A central challenge of quantum simulation of electronic structure is choosing a representation that scales gracefully with system size while remaining compatible with both near-term variational and fault-tolerant phase-estimation algorithms. Standard second-quantized formulations require \(\mathcal{O}(N)\) qubits and \(\mathcal{O}(N^{4})\) two-body matrix elements for \(N\) basis functions, whereas real-space first-quantized representations need only \(\mathcal{O}(\eta \log_{2} m)\) qubits for \(\eta\) electrons on an \(m\)-point mesh, with a diagonal electron-electron interaction operator.

We propose a first-quantized real-space mesh as a natural representation of the electronic structure of atoms and molecules on a quantum register. Position and shift operators on the discretized configuration space are expressed directly as Pauli words on \(w = \log_{2}(m)\) qubits per axis per electron, the kinetic operator becomes a Hermitian-symmetrized finite-volume Laplacian, and electron-nuclear and electron-electron interactions remain diagonal in the position basis. The representation scales favourably with both spatial resolution and particle number, supports fermionic antisymmetrization, and accommodates non-uniform grids that concentrate resources near the nuclei.

We then present three complementary algorithms implemented on top of this representation -- variational ground-state preparation (VQE), real-time evolution via Strang-split Trotterization, and single-ancilla iterative phase estimation -- and benchmark them on the hydrogen and helium atoms.