed \emph{classically} even in the quantum setting. This can be thought of one limitation of this work. As a direct consequence of our results, we obtain the following: (A) The first known asymptotic separation of the randomized query complexity. Specifically, we show that there is a function $f: \{0, 1\}^k \to \{0, 1\}$ and small error $\varepsilon > 0$ such that solving $n$ instances simultaneously requires the query complexity $\tilde{O}(n\sqrt{k})$ but solving one instance with the same error has the complexity $\tildeΩ(k)$. In communication complexity this type of separation was previously given in~Feder, Kushilevitz, Naor and Nisan (1995). (B) The query complexity counterpart of the ``information = amortized communication" relation, one of the most influential results in communication complexity shown by Braverman and Rao (2011) and further investigated by Braverman (2015). We hope that our results will provide further interesting applications in the future.

DOI
10.48550/arxiv.2408.15570

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This paper investigates the direct sum question for expected randomized and distributional query complexity. Our main result gives an exact characterization of the amortized expected randomized query complexity. For any total relation fff and any error tolerance ε∈[0,1]\varepsilon \in [0,1]ε∈[0,1], we prove lim⁡n→∞R‾ε(fn)n=(1−ε)R‾0(f). \lim_{n \to \infty} \frac{\overline{R}_\varepsilon(f^n)}{n} = (1 - \varepsilon) \overline{R}_0(f). limn→∞​nRε​(fn)​=(1−ε)R0​(f). Thus the amortization converts bounded-error into zero error with the exact multiplicative factor 1−ε1-\varepsilon1−ε. We also prove corresponding liminf/limsup bounds for worst-case randomized and distributional query complexity. These results improve prior direct-sum bounds that were known only up to constant factors or in restricted error regimes, and they resolve an open question posed by Blais and Brody (2019). Additionally for one-sided computation of the function OR⁡n∘f\operatorname{OR}_n \circ fORn​∘f, we obtain analogous exact amortized identities for both expected and worst-case cost. As applications, we obtain separations between amortized and single-instance costs, including unbounded separations for distributional complexity and randomized relations, and a quadratic barrier for randomized total functions.
Quantum query complexity enjoys a variety of pleasing joint computation properties: for example, a composition theorem asserting Q(f∘g)=Θ(Q(f)Q(g))Q(f\circ g)=Θ(Q(f)Q(g))Q(f∘g)=Θ(Q(f)Q(g)) for all Boolean functions fff and ggg; a direct sum theorem asserting that computing kkk copies of a function (or search problem) costs Ω(k)Ω(k)Ω(k) times as much as the cost of computing one copy; and a direct product theorem asserting that for Boolean functions, even succeeding at the direct sum problem with exponentially small probability still requires Ω(k)Ω(k)Ω(k) times the cost of computing one copy to bounded error. However, all of these results are strictly for worst-case quantum query complexity. For example, if we have a fixed distribution μμμ over inputs, the direct sum theorem says nothing about the quantum query complexity of computing kkk copies of fff when the input comes from the product distribution μkμ^kμk instead of being worst-case. (Note that while a standard Yao-type minimax theorem guarantees a hard distribution for the direct sum problem, there's no guarantee that this hard distribution is a product distribution.) A similar problem occurs for the direct product theorem and the composition theorem: none of these results respect distributions. In this work, we give distributional joint computation lower bounds for the composition, direct sum, and direct product problems. Along the way, we introduce some new tools for handling quantum query lower bounds, including (a) a new ``multiplicative'' variant of the γ2γ_2γ2​ norm (which we use in place of the multiplicative adversary method for proving the direct product theorem), and (b) a new ``Shaltiel-free'' measure of quantum query complexity, which we show characterizes the composition behavior of distributional quantum query complexity and satisfies pleasing properties.
Consider the expected query complexity of computing the k-fold direct product f⊗kf^{\otimes k}f⊗k of a function f to error ε\varepsilon ε with respect to a distribution μk\mu ^kμk . One strategy is to sequentially compute each of the k copies to error ε/k\varepsilon /kε/k with respect to μ\mu μ and apply the union bound. We prove a strong direct sum theorem showing that this naive strategy is essentially optimal. In particular, computing a direct product necessitates a blowup in both query complexity and error. Strong direct sum theorems contrast with results that only show a blowup in query complexity or error but not both. There has been a long line of such results for distributional query complexity, dating back to (Impagliazzo, Raz, Wigderson 1994) and (Nisan, Rudich, Saks 1994), but a strong direct sum theorem had been elusive. A key idea in our work is the first use of the Hardcore Theorem (Impagliazzo 1995) in the context of query complexity. We prove a new “resilience lemma" that accompanies it, showing that the hardcore of f⊗kf^{\otimes k}f⊗k is likely to remain dense under arbitrary partitions of the input space.
In communication complexity the input of a function f:X×Y→Zf:X\times Y\rightarrow Zf:X×Y→Z is distributed between two players Alice and Bob. If Alice knows only x∈Xx\in Xx∈X and Bob only y∈Yy\in Yy∈Y, how much information must Alice and Bob share to be able to elicit the value of f(x,y)f(x,y)f(x,y)? Do we need ℓ\ellℓ more resources to solve ℓ\ellℓ instances of a problem? This question is the direct sum question and has been studied in many computational models. In this paper we focus on the case of 2-party deterministic communication complexity and give a counterexample to the direct sum conjecture in its strongest form. To do so we exhibit a family of functions for which the complexity of solving ℓ\ellℓ instances is less than (1−ε)ℓ(1 -ε)\ell(1−ε)ℓ times the complexity of solving one instance for some small enough ε>0ε>0ε>0. We use a customised method in the analysis of our family of total functions, showing that one can force the alternation of rounds between players. This idea allows us to exploit the integrality of the complexity measure to create an increasing gap between the complexity of solving the instances independently with that of solving them together.
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 initiate a systematic study of pseudo-deterministic quantum algorithms. These are quantum algorithms that, for any input, output a canonical solution with high probability. Focusing on the query complexity model, our main contributions include the following complexity separations, which require new lower bound techniques specifically tailored to pseudo-determinism: - We exhibit a problem, Avoid One Encrypted String (AOES), whose classical randomized query complexity is O(1)O(1)O(1) but is maximally hard for pseudo-deterministic quantum algorithms (Ω(N)Ω(N)Ω(N) query complexity). - We exhibit a problem, Quantum-Locked Estimation (QL-Estimation), for which pseudo-deterministic quantum algorithms admit an exponential speed-up over classical pseudo-deterministic algorithms (O(log⁡(N))O(\log(N))O(log(N)) vs. Θ(N)Θ(\sqrt{N})Θ(N​)), while the randomized query complexity is O(1)O(1)O(1). Complementing these separations, we show that for any total problem RRR, pseudo-deterministic quantum algorithms admit at most a quintic advantage over deterministic algorithms, i.e., D(R)=O~(psQ(R)5)D(R) = \tilde O(psQ(R)^5)D(R)=O~(psQ(R)5). On the algorithmic side, we identify a class of quantum search problems that can be made pseudo-deterministic with small overhead, including Grover search, element distinctness, triangle finding, kkk-sum, and graph collision.