A quantum algorithm for estimating the determinant
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915268785078272 |
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| author | Giovannetti, Vittorio Lloyd, Seth Maccone, Lorenzo |
| author_facet | Giovannetti, Vittorio Lloyd, Seth Maccone, Lorenzo |
| contents | We present a quantum algorithm for estimating the matrix determinant based on quantum spectral sampling. The algorithm estimates the logarithm of the determinant of an $n \times n$ positive sparse matrix to an accuracy $ε$ in time ${\cal O}(\log n/ε^3)$, exponentially faster than previously existing classical or quantum algorithms that scale linearly in $n$. The quantum spectral sampling algorithm generalizes to estimating any quantity $\sum_j f(λ_j)$, where $λ_j$ are the matrix eigenvalues. For example, the algorithm allows the efficient estimation of the partition function $Z(β) =\sum_j e^{-βE_j}$ of a Hamiltonian system with energy eigenvalues $E_j$, and of the entropy $ S =-\sum_j p_j \log p_j$ of a density matrix with eigenvalues $p_j$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_11049 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A quantum algorithm for estimating the determinant Giovannetti, Vittorio Lloyd, Seth Maccone, Lorenzo Quantum Physics We present a quantum algorithm for estimating the matrix determinant based on quantum spectral sampling. The algorithm estimates the logarithm of the determinant of an $n \times n$ positive sparse matrix to an accuracy $ε$ in time ${\cal O}(\log n/ε^3)$, exponentially faster than previously existing classical or quantum algorithms that scale linearly in $n$. The quantum spectral sampling algorithm generalizes to estimating any quantity $\sum_j f(λ_j)$, where $λ_j$ are the matrix eigenvalues. For example, the algorithm allows the efficient estimation of the partition function $Z(β) =\sum_j e^{-βE_j}$ of a Hamiltonian system with energy eigenvalues $E_j$, and of the entropy $ S =-\sum_j p_j \log p_j$ of a density matrix with eigenvalues $p_j$. |
| title | A quantum algorithm for estimating the determinant |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2504.11049 |