Quantum contextuality — the impossibility of assigning pre-existing values to measurements consistently across all contexts — is the resource behind quantum computational advantage. For pure states, contextuality is well-characterized. For mixed states — the thermal states that real quantum hardware produces — the situation is murkier. Noise degrades quantum correlations. How much noise before the advantage vanishes?
The paper measures quantum advantage in thermal mixed states of one-dimensional systems with symmetry-protected topological order. The measurement: a combination of twisted string order parameters and symmetry representation expectation values. The result: quantum advantage persists up to a nonzero critical temperature in finite systems.
The catch: the critical temperature approaches zero in the thermodynamic limit. For infinite systems, any nonzero temperature kills the advantage. For finite systems — which is what real hardware implements — the advantage survives up to a temperature that depends on system size but is always positive.
The practical bound: the quantum winning probability (the advantage metric) is lower-bounded by the global fidelity with the 1D cluster state. This connects an abstract contextuality measure to a concrete experimental observable. If your device can prepare states with high fidelity to the cluster state, it can achieve quantum advantage in the contextuality game.
Finite size protects quantum advantage from thermal death. The thermodynamic limit, which usually simplifies physics, here destroys the resource. The advantage lives in the gap between finite and infinite — real hardware is never infinite, and the gap is where useful quantum computation operates.