Distributed Quantum Property Testing with Communication Constraints

TL;DR

This paper develops a framework for distributed quantum inference under communication constraints, analyzing quantum state certification with limited quantum communication and shared entanglement, and establishing fundamental bounds.

Contribution

It introduces a general framework for distributed quantum inference with communication limits and provides tight bounds for quantum state certification.

Findings

Sample complexity is $oxed{ extstyle rac{d^2}{2^{n_q}\, ext{epsilon}^2}}$ with public randomness.

Matching lower bounds are proved under mixedness-preserving channels.

Shared randomness is shown to be necessary for optimal complexity.

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 $\rho$, before communicating via a constrained one-way communication channel with a central node, which aims to infer some property of $\rho$. 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 $\sigma$, decide whether $\rho=\sigma$ or $\|\rho-\sigma\|_1\geq \epsilon$. 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 $\mathcal{O}(\frac{d^2}{2^{n_q}\epsilon^2})$ 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 $\Omega(\frac{d^3}{4^{n_q} \epsilon^2})$ 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.