  We present a quantum algorithm for implementing $\phi^4$ lattice scalar field
theory on qubit computers. The field is represented in the discretized field
amplitude basis. The number of qubits and elementary gates required by the
implementation of the evolution operator is proportional to the lattice size.
The algorithm allows efficient $\phi^4$ state preparation for a large range of
input parameters in both the normal and broken-symmetry phases. The states are
prepared using a combination of variational and adiabatic evolution methods.
First, the ground state of a local Hamiltonian, which includes the $\phi^4$
self-interaction, is prepared using short variational circuits. Next, this
state is evolved by switching on the coupling between the lattice sites
adiabatically. The parameters defining the local Hamiltonian are adjustable and
constitute the input of our algorithm. We present a method to optimize these
parameters in order to reduce the adiabatic time required for state
preparation. For preparing broken-symmetry states, the adiabatic evolution
problems caused by crossing the phase transition critical line and by the
degeneracy of the broken-symmetry ground state can be addressed using an
auxiliary external field which gradually turns off during the adiabatic
process. We show that the time dependence of the external field during the
adiabatic evolution is important for addressing the broken-symmetry ground
state degeneracy. The adiabatic time dependence on the inverse error tolerance
can be reduced from quadratic to linear by using a field strength that
decreases exponentially in time relative to one that decreases linearly.
