Distributed Quantum Property Testing with Communication Constraints

2026-04-07

AI Summary

This work addresses the problem of distributed quantum state certification under communication constraints, where multiple parties each hold a copy of an unknown quantum state and must collaboratively determine, via one-way quantum channels to a central referee, whether the state equals a target state. The paper establishes the first characterization of sample complexity under limited quantum communication, introducing a general framework for distributed quantum inference and highlighting the pivotal role of shared randomness. Leveraging the quantum Ingster–Suslina method, it proves that with at most $\log d$ qubits transmitted per channel, the sample complexity under public randomness is $\mathcal{O}(d^2/(2^{n_q}\varepsilon^2))$, a bound shown to be tight under mixed-state-preserving channels; in contrast, private randomness necessitates $\Omega(d^3/(4^{n_q}\varepsilon^2))$ samples.

Abstract

We introduce a framework for distributed quantum inference under communication constraints. In our model, $m$ distributed nodes each receive one copy of an unknown $d$-dimensional quantum state $ρ$, before communicating via a constrained one-way communication channel with a central node, which aims to infer some property of $ρ$. This framework generalizes the classical distributed inference framework introduced by Acharya, Canonne, and Tyagi [COLT2019], by allowing quantum resources such as quantum communication and shared entanglement. Within this setting, we focus on the fundamental problem of quantum state certification: Given a complete description of some state $σ$, decide whether $ρ=σ$ or $\|ρ-σ\|_1\geq ε$. Additionally, we focus on the case of limited quantum communication between distributed nodes and the central node. We show that when each communication channel is limited to only $n_q\leq \log d$ qubits, then the sample complexity of distributed state certification is $\mathcal{O}(\frac{d^2}{2^{n_q}ε^2})$ when public randomness is available to all nodes. Moreover, under the assumption that the channels used by the distributed nodes are mixedness-preserving, we prove a matching lower bound. We further demonstrate that shared randomness is necessary to achieve the above complexity, by proving an $Ω(\frac{d^3}{4^{n_q} ε^2})$ lower bound in the private-coin setting under the same assumption as above. Our lower bounds leverage a recently introduced quantum analogue of the celebrated Ingster-Suslina method and generalize arguments from the classical setting. Together, our work provides the first characterization of distributed quantum state certification in the regime of limited quantum communication and establishes a general framework for distributed quantum inference with communication constraints.

Problem

• distributed quantum inference
• quantum state certification
• communication constraints
• sample complexity
• quantum communication

Innovation

• distributed quantum inference
• quantum state certification
• communication constraints
• sample complexity
• shared entanglement