  We construct an efficient autonomous quantum-circuit design algorithm for
creating efficient quantum circuits to simulate Hamiltonian many-body quantum
dynamics for arbitrary input states. The resultant quantum circuits have
optimal space complexity and employ a sequence of gates that is close to
optimal with respect to time complexity. We also devise an algorithm that
exploits commutativity to optimize the circuits for parallel execution. As
examples, we show how our autonomous algorithm constructs circuits for
simulating the dynamics of Kitaev's honeycomb model and the
Bardeen-Cooper-Schrieffer model of superconductivity. Furthermore we provide
numerical evidence that the rigorously proven upper bounds for the simulation
error here and in previous work may sometimes overestimate the error by orders
of magnitude compared to the best achievable performance for some
physics-inspired simulations.
