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Quantum computers have the potential for an exponential speedup of classicalmolecular computations. However, existing algorithms have limitations; quantumphase estimation (QPE) algorithms are intractable on current hardware whilevariational quantum eigensolvers (VQE) are dependent upon approximate wavefunctions without guaranteed convergence. In this Article we present analgorithm that yields verifiably exact solutions of the many-electronSchr\"odinger equation. Rather than solve the Schr\"odinger equation directly,we solve its contraction over all electrons except two, known as the contractedSchr\"odinger equation (CSE). The CSE generates an exact wave function ansatz,constructed from a product of two-body-based non-unitary transformations, thatscales polynomially with molecular size and hence, provides a potentiallyexponential acceleration of classical molecular electronic structurecalculations on ideal quantum devices. We demonstrate the algorithm on bothquantum simulators and noisy quantum computers with applications to H$_{2}$dissociation and the rectangle-to-square transition in H$_{4}$. The CSE quantumalgorithm, which is a type of contracted quantum eigensolver (CQE), provides asignificant step towards realizing verifiably accurate but scalable molecularsimulations on quantum devices.

Verifiably Exact Solution of the Electronic Schr\"odinger Equation on Quantum Devices