  The accurate computation of ground and excited states of many-fermion quantum
systems is one of the most consequential, contemporary challenges in the
physical and computational sciences whose solution stands to benefit
significantly from the advent of quantum computing devices. Existing
methodologies using phase estimation or variational algorithms have potential
drawbacks such as deep circuits requiring substantial error correction or
non-trivial high-dimensional classical optimization. Here we introduce a
quantum solver of contracted eigenvalue equations, the quantum analogue of
classical methods for the energies and reduced density matrices of ground and
excited states. The solver does not require deep circuits or difficult
classical optimization and achieves an exponential speed-up of the exact
classical algorithms. We demonstrate the algorithm though computations on both
a quantum simulator and two IBM quantum processing units.
