  Simulating time evolution of generic quantum many-body systems using
classical numerical approaches has an exponentially growing cost either with
evolution time or with the system size. In this work, we present a polynomially
scaling hybrid quantum-classical algorithm for time evolving a one-dimensional
uniform system in the thermodynamic limit. This algorithm uses a layered
uniform sequential quantum circuit as a variational ansatz to represent
infinite translation-invariant quantum states. We show numerically that this
ansatz requires a number of parameters polynomial in the simulation time for a
given accuracy. Furthermore, this favourable scaling of the ansatz is
maintained during our variational evolution algorithm. All steps of the hybrid
optimization are designed with near-term digital quantum computers in mind.
After benchmarking the evolution algorithm on a classical computer, we
demonstrate the measurement of observables of this uniform state using a finite
number of qubits on a cloud-based quantum processing unit. With more efficient
tensor contraction schemes, this algorithm may also offer improvements as a
classical numerical algorithm.
