  Traditional methods in quantum chemistry rely on Hartree-Fock-based
Slater-determinant (SD) representations, whose underlying zeroth-order picture
assumes separability by particle. Here, we explore a radically different
approach, based on separability by Cartesian component, rather than by particle
[J. Chem. Phys., 2018, 148, 104101]. The approach appears to be very well
suited for 3D grid-based methods in quantum chemistry, and thereby also for
so-called "first-quantized" quantum computing. We first present an overview of
the approach as implemented on classical computers, including numerical results
that justify performance claims. In particular, we perform numerical
calculations with four explicit electrons that are equivalent to full-CI matrix
diagonalization with nearly $10^{15}$ SDs. We then present an implementation
for quantum computers, for which both the number of qubits, and the number of
quantum gates, may be substantially reduced in comparison with other quantum
circuitry that has been envisioned for implementing first-quantized "quantum
computational chemistry" (QCC).
