  Quantum computation of the energy of molecules and materials is one of the
most promising applications of fault-tolerant quantum computers. However,
practical applications require algorithms with reduced resource requirements.
Previous work has mainly represented the Hamiltonian of the system in second
quantization. Existing methods in first quantization are limited to grid-based
approaches that do not allow for active space calculations. In this work, we
present a method to solve the generic ground-state chemistry problem in first
quantization on a fault-tolerant quantum computer using any basis set. This
allows for calculations in the active space using modern quantum chemistry
basis sets. We derive a linear-combination-of-unitaries decomposition for a
chemical Hamiltonian in first quantization and then construct an efficient
block encoding, exploiting sparsity of the Hamiltonian. For active space
calculations using a molecular orbital basis set, we achieve an asymptotic
speed up in Toffoli-gate count compared to the equivalent method in second
quantization [Berry, et. al. Quantum 3, 208 (2019)]. We also consider the dual
plane waves for materials simulations and find that in physically interesting
regimes we achieve orders of magnitude improvement in quantum resources
compared to the second quantization counterpart. In some instances, our
approach provides similar or even lower resources compared to the first
quantization plane wave algorithm of Refs.[Babbush, et. al npj Quantum Inf 5(1)
92 (2019), Su et. al PRX Quantum 2(4), 040332 (2021)] that, unlike our
approach, avoids loading the classical data from quantum memory. This work
opens up possibilities to reduce quantum resources even further using
factorization methods of a Hamiltonian or modern pseudopotentials. Furthermore,
our approach can be adapted to other applications, such as the vibrational
properties of chemical systems.
