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Variational quantum state eigensolver
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DOI：10.1038/s41534-022-00611-6.png)
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Abstract
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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 p 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((rho) over tildeH) where H. is a non-degenerate Hamiltonian. Due to Schur-concavity, C is minimized when (rho) over tilde = 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.
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