Asymptotically optimal unitary estimation in $\mathrm{SU}(3)$ by the analysis of graph Laplacian
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912604077686784 |
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| author | Yoshida, Satoshi Yoshida, Hironobu Murao, Mio |
| author_facet | Yoshida, Satoshi Yoshida, Hironobu Murao, Mio |
| contents | Unitary estimation is the task to estimate an unknown unitary operator $U\in\mathrm{SU}(d)$ with $n$ queries to the corresponding unitary operation, and its accuracy is evaluated by an estimation fidelity. We show that the optimal asymptotic fidelity of $3$-dimensional unitary estimation is given by $F_\mathrm{est}(n,d=3) = 1-\frac{56π^2}{9n^2} + O(n^{-3})$ by the analysis of the graph Laplacian based on the finite element method. We also show the lower bound on the fidelity of $d$-dimensional unitary estimation for an arbitrary $d$ given by $F_\mathrm{est}(n,d) \geq 1- \frac{(d+1)(d-1)(3d-2)(3d-1)}{6n^2} + O(n^{-3})$ achieving the best known lower bound and tight scaling with respect to $n$ and $d$. This lower bound is derived based on the unitary estimation protocol shown in [J. Kahn, Phys. Rev. A 75, 022326, 2007]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_20608 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotically optimal unitary estimation in $\mathrm{SU}(3)$ by the analysis of graph Laplacian Yoshida, Satoshi Yoshida, Hironobu Murao, Mio Quantum Physics Unitary estimation is the task to estimate an unknown unitary operator $U\in\mathrm{SU}(d)$ with $n$ queries to the corresponding unitary operation, and its accuracy is evaluated by an estimation fidelity. We show that the optimal asymptotic fidelity of $3$-dimensional unitary estimation is given by $F_\mathrm{est}(n,d=3) = 1-\frac{56π^2}{9n^2} + O(n^{-3})$ by the analysis of the graph Laplacian based on the finite element method. We also show the lower bound on the fidelity of $d$-dimensional unitary estimation for an arbitrary $d$ given by $F_\mathrm{est}(n,d) \geq 1- \frac{(d+1)(d-1)(3d-2)(3d-1)}{6n^2} + O(n^{-3})$ achieving the best known lower bound and tight scaling with respect to $n$ and $d$. This lower bound is derived based on the unitary estimation protocol shown in [J. Kahn, Phys. Rev. A 75, 022326, 2007]. |
| title | Asymptotically optimal unitary estimation in $\mathrm{SU}(3)$ by the analysis of graph Laplacian |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2509.20608 |