  It has been demonstrated that the critical point of the phase transition in
scalar quantum field theory with a quartic interaction in one space dimension
can be approximated via a Gaussian Effective Potential (GEP). We discuss how
this critical point can be estimated using quantum hardware. We perform quantum
computations with various lattice sizes and obtain evidence of a transition
from a symmetric to a symmetry-broken phase. We use both discrete- and
continuous-variable quantum computation. We implement the ten-site case on IBM
quantum hardware using the Variational Quantum Eigensolver (VQE) algorithm to
minimize the GEP and identify lattice level-crossings. These are extrapolated
via simulations to find the continuum critical point.
