  Quantum subspace methods (QSMs) are a class of quantum computing algorithms
where the time-independent Schrodinger equation for a quantum system is
projected onto a subspace of the underlying Hilbert space. This projection
transforms the Schrodinger equation into an eigenvalue problem determined by
measurements carried out on a quantum device. The eigenvalue problem is then
solved on a classical computer, yielding approximations to ground- and
excited-state energies and wavefunctions. QSMs are examples of hybrid
quantum-classical methods, where a quantum device supported by classical
computational resources is employed to tackle a problem. QSMs are rapidly
gaining traction as a strategy to simulate electronic wavefunctions on quantum
computers, and thus their design, development, and application is a key
research field at the interface between quantum computation and electronic
structure. In this review, we provide a self-contained introduction to QSMs,
with emphasis on their application to the electronic structure of molecules. We
present the theoretical foundations and applications of QSMs, and we discuss
their implementation on quantum hardware, illustrating the impact of noise on
their performance.
