State-to-state fermionic mapping to quantum processors using a combinatoric representation

Emulating chemical and material systems with quantum computers is a tractable path for obtaining low energy observables for systems that are strongly correlated or require accurate treatment of large active spaces. Current techniques for mapping fermions to spin-1/2 systems center around mapping orbital operators to spin operators. In this representation, despite current techniques to remove qubits using symmetries, the size of the Hilbert space accessed on the quantum processor includes irrelevant states for the properties a quantum computationalist may be interested in. By considering the many-body fermion problem with a combinatoric representation, we demonstrate how fermionic states can be bijectively mapped to a compressed quantum Hilbert space with an analytic function. Our technique yields a significant reduction in the number of qubits required to represent the Hamiltonian subspace. We develop algorithms for near-term and fault-tolerant devices while demonstrating the effectiveness of our technique by implementing the Variational Quantum Eigensolver for molecular Hamiltonians on a compressed basis.