  We propose a general-purpose, self-adaptive approach to construct variational
wavefunction ans\"atze for highly accurate quantum dynamics simulations based
on McLachlan's variational principle. The key idea is to dynamically expand the
variational ansatz along the time-evolution path such that the ``McLachlan
distance'', which is a measure of the simulation accuracy, remains below a set
threshold. We apply this adaptive variational quantum dynamics simulation
(AVQDS) approach to the integrable Lieb-Schultz-Mattis spin chain and the
nonintegrable mixed-field Ising model, where it captures both finite-rate and
sudden post-quench dynamics with high fidelity. The AVQDS quantum circuits that
prepare the time-evolved state are much shallower than those obtained from
first-order Trotterization and contain up to two orders of magnitude fewer CNOT
gate operations. We envision that a wide range of dynamical simulations of
quantum many-body systems on near-term quantum computing devices will be made
possible through the AVQDS framework.
