  A successful approach to understand field theories is to resolve the physics
into different length or energy scales using the renormalization group
framework. We propose a quantum simulation of quantum field theory which
encodes field degrees of freedom in a wavelet basis---a multi-scale description
of the theory. Since wavelets are compact wavefunctions, this encoding allows
for quantum simulations to create particle excitations with compact support and
provides a natural way to associate observables in the theory to finite
resolution detectors. We show that the wavelet basis is well suited to compute
subsystem entanglement entropy by dividing the field into contributions from
short-range wavelet degrees of freedom and long-range scale degrees of freedom,
of which the latter act as renormalized modes which capture the essential
physics at a renormalization fixed point.
