  Methods for electronic structure based on Gaussian and molecular orbital
discretizations offer a well established, compact representation that forms
much of the foundation of correlated quantum chemistry calculations on both
classical and quantum computers. Despite their ability to describe essential
physics with relatively few basis functions, these representations can suffer
from a quartic growth of the number of integrals. Recent results have shown
that, for some quantum and classical algorithms, moving to representations with
diagonal two-body operators can result in dramatically lower asymptotic costs,
even if the number of functions required increases significantly. We introduce
a way to interpolate between the two regimes in a systematic and controllable
manner, such that the number of functions is minimized while maintaining a
block diagonal structure of the two-body operator and desirable properties of
an original, primitive basis. Techniques are analyzed for leveraging the
structure of this new representation on quantum computers. Empirical results
for hydrogen chains suggest a scaling improvement from $O(N^{4.5})$ in
molecular orbital representations to $O(N^{2.6})$ in our representation for
quantum evolution in a fault-tolerant setting, and exhibit a constant factor
crossover at 15 to 20 atoms. Moreover, we test these methods using modern
density matrix renormalization group methods classically, and achieve excellent
accuracy with respect to the complete basis set limit with a speedup of 1-2
orders of magnitude with respect to using the primitive or Gaussian basis sets
alone. These results suggest our representation provides significant cost
reductions while maintaining accuracy relative to molecular orbital or strictly
diagonal approaches for modest-sized systems in both classical and quantum
computation for correlated systems.
