Minimizing the negativity of quantum circuits in overcomplete quasiprobability representations

The problem of simulatability of quantum processes using classical resources plays a cornerstone role for quantum computing. Quantum circuits can be simulated classically, e.g., using Monte Carlo sampling techniques applied to quasiprobability representations of circuits' basic elements, i.e., states, gates, and measurements. The effectiveness of the simulation is determined by the amount of the negativity in the representation of these basic elements. Here we develop an approach for minimizing the total negativity of a given quantum circuit with respect to quasiprobability representations, that are overcomplete, i.e., are such that the dimensionality of corresponding quasistochastic vectors and matrices is larger than the squared dimension of quantum states. Our approach includes both optimization over equivalent quasistochastic vectors and matrices, which appear due to the overcompleteness, and optimization over overcomplete frames. We demonstrate the performance of the developed approach on some illustrative cases, and show its significant advantage compared to the standard overcomplete quasistochastic representations. We also study the negativity minimization of noisy brick-wall random circuits via a combination of increasing frame dimension and applying gate merging technique. We demonstrate that the former approach appears to be more efficient in the case of a strong decoherence.