  We show that quantum circuit complexity for the unitary time evolution
operator of any time-independent Hamiltonian is linear in time at early times,
independent of any choices of the fundamental gates or cost metric. Deviations
from linear early-time growth arise from the commutation algebra of the gates
and are manifestly negative for any circuit, decreasing the linear growth rate
and leading to a bound on the growth rate of complexity of a circuit at early
times. We illustrate this general result by applying it to qubit and harmonic
oscillator systems, including the coupled and anharmonic oscillator. By
discretizing free and interacting scalar field theories on a lattice, we are
also able to extract the early-time behavior and dependence on the lattice
spacing of complexity of these field theories in the continuum limit,
demonstrating how this approach applies to systems that have been previously
difficult to study using existing techniques for quantum circuit complexity.
