Nonlocal games using mixed-state symmetry-protected topological order

A violation of scalable Bell inequalities serves as a benchmark for quantum advantage, showing that a quantum computer outperforms a classical computer for a certain problem. However, in practice near term devices do not perfectly prepare pure states but rather mixed states resulted from a combination of noise channels. We investigate quantum advantage by considering thermal mixed states of one-dimensional many-body states with symmetry-protected topological (SPT) order, a known resource for measurement-based quantum computation. In the pure-state (or zero-temperature) limit, these states are known to outperform classical computers in a many-body extension of nonlocal games, provided they have a sufficiently high SPT order parameter. In contrast, we find that quantum advantage in the mixed-state case is determined by a combination of the order parameter and symmetry representations. Using the Minimally Entangled Typical Thermal States algorithm to efficiently approximate thermal expectation values, we show that quantum advantage persists in finite-sized instances of the nonlocal game up to an exactly solvable critical temperature. This temperature is relatively robust to system size, suggesting these states are useful candidates for demonstrating quantumness on noisy hardware.