  Variational quantum algorithms are promising applications of noisy
intermediate-scale quantum (NISQ) computers. These algorithms consist of a
number of separate prepare-and-measure experiments that estimate terms in a
Hamiltonian. The number of terms can become overwhelmingly large for problems
at the scale of NISQ hardware that may soon be available. We approach this
problem from the perspective of contextuality, and use unitary partitioning
(developed independently by Izmaylov et al. [J. Chem. Theory Comput. 16, 190
(2020)]) to define variational quantum eigensolver procedures in which
additional unitary operations are appended to the ansatz preparation to reduce
the number of terms. This approach may be scaled to use all coherent resources
available after ansatz preparation. We also study the use of asymmetric
qubitization to implement the additional coherent operations with lower circuit
depth. We investigate this technique for lattice Hamiltonians, random Pauli
Hamiltonians, and electronic structure Hamiltonians. Using this technique, we
find a constant factor speedup for lattice and random Pauli Hamiltonians. For
electronic structure Hamiltonians, we prove that linear term reduction with
respect to the number of orbitals, which has been previously observed in
numerical studies, is always achievable. For systems represented on 10--30
qubits, we find that there is a reduction in the number of terms by
approximately an order of magnitude. Applied to the plane-wave dual basis
representation of fermionic Hamiltonians, however, unitary partitioning offers
only a constant factor reduction. Finally, we show that noncontextual
Hamiltonians may be reduced to effective commuting Hamiltonians using unitary
partitioning.
