Variational quantum state eigensolver

Abstract Extracting eigenvalues and eigenvectors of exponentially large matrices will be an important application of near-term quantum computers. The variational quantum eigensolver (VQE) treats the case when the matrix is a Hamiltonian. Here, we address the case when the matrix is a density matrix ρ. We introduce the variational quantum state eigensolver (VQSE), which is analogous to VQE in that it variationally learns the largest eigenvalues of ρ as well as a gate sequence V that prepares the corresponding eigenvectors. VQSE exploits the connection between diagonalization and majorization to define a cost function $$C={{{\rm{Tr}}}}(\tilde{\rho }H)$$ C = Tr ( ρ ̃ H ) where H is a non-degenerate Hamiltonian. Due to Schur-concavity, C is minimized when $$\tilde{\rho }=V\rho {V}^{{\dagger} }$$ ρ ̃ = V ρ V † is diagonal in the eigenbasis of H. VQSE only requires a single copy of ρ (only n qubits) per iteration of the VQSE algorithm, making it amenable for near-term implementation. We heuristically demonstrate two applications of VQSE: (1) Principal component analysis, and (2) Error mitigation.