A quantum central path algorithm for linear optimization
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866929545385345024 |
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| author | Augustino, Brandon Leng, Jiaqi Nannicini, Giacomo Terlaky, Tamás Wu, Xiaodi |
| author_facet | Augustino, Brandon Leng, Jiaqi Nannicini, Giacomo Terlaky, Tamás Wu, Xiaodi |
| contents | We propose a novel quantum algorithm for solving linear optimization problems by quantum-mechanical simulation of the central path. While interior point methods follow the central path with an iterative algorithm that works with successive linearizations of the perturbed KKT conditions, we perform a single simulation working directly with the nonlinear complementarity equations. This approach yields an algorithm for solving linear optimization problems involving $m$ constraints and $n$ variables to $\varepsilon$-optimality using $\mathcal{O} \left( \sqrt{m + n} \frac{R_{1}}{\varepsilon}\right)$ queries to an oracle that evaluates a potential function, where $R_{1}$ is an $\ell_{1}$-norm upper bound on the size of the optimal solution. In the standard gate model (i.e., without access to quantum RAM) our algorithm can obtain highly-precise solutions to LO problems using at most $$\mathcal{O} \left( \sqrt{m + n} \textsf{nnz} (A) \frac{R_1}{\varepsilon}\right)$$ elementary gates, where $\textsf{nnz} (A)$ is the total number of non-zero elements found in the constraint matrix. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_03977 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A quantum central path algorithm for linear optimization Augustino, Brandon Leng, Jiaqi Nannicini, Giacomo Terlaky, Tamás Wu, Xiaodi Quantum Physics Data Structures and Algorithms Optimization and Control We propose a novel quantum algorithm for solving linear optimization problems by quantum-mechanical simulation of the central path. While interior point methods follow the central path with an iterative algorithm that works with successive linearizations of the perturbed KKT conditions, we perform a single simulation working directly with the nonlinear complementarity equations. This approach yields an algorithm for solving linear optimization problems involving $m$ constraints and $n$ variables to $\varepsilon$-optimality using $\mathcal{O} \left( \sqrt{m + n} \frac{R_{1}}{\varepsilon}\right)$ queries to an oracle that evaluates a potential function, where $R_{1}$ is an $\ell_{1}$-norm upper bound on the size of the optimal solution. In the standard gate model (i.e., without access to quantum RAM) our algorithm can obtain highly-precise solutions to LO problems using at most $$\mathcal{O} \left( \sqrt{m + n} \textsf{nnz} (A) \frac{R_1}{\varepsilon}\right)$$ elementary gates, where $\textsf{nnz} (A)$ is the total number of non-zero elements found in the constraint matrix. |
| title | A quantum central path algorithm for linear optimization |
| topic | Quantum Physics Data Structures and Algorithms Optimization and Control |
| url | https://arxiv.org/abs/2311.03977 |