A shortcut to an optimal quantum linear system solver
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866911577568968704 |
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| author | Dalzell, Alexander M. |
| author_facet | Dalzell, Alexander M. |
| contents | Given a linear system of equations $A\boldsymbol{x}=\boldsymbol{b}$, quantum linear system solvers (QLSSs) approximately prepare a quantum state $|\boldsymbol{x}\rangle$ for which the amplitudes are proportional to the solution vector $\boldsymbol{x}$. Asymptotically optimal QLSSs have query complexity $O(κ\log(1/\varepsilon))$, where $κ$ is the condition number of $A$, and $\varepsilon$ is the approximation error. However, runtime guarantees for existing optimal and near-optimal QLSSs do not have favorable constant prefactors, in part because they rely on complex or difficult-to-analyze techniques like variable-time amplitude amplification and adiabatic path-following. Here, we give a conceptually simple QLSS that does not use these techniques. If the solution norm $\lVert\boldsymbol{x}\rVert$ is known exactly, our QLSS requires only a single application of kernel reflection (a straightforward extension of the eigenstate filtering (EF) technique of previous work) and the query complexity of the QLSS is $(1+O(\varepsilon))κ\ln(2\sqrt{2}/\varepsilon)$. If the norm is unknown, our method allows it to be estimated up to a constant factor using $O(\log\log(κ))$ applications of kernel projection (a direct generalization of EF) yielding a straightforward QLSS with near-optimal $O(κ\log\log(κ)\log\log\log(κ)+κ\log(1/\varepsilon))$ total complexity. Alternatively, by reintroducing a concept from the adiabatic path-following technique, we show that $O(κ)$ complexity can be achieved for norm estimation, yielding an optimal QLSS with $O(κ\log(1/\varepsilon))$ complexity while still avoiding the need to invoke the adiabatic theorem. Finally, we compute an explicit upper bound of $56κ+1.05κ\ln(1/\varepsilon)+o(κ)$ for the complexity of our optimal QLSS. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_12086 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A shortcut to an optimal quantum linear system solver Dalzell, Alexander M. Quantum Physics Given a linear system of equations $A\boldsymbol{x}=\boldsymbol{b}$, quantum linear system solvers (QLSSs) approximately prepare a quantum state $|\boldsymbol{x}\rangle$ for which the amplitudes are proportional to the solution vector $\boldsymbol{x}$. Asymptotically optimal QLSSs have query complexity $O(κ\log(1/\varepsilon))$, where $κ$ is the condition number of $A$, and $\varepsilon$ is the approximation error. However, runtime guarantees for existing optimal and near-optimal QLSSs do not have favorable constant prefactors, in part because they rely on complex or difficult-to-analyze techniques like variable-time amplitude amplification and adiabatic path-following. Here, we give a conceptually simple QLSS that does not use these techniques. If the solution norm $\lVert\boldsymbol{x}\rVert$ is known exactly, our QLSS requires only a single application of kernel reflection (a straightforward extension of the eigenstate filtering (EF) technique of previous work) and the query complexity of the QLSS is $(1+O(\varepsilon))κ\ln(2\sqrt{2}/\varepsilon)$. If the norm is unknown, our method allows it to be estimated up to a constant factor using $O(\log\log(κ))$ applications of kernel projection (a direct generalization of EF) yielding a straightforward QLSS with near-optimal $O(κ\log\log(κ)\log\log\log(κ)+κ\log(1/\varepsilon))$ total complexity. Alternatively, by reintroducing a concept from the adiabatic path-following technique, we show that $O(κ)$ complexity can be achieved for norm estimation, yielding an optimal QLSS with $O(κ\log(1/\varepsilon))$ complexity while still avoiding the need to invoke the adiabatic theorem. Finally, we compute an explicit upper bound of $56κ+1.05κ\ln(1/\varepsilon)+o(κ)$ for the complexity of our optimal QLSS. |
| title | A shortcut to an optimal quantum linear system solver |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2406.12086 |