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arXiv · 2004.01372

Variational Quantum State Eigensolver

M. Cerezo·Kunal Sharma·Andrew Arrasmith·Patrick J. Coles
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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 $\rho$. 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.

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M. Cerezo, Kunal Sharma, Andrew Arrasmith, Patrick J. Coles. 2020-04-03. Variational Quantum State Eigensolver. https://doi.org/10.1038/s41534-022-00611-6

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