Variational Quantum Linear Solver
Previously proposed quantum algorithms for solving linear systems of equations cannot be implemented in the near term due to the required circuit depth. Here, we propose a hybrid quantum-classical algorithm, called Variational Quantum Linear Solver (VQLS), for solving linear systems on near-term qua...
| Main Authors: | , , , , , |
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| Format: | Article |
| Language: | English |
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Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften
2023-11-01
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| Series: | Quantum |
| Online Access: | https://quantum-journal.org/papers/q-2023-11-22-1188/pdf/ |
| _version_ | 1797520800282050560 |
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| author | Carlos Bravo-Prieto Ryan LaRose M. Cerezo Yigit Subasi Lukasz Cincio Patrick J. Coles |
| author_facet | Carlos Bravo-Prieto Ryan LaRose M. Cerezo Yigit Subasi Lukasz Cincio Patrick J. Coles |
| author_sort | Carlos Bravo-Prieto |
| collection | DOAJ |
| description | Previously proposed quantum algorithms for solving linear systems of equations cannot be implemented in the near term due to the required circuit depth. Here, we propose a hybrid quantum-classical algorithm, called Variational Quantum Linear Solver (VQLS), for solving linear systems on near-term quantum computers. VQLS seeks to variationally prepare $|x\rangle$ such that $A|x\rangle\propto|b\rangle$. We derive an operationally meaningful termination condition for VQLS that allows one to guarantee that a desired solution precision $\epsilon$ is achieved. Specifically, we prove that $C \geqslant \epsilon^2 / \kappa^2$, where $C$ is the VQLS cost function and $\kappa$ is the condition number of $A$. We present efficient quantum circuits to estimate $C$, while providing evidence for the classical hardness of its estimation. Using Rigetti's quantum computer, we successfully implement VQLS up to a problem size of $1024\times1024$. Finally, we numerically solve non-trivial problems of size up to $2^{50}\times2^{50}$. For the specific examples that we consider, we heuristically find that the time complexity of VQLS scales efficiently in $\epsilon$, $\kappa$, and the system size $N$. |
| first_indexed | 2024-03-10T08:01:48Z |
| format | Article |
| id | doaj.art-30a10aba66be4cf8af68e962dbfc43fa |
| institution | Directory Open Access Journal |
| issn | 2521-327X |
| language | English |
| last_indexed | 2024-03-10T08:01:48Z |
| publishDate | 2023-11-01 |
| publisher | Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften |
| record_format | Article |
| series | Quantum |
| spelling | doaj.art-30a10aba66be4cf8af68e962dbfc43fa2023-11-22T11:21:56ZengVerein zur Förderung des Open Access Publizierens in den QuantenwissenschaftenQuantum2521-327X2023-11-017118810.22331/q-2023-11-22-118810.22331/q-2023-11-22-1188Variational Quantum Linear SolverCarlos Bravo-PrietoRyan LaRoseM. CerezoYigit SubasiLukasz CincioPatrick J. ColesPreviously proposed quantum algorithms for solving linear systems of equations cannot be implemented in the near term due to the required circuit depth. Here, we propose a hybrid quantum-classical algorithm, called Variational Quantum Linear Solver (VQLS), for solving linear systems on near-term quantum computers. VQLS seeks to variationally prepare $|x\rangle$ such that $A|x\rangle\propto|b\rangle$. We derive an operationally meaningful termination condition for VQLS that allows one to guarantee that a desired solution precision $\epsilon$ is achieved. Specifically, we prove that $C \geqslant \epsilon^2 / \kappa^2$, where $C$ is the VQLS cost function and $\kappa$ is the condition number of $A$. We present efficient quantum circuits to estimate $C$, while providing evidence for the classical hardness of its estimation. Using Rigetti's quantum computer, we successfully implement VQLS up to a problem size of $1024\times1024$. Finally, we numerically solve non-trivial problems of size up to $2^{50}\times2^{50}$. For the specific examples that we consider, we heuristically find that the time complexity of VQLS scales efficiently in $\epsilon$, $\kappa$, and the system size $N$.https://quantum-journal.org/papers/q-2023-11-22-1188/pdf/ |
| spellingShingle | Carlos Bravo-Prieto Ryan LaRose M. Cerezo Yigit Subasi Lukasz Cincio Patrick J. Coles Variational Quantum Linear Solver Quantum |
| title | Variational Quantum Linear Solver |
| title_full | Variational Quantum Linear Solver |
| title_fullStr | Variational Quantum Linear Solver |
| title_full_unstemmed | Variational Quantum Linear Solver |
| title_short | Variational Quantum Linear Solver |
| title_sort | variational quantum linear solver |
| url | https://quantum-journal.org/papers/q-2023-11-22-1188/pdf/ |
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