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arXiv:2608.04152

Zero-error expectation equals amortized query complexity

Daiki Suruga

Published Aug 4, 2026• 0 citations
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Abstract

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 $f$ and any error tolerance $\varepsilon \in [0,1]$, we prove \[ \lim_{n \to \infty} \frac{\overline{R}_\varepsilon(f^n)}{n} = (1 - \varepsilon) \overline{R}_0(f). \] Thus the amortization converts bounded-error into zero error with the exact multiplicative factor $1-\varepsilon$. 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 $\operatorname{OR}_n \circ f$, 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.

DOI
10.48550/arxiv.2608.04152

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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 \to \infty} \frac{\overline{R}_\varepsilon(f^n)}{n} = (1 - \varepsilon) \overline{R}_0(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.
A fundamental question in computer science is: Is it harder to solve nnn 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→{0,1}f: \{0, 1\}^k \to \{0, 1\}f:{0,1}k→{0,1} and small error ε>0\varepsilon > 0ε>0 such that solving nnn instances simultaneously requires the query complexity O~(nk)\tilde{O}(n\sqrt{k})O~(nk​) but solving one instance with the same error has the complexity Ω~(k)\tildeΩ(k)Ω~(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.
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.
We establish two results regarding the query complexity of bounded-error randomized algorithms. Bounded-error separation theorem. There exists a total function f : {0,1}^n -> {0,1} whose epsilon-error randomized query complexity satisfies overline{R}_epsilon(f) = Omega(R(f) * log 1/epsilon). Strong direct sum theorem. For every function f and every k >= 2, the randomized query complexity of computing k instances of f simultaneously satisfies overline{R}_epsilon(f^k) = Theta(k * overline{R}_{epsilon/k}(f)). As a consequence of our two main results, we obtain an optimal superlinear direct-sum-type theorem for randomized query complexity: there exists a function f for which R(f^k) = Theta(k log k * R(f)). This answers an open question of Drucker (2012). Combining this result with the query-to-communication complexity lifting theorem of Göös, Pitassi, and Watson (2017), this also shows that there is a total function whose public-coin randomized communication complexity satisfies R^{cc}(f^k) = Theta(k log k * R^{cc}(f)), answering a question of Feder, Kushilevitz, Naor, and Nisan (1995).
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