A Quantum Speed-Up for Approximating the Top Eigenvectors of a Matrix
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916480342294528 |
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| author | Chen, Yanlin Gilyén, András de Wolf, Ronald |
| author_facet | Chen, Yanlin Gilyén, András de Wolf, Ronald |
| contents | Finding a good approximation of the top eigenvector of a given $d\times d$ matrix $A$ is a basic and important computational problem, with many applications. We give two different quantum algorithms that, given query access to the entries of a Hermitian matrix $A$ and assuming a constant eigenvalue gap, output a classical description of a good approximation of the top eigenvector: one algorithm with time complexity $\mathcal{\tilde{O}}(d^{1.75})$ and one with time complexity $d^{1.5+o(1)}$ (the first algorithm has a slightly better dependence on the $\ell_2$-error of the approximating vector than the second, and uses different techniques of independent interest). Both of our quantum algorithms provide a polynomial speed-up over the best-possible classical algorithm, which needs $Ω(d^2)$ queries to entries of $A$, and hence $Ω(d^2)$ time. We extend this to a quantum algorithm that outputs a classical description of the subspace spanned by the top-$q$ eigenvectors in time $qd^{1.5+o(1)}$. We also prove a nearly-optimal lower bound of $\tildeΩ(d^{1.5})$ on the quantum query complexity of approximating the top eigenvector.
Our quantum algorithms run a version of the classical power method that is robust to certain benign kinds of errors, where we implement each matrix-vector multiplication with small and well-behaved error on a quantum computer, in different ways for the two algorithms. Our first algorithm estimates the matrix-vector product one entry at a time, using a new "Gaussian phase estimation" procedure. Our second algorithm uses block-encoding techniques to compute the matrix-vector product as a quantum state, from which we obtain a classical description by a new time-efficient unbiased pure-state tomography procedure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_14765 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Quantum Speed-Up for Approximating the Top Eigenvectors of a Matrix Chen, Yanlin Gilyén, András de Wolf, Ronald Quantum Physics Data Structures and Algorithms Finding a good approximation of the top eigenvector of a given $d\times d$ matrix $A$ is a basic and important computational problem, with many applications. We give two different quantum algorithms that, given query access to the entries of a Hermitian matrix $A$ and assuming a constant eigenvalue gap, output a classical description of a good approximation of the top eigenvector: one algorithm with time complexity $\mathcal{\tilde{O}}(d^{1.75})$ and one with time complexity $d^{1.5+o(1)}$ (the first algorithm has a slightly better dependence on the $\ell_2$-error of the approximating vector than the second, and uses different techniques of independent interest). Both of our quantum algorithms provide a polynomial speed-up over the best-possible classical algorithm, which needs $Ω(d^2)$ queries to entries of $A$, and hence $Ω(d^2)$ time. We extend this to a quantum algorithm that outputs a classical description of the subspace spanned by the top-$q$ eigenvectors in time $qd^{1.5+o(1)}$. We also prove a nearly-optimal lower bound of $\tildeΩ(d^{1.5})$ on the quantum query complexity of approximating the top eigenvector. Our quantum algorithms run a version of the classical power method that is robust to certain benign kinds of errors, where we implement each matrix-vector multiplication with small and well-behaved error on a quantum computer, in different ways for the two algorithms. Our first algorithm estimates the matrix-vector product one entry at a time, using a new "Gaussian phase estimation" procedure. Our second algorithm uses block-encoding techniques to compute the matrix-vector product as a quantum state, from which we obtain a classical description by a new time-efficient unbiased pure-state tomography procedure. |
| title | A Quantum Speed-Up for Approximating the Top Eigenvectors of a Matrix |
| topic | Quantum Physics Data Structures and Algorithms |
| url | https://arxiv.org/abs/2405.14765 |