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