%0 Generic %1 cerezo2020variational %A Cerezo, M. %A Sharma, Kunal %A Arrasmith, Andrew %A Coles, Patrick J. %D 2020 %K quantumcomputing %T Variational Quantum State Eigensolver %U http://arxiv.org/abs/2004.01372 %X 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 $C=Tr(H)$ where $H$ is a non-degenerate Hamiltonian. Due to Schur-concavity, $C$ is minimized when $= VV^\dagger$ is diagonal in the eigenbasis of $H$. VQSE only requires a single copy of $\rho$ (only $n$ qubits), making it amenable for near-term implementation. We demonstrate two applications of VQSE: (1) Principal component analysis, and (2) Error mitigation.

@misc{cerezo2020variational, 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 $C={\rm Tr}(\tilde{\rho} H)$ 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), making it amenable for near-term implementation. We demonstrate two applications of VQSE: (1) Principal component analysis, and (2) Error mitigation.}, added-at = {2021-01-02T22:26:17.000+0100}, author = {Cerezo, M. and Sharma, Kunal and Arrasmith, Andrew and Coles, Patrick J.}, biburl = {https://www.bibsonomy.org/bibtex/24f0a2fd514e10b8f561f180ad057e0d7/cmcneile}, description = {Variational Quantum State Eigensolver}, interhash = {252affa9686e191bcf17e80fcd6ab10a}, intrahash = {4f0a2fd514e10b8f561f180ad057e0d7}, keywords = {quantumcomputing}, note = {cite arxiv:2004.01372Comment: 10 + 4 pages, 7 figures}, timestamp = {2021-01-02T22:26:17.000+0100}, title = {Variational Quantum State Eigensolver}, url = {http://arxiv.org/abs/2004.01372}, year = 2020 }