  Constrained problems are frequently encountered in classical and quantum
optimization. Particle conservation, in particular, is commonly imposed when
studying energy spectra of chemical and solid state systems. Though particle
number-constraining techniques have been developed for fermionic (e.g.
molecular electronic structure) Hamiltonians, analogous techniques are lacking
for non-binary and non-fermionic problems, as in the case of bosonic systems or
classical optimization problems over integer variables. Here we introduce the
binary encoded multilevel particles circuit ansatz (BEMPA) -- an ansatz which
preserves particle count by construction -- for use in quantum variational
algorithms. The key insight is to build the circuit blocks by carefully
positioning a set of symmetry-preserving 2- and 3-qubit gates. We numerically
analyze the problem of finding the ground state eigenvalues -- via the
Variational Quantum Eigensolver (VQE) algorithm -- of the Bose-Hubbard
Hamiltonian. For a range of model parameters spanning from Mott insulator to
superfluid phase, we demonstrate that our proposed circuit ansatz finds the
ground state eigenvalues within drastically shorter runtimes compared to
penalty-based strategies methods. Finally, we analyze the potential resource
benefits of changing the qubit encoding at the end of the optimization routine.
Our results attest to the efficacy of BEMPA for simulating bosonic problems for
which particle number is preserved.
