  Digital simulation of quantum dynamics by evaluating the time evolution of a
Hamiltonian is the initially proposed application of quantum computing. The
large number of quantum gates required for emulating the complete second
quantization form of the Hamiltonian, however, makes such an approach
unsuitable for near-term devices with limited gate fidelities that cause high
physical errors. In addition, Trotter error caused by noncommuting terms can
accumulate and harm the overall circuit fidelity, thus causing algorithmic
errors. In this paper, we propose a new term ordering strategy, max-commute-tsp
(MCTSP), that simultaneously mitigates both algorithmic and physical errors.
First, we improve the Trotter fidelity compared with previously proposed
optimization by reordering Pauli terms and partitioning them into commuting
families. We demonstrate the practicality of this method by constructing and
evaluating quantum circuits that simulate different molecular Hamiltonians,
together with theoretical explanations for the fidelity improvements from our
term grouping method. Second, we describe a new gate cancellation technique
that reduces the high gate counts by formulating the gate cancellation problem
as a travelling salesperson problem, together with benchmarking experiments.
Finally, we also provide benchmarking results that demonstrate the combined
advantage of max-commute-tsp to mitigate both physical and algorithmic errors
via quantum circuit simulation under realistic noise models.
