Variational quantum state eigensolver
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...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
Nature Portfolio
2022-09-01
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| Series: | npj Quantum Information |
| Online Access: | https://doi.org/10.1038/s41534-022-00611-6 |
| _version_ | 1798001516494192640 |
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| author | M. Cerezo Kunal Sharma Andrew Arrasmith Patrick J. Coles |
| author_facet | M. Cerezo Kunal Sharma Andrew Arrasmith Patrick J. Coles |
| author_sort | M. Cerezo |
| collection | DOAJ |
| description | 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 $$C={{{\rm{Tr}}}}(\tilde{\rho }H)$$ C = Tr ( ρ ̃ H ) where H is a non-degenerate Hamiltonian. Due to Schur-concavity, C is minimized when $$\tilde{\rho }=V\rho {V}^{{\dagger} }$$ ρ ̃ = V ρ V † 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. |
| first_indexed | 2024-04-11T11:37:28Z |
| format | Article |
| id | doaj.art-00a0d0d146ba4fbcacd2087066c95718 |
| institution | Directory Open Access Journal |
| issn | 2056-6387 |
| language | English |
| last_indexed | 2024-04-11T11:37:28Z |
| publishDate | 2022-09-01 |
| publisher | Nature Portfolio |
| record_format | Article |
| series | npj Quantum Information |
| spelling | doaj.art-00a0d0d146ba4fbcacd2087066c957182022-12-22T04:25:55ZengNature Portfolionpj Quantum Information2056-63872022-09-018111110.1038/s41534-022-00611-6Variational quantum state eigensolverM. Cerezo0Kunal Sharma1Andrew Arrasmith2Patrick J. Coles3Theoretical Division, MS B213, Los Alamos National LaboratoryTheoretical Division, MS B213, Los Alamos National LaboratoryTheoretical Division, MS B213, Los Alamos National LaboratoryTheoretical Division, MS B213, Los Alamos National LaboratoryAbstract 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 $$C={{{\rm{Tr}}}}(\tilde{\rho }H)$$ C = Tr ( ρ ̃ H ) where H is a non-degenerate Hamiltonian. Due to Schur-concavity, C is minimized when $$\tilde{\rho }=V\rho {V}^{{\dagger} }$$ ρ ̃ = V ρ V † 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.https://doi.org/10.1038/s41534-022-00611-6 |
| spellingShingle | M. Cerezo Kunal Sharma Andrew Arrasmith Patrick J. Coles Variational quantum state eigensolver npj Quantum Information |
| title | Variational quantum state eigensolver |
| title_full | Variational quantum state eigensolver |
| title_fullStr | Variational quantum state eigensolver |
| title_full_unstemmed | Variational quantum state eigensolver |
| title_short | Variational quantum state eigensolver |
| title_sort | variational quantum state eigensolver |
| url | https://doi.org/10.1038/s41534-022-00611-6 |
| work_keys_str_mv | AT mcerezo variationalquantumstateeigensolver AT kunalsharma variationalquantumstateeigensolver AT andrewarrasmith variationalquantumstateeigensolver AT patrickjcoles variationalquantumstateeigensolver |