Nonlocal quantum games provide proof of principle that quantum resources can confer advantage at certain tasks. They also provide a compelling way to explore the computational utility of phases of matter on quantum hardware. In a recent manuscript [Hart et al., arXiv:2403.04829] we demonstrated that a toric code resource state conferred advantage at a certain nonlocal game, which remained robust to small deformations of the resource state. In this manuscript we demonstrate that this robust advantage is a generic property of resource states drawn from topological or fracton ordered phases of quantum matter. To this end, we illustrate how several other states from paradigmatic topological and fracton ordered phases can function as resources for suitably defined nonlocal games, notably the three-dimensional toric-code phase, the X-cube fracton phase, and the double-semion phase. The key in every case is to design a nonlocal game that harnesses the characteristic braiding processes of a quantum phase as a source of contextuality. We unify the strategies that take advantage of mutual statistics by relating the operators to be measured to order and disorder parameters of an underlying generalized symmetry-breaking phase transition. Finally, we massively generalize the family of games that admit perfect strategies when codewords of homological quantum error-correcting codes are used as resources.
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 nonvanishing 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 ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ SPT phase in one dimension, that do not exhibit an analogous quantum advantage away from their fixed points.
Quantum games, such as the CHSH game, are used to illustrate the puzzle and power of entanglement. These games are played over many rounds and in each round, the participants, Alice and Bob, each receive a question bit to which they each have to give an answer bit, without being able to communicate during the game. When all possible classical answering strategies are analyzed, it is found that Alice and Bob cannot win more than 75% of the rounds. A higher percentage of wins arguably requires an exploitable bias in the random generation of the question bits or access to "non-local" resources, such as entangled pairs of particles. However, in an actual game, the number of rounds has to be finite and question regimes may come up with unequal likelihood, so there is always a possibility that Alice and Bob win by pure luck. This statistical possibility has to be transparently analyzed for practical applications such as the detection of eavesdropping in quantum communication. Similarly, when Bell tests are used in macroscopic situations to investigate the connection strength between system components and the validity of proposed causal models, the available data are limited and the possible combinations of question bits (measurement settings) may not be controlled to occur with equal likelihood. In the present work, we give a fully self-contained proof for a bound on the probability to win a CHSH game by pure luck without making the usual assumption of only small biases in the random number generators. We also show bounds for the case of unequal probabilities based on results from McDiarmid and Combes and numerically illustrate certain exploitable biases.