  Many-body quantum games provide a natural perspective on phases of matter in
quantum hardware, crisply relating the quantum correlations inherent in phases
of matter to the securing of quantum advantage at a device-oriented task. In
this paper we introduce a family of multiplayer quantum games for which
topologically ordered phases of matter are a resource yielding quantum
advantage. Unlike previous examples, quantum advantage persists away from the
exactly solvable point and is robust to arbitrary local perturbations,
irrespective of system size. We demonstrate this robustness experimentally on
Quantinuum's H1-1 quantum computer by playing the game with a continuous family
of randomly deformed toric code states that can be created with constant-depth
circuits leveraging mid-circuit measurements and unitary feedback. We are thus
able to tune through a topological phase transition - witnessed by the loss of
robust quantum advantage - on currently available quantum hardware. This
behavior is contrasted with an analogous family of deformed GHZ states, for
which arbitrarily weak local perturbations destroy quantum advantage in the
thermodynamic limit. Finally, we discuss a topological interpretation of the
game, which leads to a natural generalization involving an arbitrary number of
players.
