Playing nonlocal games with phases of quantum matter

TL;DR

This paper investigates how ground states of quantum Ising models, especially in the ferromagnetic phase, can be used to achieve quantum advantage in nonlocal parity games, extending beyond ideal GHZ states.

Contribution

It demonstrates that ground states of quantum Ising models can provide quantum advantage in nonlocal games, especially in the ferromagnetic phase, and explores the limitations in topological and SPT phases.

Findings

Quantum advantage in parity games is achievable in the ferromagnetic phase of the quantum Ising model.

The ground state of the 1+1D transverse-field Ising model offers advantage across a broad region including critical points.

Topological and SPT phases do not exhibit quantum advantage away from their fixed points.

Abstract

The parity game is an example of a nonlocal game: by sharing a Greenberger-Horne-Zeilinger (GHZ) state before playing this game, the players can win with a higher probability than is allowed by classical physics. The GHZ state of $N$ qubits is also the ground state of the ferromagnetic quantum Ising model on $N$ qubits in the limit of vanishingly weak quantum fluctuations. Motivated by this observation, we examine the probability that $N$ players who share the ground state of a generic quantum Ising model, which exhibits non-vanishing quantum fluctuations, still win the parity game using the protocol optimized for the GHZ state. Our main result is a modified parity game for which this protocol asymptotically exhibits quantum advantage in precisely the ferromagnetic phase of the quantum Ising model. We further prove that the ground state of the exactly soluble $d=1+1$ transverse-field Ising model can provide a quantum advantage for the parity game over an even wider region, which includes the entire ferromagnetic phase, the critical point and part of the paramagnetic phase. By contrast, we find examples of topological phases and symmetry-protected topological (SPT) phases of matter, namely the deconfined phase of the toric code Hamiltonian and the $\mathbb{Z}_2 \times \mathbb{Z}_2$ SPT phase in one dimension, that do not exhibit an analogous quantum advantage away from their fixed points.