Not yet reviewed by Pith; the record is open.

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.