  We introduce a new method for representing the low energy subspace of a
bosonic field theory on the qubit space of digital quantum computers. This
discretization leads to an exponentially precise description of the subspace of
the continuous theory thanks to the Nyquist-Shannon sampling theorem. The
method makes the implementation of quantum algorithms for purely bosonic
systems as well as fermion-boson interacting systems feasible. We present
algorithmic circuits for computing the time evolution of these systems. The
complexity of the algorithms scales polynomially with the system size. The
algorithm is a natural extension of the existing quantum algorithms for
simulating fermion systems in quantum chemistry and condensed matter physics to
systems involving bosons and fermion-boson interactions and has a broad variety
of potential applications in particle physics, condensed matter, etc. Due to
the relatively small amount of additional resources required by the inclusion
of bosons in our algorithm, the simulation of electron-phonon and similar
systems can be placed in the same near-future reach as the simulation of
interacting electron systems. We benchmark our algorithm by implementing it for
a $2$-site Holstein polaron problem on an Atos Quantum Learning Machine (QLM)
quantum simulator. The polaron quantum simulations are in excellent agreement
with the results obtained by exact diagonalization.
