The dimension of the Hilbert space describing a quantum system scales exponentially with system size. For example, consider a system of N spins localized on fixed lattice sites. The dimension of the on-site basis is 2, because the state of each spin can be described as a superposition of spin-up and spin-down, denoted | ↑ ↑ ⟩ and | ↓ ↓ ⟩ . The full system has dimension 2 N , and the Hamiltonian represented as a matrix has size 2 N × × 2 N . This implies that computation time and memory requirements scale very unfavorably in exact diagonalization. In practice, the memory requirements can be reduced by taking advantage of symmetry of the problem, imposing conservation laws, working with sparse matrices, or using other techniques.