  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 $\rho$. We
introduce the Variational Quantum State Eigensolver (VQSE), which is analogous
to VQE in that it variationally learns the largest eigenvalues of $\rho$ 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 $C=\Tr(\tilde{\rho} H)$ where $H$ is a non-degenerate
Hamiltonian. Due to Schur-concavity, $C$ is minimized when $\tilde{\rho} =
V\rho V^\dagger$ is diagonal in the eigenbasis of $H$. VQSE only requires a
single copy of $\rho$ (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.
