arXiv subjects

Daiki Suruga

Publications and source records attributed to Daiki Suruga.

7 recordsLinked to original sources

arXiv2026

Zero-error information equals amortized communication complexity

The direct sum problem in computational complexity asks whether solving $n$ independent instances of a computational task inherently requires $n$ times the resources needed to solve a single instance. In this paper, we resolve a central form of the direct sum conjecture in randomized communication complexity. Specifically, we prove that the "amortized expected randomized communication complexity" of any function is exactly equal to its "zero-error information complexity"---a measure of the precise amount of information the communicating parties must reveal about their inputs to compute the function without error. This result also provides a tight characterization of the amortized "worst-case" randomized communication complexity up to a constant factor. To achieve our exact characterization, we introduce a new single-instance protocol embedding equipped with a prefix-verification mechanism to accurately localize global errors. Furthermore, we apply our new structural theorems to the fundamental Set-Disjointness problem. Our resulting exact asymptotic bounds for Set-Disjointness successfully refute a conjecture in DFHL18 regarding its scaling behavior.

cs.CC↗

arXiv2026

Zero-error expectation equals amortized query complexity

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.

cs.CC↗

arXiv2025

Another generalization of Hadamard test: Optimal sample complexities for learning functions on the unitary group

Estimating properties of unknown unitary operations is a fundamental task in quantum information science. While full unitary tomography requires a number of samples to the unknown unitary scaling linearly with the dimension (implying exponentially with the number of qubits), estimating specific functions of a unitary can be significantly more efficient. In this paper, we present a unified framework for the sample-efficient estimation of arbitrary square integrable functions $f: \mathbf{U}(d) \to \mathbb{C}$, using only access to the controlled-unitary operation. We first provide a tight characterization of the optimal sample complexity when the accuracy is measured by the averaged bias over the unitary $\mathbf{U}(d)$. We then construct a sample-efficient estimation algorithm that becomes optimal under the Probably Approximately Correct (PAC) learning criterion for various classes of functions. Applications include optimal estimation of matrix elements of irreducible representations, the trace, determinant, and general polynomial functions on $\mathbf{U}(d)$. Our technique generalize the Hadamard test and leverage tools from representation theory, yielding both lower and upper bound on sample complexity.

quant-ph↗

arXiv2024

Direct sum theorems beyond query complexity

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 $\tilde{\Omega}(k)$. 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.

cs.CC↗

arXiv2024

A probabilistic analysis on general probabilistic scheduling problems

The scheduling problem is a key class of optimization problems and has various kinds of applications both in practical and theoretical scenarios. In the scheduling problem, probabilistic analysis is a basic tool for investigating performance of scheduling algorithms, and therefore has been carried out by plenty amount of prior works. However, probabilistic analysis has several potential problems. For example, current research interest in the scheduling problem is limited to i.i.d. scenarios, due to its simplicity for analysis. This paper provides a new framework for probabilistic analysis in the scheduling problem and aims to deal with such problems. As a consequence, we obtain several theorems including a theoretical limit of the scheduling problem which can be applied to \emph{general, non-i.i.d. probability distributions}. Several information theoretic techniques, such as \emph{information-spectrum method}, turned out to be useful to prove our results. Since the scheduling problem has relations to many other research fields, our framework hopefully yields other interesting applications in the future.

cs.IT↗

arXiv2023

Matching upper bounds on symmetric predicates in quantum communication complexity

In this paper, we focus on the quantum communication complexity of functions of the form $f \circ G = f(G(X_1, Y_1), \ldots, G(X_n, Y_n))$ where $f: \{0, 1\}^n \to \{0, 1\}$ is a symmetric function, $G: \{0, 1\}^j \times \{0, 1\}^k \to \{0, 1\}$ is any function and Alice (resp. Bob) is given $(X_i)_{i \leq n}$ (resp. $(Y_i)_{i \leq n}$). Recently, Chakraborty et al. [STACS 2022] showed that the quantum communication complexity of $f \circ G$ is $O(Q(f)\mathrm{QCC}_\mathrm{E}(G))$ when the parties are allowed to use shared entanglement, where $Q(f)$ is the query complexity of $f$ and $\mathrm{QCC}_\mathrm{E}(G)$ is the exact communication complexity of $G$. In this paper, we first show that the same statement holds without shared entanglement, which generalizes their result. Based on the improved result, we next show tight upper bounds on $f \circ \mathrm{AND}_2$ for any symmetric function $f$ (where $\textrm{AND}_2 : \{0, 1\} \times \{0, 1\} \to \{0, 1\}$ denotes the 2-bit AND function) in both models: with shared entanglement and without shared entanglement. This matches the well-known lower bound by Razborov~[Izv. Math. 67(1) 145, 2003] when shared entanglement is allowed and improves Razborov's bound when shared entanglement is not allowed.

quant-ph↗

arXiv2022

Bounds on oblivious multiparty quantum communication complexity

The main conceptual contribution of this paper is investigating quantum multiparty communication complexity in the setting where communication is \emph{oblivious}. This requirement, which to our knowledge is satisfied by all quantum multiparty protocols in the literature, means that the communication pattern, and in particular the amount of communication exchanged between each pair of players at each round is fixed \emph{independently of the input} before the execution of the protocol. We show, for a wide class of functions, how to prove strong lower bounds on their oblivious quantum $k$-party communication complexity using lower bounds on their \emph{two-party} communication complexity. We apply this technique to prove tight lower bounds for all symmetric functions with \textsf{AND} gadget, and in particular obtain an optimal $\Omega(k\sqrt{n})$ lower bound on the oblivious quantum $k$-party communication complexity of the $n$-bit Set-Disjointness function. We also show the tightness of these lower bounds by giving (nearly) matching upper bounds.

quant-ph↗