Variational quantum state eigensolver

Abstract

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 $$C={{{\rm{Tr}}}}(\tilde{\rho }H)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo>=</mml:mo> <mml:mi>Tr</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:mover> <mml:mrow> <mml:mi>ρ</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>̃</mml:mo> </mml:mrow> </mml:mover> <mml:mi>H</mml:mi> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> where H is a non-degenerate Hamiltonian. Due to Schur-concavity, C is minimized when $$\tilde{\rho }=V\rho {V}^{{\dagger} }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mover> <mml:mrow> <mml:mi>ρ</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>̃</mml:mo> </mml:mrow> </mml:mover> <mml:mo>=</mml:mo> <mml:mi>V</mml:mi> <mml:mi>ρ</mml:mi> <mml:msup> <mml:mrow> <mml:mi>V</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>†</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> 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.