Gauge-invariant QMETTS with mutually unbiased physical bases for $Z_2$ lattice gauge theories at finite temperature and density
In quantum computations of gauge theories at finite temperature and finite density, enforcing Gauss's law for all states contributing to the thermal ensemble is a nontrivial challenge. In this work, we adopt the Quantum Minimally Entangled Typical Thermal States (QMETTS) algorithm for $Z_2$ gauge-constrained systems and propose a method for computing finite-temperature and finite-density expectation values without eliminating gauge degrees of freedom. To preserve gauge invariance while maintaining efficient sampling, we introduce measurement bases that are gauge invariant and mutually unbiased within the physical subspace. We show that such measurement bases can be constructed efficiently for $Z_2$ lattice gauge theories in general dimensions and arbitrary boundary conditions by exploiting the correspondence between $Z_2$ lattice gauge theories and the stabilizer formalism. Furthermore, since expectation-value estimation on quantum hardware is inherently affected by shot noise, we explicitly incorporate shot noise into the analysis. We find that the single-shot strategy is near optimal under a fixed total number of circuit executions in terms of the variance. This result indicates that it is generally more efficient to generate more QMETTS samples than to accurately estimate the expectation value for each individual pure state. We validate the proposed method numerically in a $(1+1)$-dimensional $Z_2$ lattice gauge theory coupled to staggered fermions. Our results provide a gauge-invariant framework for finite-temperature and finite-density calculations on quantum devices.