  Classical simulation of real-space quantum dynamics is challenging due to the
exponential scaling of computational cost with system dimensions. Quantum
computer offers the potential to simulate quantum dynamics with polynomial
complexity; however, existing quantum algorithms based on the split-operator
techniques require large-scale fault-tolerant quantum computers that remain
elusive in the near future. Here we present variational simulations of
real-space quantum dynamics suitable for implementation in Noisy
Intermediate-Scale Quantum (NISQ) devices. The Hamiltonian is first encoded
onto qubits using a discrete variable representation (DVR) and binary encoding
scheme. We show that direct application of real-time variational quantum
algorithm based on the McLachlan's principle is inefficient as the measurement
cost grows exponentially with the qubit number for general potential energy and
extremely small time-step size is required to achieve accurate results.
Motivated by the insights that most chemical dynamics occur in the low energy
subspace, we propose a subspace expansion method by projecting the total
Hamiltonian, including the time-dependent driving field, onto the system
low-energy eigenstate subspace using quantum computers, the exact quantum
dynamics within the subspace can then be solved classically. We show that the
measurement cost of the subspace approach grows polynomially with
dimensionality for general potential energy. Our numerical examples demonstrate
the capability of our approach, even under intense laser fields. Our work opens
the possibility of simulating chemical dynamics with NISQ hardware.
