  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.
