arXiv subjects

Masayuki Miyamoto

Publications and source records attributed to Masayuki Miyamoto.

9 recordsLinked to original sources

arXiv2026

Near-Optimal Gap Amplification for Nonnegative Unentangled Quantum Proofs

We study gap amplification of the class $\mathsf{QMA}^{+}(2)$ characterized by unentangled quantum proofs whose amplitudes are nonnegative in the computational basis. This class was recently introduced by Jeronimo and Wu (STOC 2023), and its behavior depends sharply on the completeness-soundness gap: although it captures the power of $\mathsf{NEXP}$ for some small constant gap, it is equal to $\mathsf{QMA}(2)$ for larger constant gap. This is in stark contrast to $\mathsf{QMA}(2)$ where strong gap amplification is known due to the product test by Harrow and Montanaro (FOCS 2010, JACM 2013). In this paper, we prove for every completeness $c$ and soundness $s$ with $c-s=1/\mathrm{poly}(n)$, \[ \mathsf{NEXP} = \mathsf{QMA}^{+}(2,c,s) = \mathsf{QMA}^{+}\left(2,1-\frac1{\mathrm{poly}(n)},\frac14+\frac1{\mathrm{poly}(n)}\right). \] Our result gives a clean complexity phase transition for $\mathsf{QMA}^{+}(2)$ since we have \[ \mathsf{QMA}^{\mathbb R}(2) = \mathsf{QMA}^{+}\left(2,1-\frac1{\mathrm{poly}(n)},\frac14-\frac1{\mathrm{poly}(n)}\right), \] where $\mathsf{QMA}^{\mathbb R}(2)$ denotes $\mathsf{QMA}(2)$ with witnesses restricted to real amplitudes. Our amplification is thus optimal in the sense that a slight improvement of our soundness would have the collapse \[ \mathsf{QMA}^{\mathbb R}(2)=\mathsf{NEXP}. \] Our proof combines symmetric-subspace projections with the relation $\mathsf{QMA}^{+}(1)=\mathsf{NEXP}$ of Bassirian, Fefferman, and Marwaha (ITCS 2024). The main technical ingredient is a dimension-independent de Finetti theorem in Hilbert-Schmidt norm that applies when the number of registers under consideration grows logarithmically.

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arXiv2026

Distributed Statistical Zero-Knowledge Proofs via Sumcheck

We study distributed zero-knowledge proofs, introduced by Bick, Kol, and Oshman (SODA 2022). While distributed interactive proofs have advanced rapidly, general-purpose techniques for distributed zero-knowledge remain limited and mostly problem-specific. We address this gap by introducing distributed statistical zero-knowledge, requiring that each node's view be simulatable within negligible statistical distance, and by lifting the classical Sumcheck protocol (Lund, Fortnow, Karloff, and Nisan, FOCS 1990) into a modular primitive for distributed zero-knowledge proofs. Our main contribution is a distributed zero-knowledge implementation of Sumcheck. Given oracle access to a polynomial F over a finite field $\mathbb{F}$ with N variables, we design a protocol verifying claims of the form $\sum_{x\in\mathbb{F}} F(x)=a$ using $O(N)$ rounds of $O(\log |\mathbb{F}|)$-bit messages, while achieving statistical zero-knowledge and small soundness error. We apply this primitive to two problems. For non-k-colorability, we obtain an $O(n)$-round distributed statistical zero-knowledge proof deciding whether a graph is not k-colorable, for any constant k, using $O(log^{1+o(1)} n)$-bit messages. This is the first nontrivial distributed interactive proof for this problem, even without zero-knowledge guarantees. For Subgraph Counting, we obtain an $O(k \log n)$-round, $O(k \log n)$-bit distributed statistical zero-knowledge proof for counting copies of a given k-node pattern, improving previous distributed interactive proofs while additionally providing statistical zero-knowledge. Finally, we show that additional round compression of Sumcheck is problem-dependent: for non-3-colorability on constant-degree graphs, we prove a lower bound excluding $o(n/\log n)$ rounds under polynomial-time local computation.

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arXiv2026

Classical simulability of quantum circuits followed by sparse classical post-processing

We study the classical simulability of a polynomial-size quantum circuit $C_n$ on $n$ qubits followed by sparse classical post-processing (SCP) on $m$ bits, where $m \leq n \leq {\rm poly}(m)$. The SCP is described by a non-zero Boolean function $f_m$ that is classically computable in polynomial time and is sparse, i.e., has a peaked Fourier spectrum. First, we provide a necessary and sufficient condition on $C_n$ such that, for any SCP $f_m$, $C_n$ followed by $f_m$ is classically simulable. This characterization extends the result of Van den Nest and implies that various quantum circuits followed by SCP are classically simulable. Examples include IQP circuits, Clifford Magic circuits, and the quantum part of Simon's algorithm, even though these circuits alone are hard to simulate classically. Then, we consider the case where $C_n$ has constant depth $d$. While it is unlikely that, for any SCP $f_m$, $C_n$ followed by $f_m$ is classically simulable, we show that it is simulable by a polynomial-time probabilistic algorithm with access to commuting quantum circuits on $n+1$ qubits. Each such circuit consists of at most deg($f_m$) commuting gates and each commuting gate acts on at most $2^d+1$ qubits, where deg($f_m$) is the Fourier degree of $f_m$. This provides a better understanding of the hardness of simulating constant-depth quantum circuits followed by SCP.

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arXiv2025

Quantum Merlin-Arthur proof systems for synthesizing quantum states

Complexity theory typically focuses on the difficulty of solving computational problems using classical inputs and outputs, even with a quantum computer. In the quantum world, it is natural to apply a different notion of complexity, namely the complexity of synthesizing quantum states. We investigate a state-synthesizing counterpart of the class NP, referred to as stateQMA, which is concerned with preparing certain quantum states through a polynomial-time quantum verifier with the aid of a single quantum message from an all-powerful but untrusted prover. This is a subclass of the class stateQIP recently introduced by Rosenthal and Yuen (ITCS 2022), which permits polynomially many interactions between the prover and the verifier. Our main result consists of error reduction of this class and its variants with an exponentially small gap or bounded space, as well as how this class relates to other fundamental state synthesizing classes, i.e., states generated by uniform polynomial-time quantum circuits (stateBQP) and space-uniform polynomial-space quantum circuits (statePSPACE). Furthermore, we establish that the family of UQMA witnesses, considered as one of the most natural candidates for stateQMA containments, is in stateQMA. Additionally, we demonstrate that stateQCMA achieves perfect completeness.

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arXiv2024

Distributed Complexity of $P_k$-freeness: Decision and Certification

The class of graphs that do not contain a path on $k$ nodes as an induced subgraph ($P_k$-free graphs) has rich applications in the theory of graph algorithms. This paper explores the problem of deciding $P_k$-freeness from the viewpoint of distributed computing. For specific small values of $k$, we present the \textit{first} $\mathsf{CONGEST}$ algorithms specified for $P_k$-freeness, utilizing structural properties of $P_k$-free graphs in a novel way. Specifically, we show that $P_k$-freeness can be decided in $\tilde{O}(1)$ rounds for $k=4$ in the $\mathsf{broadcast\;CONGEST}$ model, and in $\tilde{O}(n)$ rounds for $k=5$ in the $\mathsf{CONGEST}$ model, where $n$ is the number of nodes in the network and $\tilde{O}(\cdot)$ hides a $\mathrm{polylog}(n)$ factor. These results significantly improve the previous $O(n^{2-2/(3k+2)})$ upper bounds by Eden et al. (Dist.~Comp.~2022). We also construct a local certification of $P_5$-freeness with certificates of size $\tilde{O}(n)$. This is nearly optimal, given our $Ω(n^{1-o(1)})$ lower bound on certificate size, and marks a significant advancement as no nontrivial bounds for proof-labeling schemes of $P_5$-freeness were previously known. For general $k$, we establish the first $\mathsf{CONGEST}$ lower bound, which is of the form $n^{2-1/Θ(k)}$. The $n^{1/Θ(k)}$ factor is unavoidable, in view of the $O(n^{2-2/(3k+2)})$ upper bound mentioned above. Additionally, our approach yields the \textit{first} superlinear lower bound on certificate size for local certification. This partially answers the conjecture on the optimal certificate size of $P_k$-freeness, asked by Bousquet et al. (arXiv:2402.12148). Finally, we propose a novel variant of the problem called ordered $P_k$ detection, and show a linear lower bound and its nontrivial connection to distributed subgraph detection.

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arXiv2023

Distributed Merlin-Arthur Synthesis of Quantum States and Its Applications

The generation and verification of quantum states are fundamental tasks for quantum information processing that have recently been investigated by Irani, Natarajan, Nirkhe, Rao and Yuen [CCC 2022], Rosenthal and Yuen [ITCS 2022], Metger and Yuen [FOCS 2023] under the term \emph{state synthesis}. This paper studies this concept from the viewpoint of quantum distributed computing, and especially distributed quantum Merlin-Arthur (dQMA) protocols. We first introduce a novel task, on a line, called state generation with distributed inputs (SGDI). In this task, the goal is to generate the quantum state $U\ketψ$ at the rightmost node of the line, where $\ketψ$ is a quantum state given at the leftmost node and $U$ is a unitary matrix whose description is distributed over the nodes of the line. We give a dQMA protocol for SGDI and utilize this protocol to construct a dQMA protocol for the Set Equality problem studied by Naor, Parter and Yogev [SODA 2020], and complement our protocol by showing classical lower bounds for this problem. Our second contribution is a dQMA protocol, based on a recent work by Zhu and Hayashi [Physical Review A, 2019], to create EPR-pairs between adjacent nodes of a network without quantum communication. As an application of this dQMA protocol, we prove a general result showing how to convert any dQMA protocol on an arbitrary network into another dQMA protocol where the verification stage does not require any quantum communication.

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arXiv2022

Distributed Quantum Interactive Proofs

The study of distributed interactive proofs was initiated by Kol, Oshman, and Saxena [PODC 2018] as a generalization of distributed decision mechanisms (proof-labeling schemes, etc.), and has received a lot of attention in recent years. In distributed interactive proofs, the nodes of an $n$-node network $G$ can exchange short messages (called certificates) with a powerful prover. The goal is to decide if the input (including $G$ itself) belongs to some language, with as few turns of interaction and as few bits exchanged between nodes and the prover as possible. There are several results showing that the size of certificates can be reduced drastically with a constant number of interactions compared to non-interactive distributed proofs. In this paper, we introduce the quantum counterpart of distributed interactive proofs: certificates can now be quantum bits, and the nodes of the network can perform quantum computation. The first result of this paper shows that by using quantum distributed interactive proofs, the number of interactions can be significantly reduced. More precisely, our result shows that for any constant~$k$, the class of languages that can be decided by a $k$-turn classical (i.e., non-quantum) distributed interactive protocol with $f(n)$-bit certificate size is contained in the class of languages that can be decided by a $5$-turn distributed quantum interactive protocol with $O(f(n))$-bit certificate size. We also show that if we allow to use shared randomness, the number of turns can be reduced to 3-turn. Since no similar turn-reduction \emph{classical} technique is currently known, our result gives evidence of the power of quantum computation in the setting of distributed interactive proofs as well.

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arXiv2021

Lower Bounds for Induced Cycle Detection in Distributed Computing

The distributed subgraph detection asks, for a fixed graph $H$, whether the $n$-node input graph contains $H$ as a subgraph or not. In the standard CONGEST model of distributed computing, the complexity of clique/cycle detection and listing has received a lot of attention recently. In this paper we consider the induced variant of subgraph detection, where the goal is to decide whether the $n$-node input graph contains $H$ as an \emph{induced} subgraph or not. We first show a $\tildeΩ(n)$ lower bound for detecting the existence of an induced $k$-cycle for any $k\geq 4$ in the CONGEST model. This lower bound is tight for $k=4$, and shows that the induced variant of $k$-cycle detection is much harder than the non-induced version. This lower bound is proved via a reduction from two-party communication complexity. We complement this result by showing that for $5\leq k\leq 7$, this $\tildeΩ(n)$ lower bound cannot be improved via the two-party communication framework. We then show how to prove stronger lower bounds for larger values of $k$. More precisely, we show that detecting an induced $k$-cycle for any $k\geq 8$ requires $\tildeΩ(n^{2-Θ{(1/k)}})$ rounds in the CONGEST model, nearly matching the known upper bound $\tilde{O}(n^{2-Θ{(1/k)}})$ of the general $k$-node subgraph detection (which also applies to the induced version) by Eden, Fiat, Fischer, Kuhn, and Oshman~[DISC 2019]. Finally, we investigate the case where $H$ is the diamond (the diamond is obtained by adding an edge to a 4-cycle, or equivalently removing an edge from a 4-clique), and show non-trivial upper and lower bounds on the complexity of the induced version of diamond detecting and listing.

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arXiv2020

Quantum Speedup for the Minimum Steiner Tree Problem

A recent breakthrough by Ambainis, Balodis, Iraids, Kokainis, Prūsis and Vihrovs (SODA'19) showed how to construct faster quantum algorithms for the Traveling Salesman Problem and a few other NP-hard problems by combining in a novel way quantum search with classical dynamic programming. In this paper, we show how to apply this approach to the minimum Steiner tree problem, a well-known NP-hard problem, and construct the first quantum algorithm that solves this problem faster than the best known classical algorithms. More precisely, the complexity of our quantum algorithm is $\mathcal{O}(1.812^k\poly(n))$, where $n$ denotes the number of vertices in the graph and $k$ denotes the number of terminals. In comparison, the best known classical algorithm has complexity $\mathcal{O}(2^k\poly(n))$.

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