  The developments of quantum computing algorithms and experiments for atomic
scale simulations have largely focused on quantum chemistry for molecules,
while their application in condensed matter systems is scarcely explored. Here
we present a quantum algorithm to perform dynamical mean field theory (DMFT)
calculations for condensed matter systems on currently available quantum
computers, and demonstrate it on two quantum hardware platforms. DMFT is
required to properly describe the large class of materials with strongly
correlated electrons. The computationally challenging part arises from solving
the effective problem of an interacting impurity coupled to a bath, which
scales exponentially with system size on conventional computers. An exponential
speedup is expected on quantum computers, but the algorithms proposed so far
are based on real time evolution of the wavefunction, which requires high-depth
circuits and hence very low noise levels in the quantum hardware. Here we
propose an alternative approach, which uses the variational quantum eigensolver
(VQE) method for ground and excited states to obtain the needed quantities as
part of an exact diagonalization impurity solver. We present the algorithm for
a two site DMFT system, which we benchmark using simulations on conventional
computers as well as experiments on superconducting and trapped ion qubits,
demonstrating that this method is suitable for running DMFT calculations on
currently available quantum hardware.
