Concrete Functions in Private Simultaneous Quantum Messages: Quantum Advantages and Structural Limitations
Year: 2026
Type: preprint
Abstract: Private simultaneous quantum messages (PSQM) provide a minimal model of noninteractive secure multiparty quantum computation, yet their communication complexity for concrete functions remains poorly understood. We show that familiar total functions already exhibit both genuine quantum advantages and strong structural limitations. For three-party equality on $n$-bit inputs, which determines whether the players' $n$-bit strings are identical, we construct an entanglement-assisted protocol with asymptotic communication $(1.633\ldots)n$ qubits, whereas every perfectly correct protocol without prior entanglement requires at least $3n$ qubits. For three-bit majority, one shared GHZ state and one transmitted qubit per player suffice for perfect correctness and perfect privacy, reducing the communication from the optimal classical cost of six bits to three qubits. Three-bit AND behaves differently. No perfectly correct and perfectly private protocol exists when the players share an arbitrary three-qubit state, apply arbitrary local unitary encodings, and transmit one qubit each. However, increasing the local dimension overcomes this obstruction: one shared three-qutrit GHZ state and one transmitted qutrit per player yield a one-shot protocol with total communication $3\log_2 3$. Together with the six-bit classical lower bound, this gives a strict quantum communication advantage. More generally, for every $k\geq 3$, we construct a perfectly correct and perfectly private one-shot protocol for $k$-bit AND using one qudit of dimension at most $k$ per player, and hence at most $k\log_2 k$ qubits in total.
Source: arXiv (Cornell University)
Author: Nishimura, Harumichi
Institution: Nagoya University
Language: English
Subfield: Atomic and Molecular Physics, and Optics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
Awards: Foundational Framework for designing resource-efficient quantum algorithms with security and performance guarantees, G3578047007, G4627739467, Foundations of Secure Distributed Quantum Computing on Medium-Scale Quantum Computers, Constructing theory of efficient verifications on quantum networks
OpenAlex ID: W7222981377