Asymptotically optimal unitary estimation in $\mathrm{SU}(3)$ by the analysis of graph Laplacian

• URL: http://arxiv.org/abs/2509.20608v1
• Date: Wed, 24 Sep 2025 23:02:33 GMT
• Title: Asymptotically optimal unitary estimation in $\mathrm{SU}(3)$ by the analysis of graph Laplacian
• Abstract summary: Unitary estimation is the task to estimate an unknown unitary operator $UinmathrmSU(d)$ with $n$ queries to the corresponding unitary operation.<n>We show that the optimal fidelity of $3$-dimensional unitary estimation is given by $F_mathrmest(n,d) = 1-frac56pi29n2 + O(n-3)$ by the analysis of the graph Laplacian.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 $F_\mathrm{est}(n,d=3) = 1-\frac{56\pi^2}{9n^2} + O(n^{-3})$ 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].

Related papers

• Finite-Sample Performance of Gradient Descent in Logistic Regression with Gaussian Design [28.74829605004241]
  We characterize the finite-sample estimation performance and convergence behavior of gradient descent (GD) on the maximum likelihood objective (i.e., the logistic loss)<n>We show that GD linearly converges to a small neighborhood of $*$ achieving an $ell$ error of order $O(sqrt|*|5d/n)$.<n>We also establish a faster local linear convergence to the same statistical error under a large $(|*|_2)$ stepsize.
  arXiv  Detail & Related papers  (2026-06-19T18:43:41Z)
• Optimal Scalar Quantization for Matrix Multiplication: Closed-Form Density and Phase Transition [50.36362492608702]
  We study entrywise scalar quantization of two matrices prior to multiplication.<n>We obtain a closed-form optimal point density [ star(u) propto exp!left(-fracu26right)bigl( (1-2)+2u22bigr), qquad u=fracx_X,, and prove a correlation-driven phase transition.
  arXiv  Detail & Related papers  (2026-03-20T01:53:44Z)
• On the $O(\rac{\sqrt{d}}{K^{1/4}})$ Convergence Rate of AdamW Measured by $\ell_1$ Norm [52.95596504632859]
  This paper establishes the convergence rate $frac1Ksum_k=1KEleft[||nabla f(xk)||_1right]leq O(fracsqrtdCK1/4)$ for AdamW measured by $ell_1$ norm.<n>We extend our result to NAdamW, an AdamW variant that employs a double-momentum mechanism, and demonstrate that it maintains the same convergence rate.
  arXiv  Detail & Related papers  (2025-05-17T05:02:52Z)
• Approximation and Generalization Abilities of Score-based Neural Network Generative Models for Sub-Gaussian Distributions [16.42916299495508]
  We study the approximation and abilities of score-based neural network generative models (SGMs)<n>Our framework is universal and can be used to establish convergence rates for SGMs under milder assumptions than previous work.
  arXiv  Detail & Related papers  (2025-05-16T05:38:28Z)
• Estimating the Mixing Coefficients of Geometrically Ergodic Markov Processes [5.00389879175348]
  We estimate the individual $beta$-mixing coefficients of a real-valued geometrically ergodic Markov process from a single sample-path. Naturally no density assumptions are required in this setting; the expected error rate is shown to be of order $mathcal O(log(n) n-1/2)$.
  arXiv  Detail & Related papers  (2024-02-11T20:17:10Z)
• On the $O(\frac{\sqrt{d}}{T^{1/4}})$ Convergence Rate of RMSProp and Its Momentum Extension Measured by $\ell_1$ Norm [54.28350823319057]
  This paper considers the RMSProp and its momentum extension and establishes the convergence rate of $frac1Tsum_k=1T.<n>Our convergence rate matches the lower bound with respect to all the coefficients except the dimension $d$.<n>Our convergence rate can be considered to be analogous to the $frac1Tsum_k=1T.
  arXiv  Detail & Related papers  (2024-02-01T07:21:32Z)
• Estimation and Inference in Distributional Reinforcement Learning [28.253677740976197]
  We show that a dataset of size $widetilde Oleft(frac|mathcalS||mathcalA|epsilon2 (1-gamma)4right)$ suffices to ensure the Kolmogorov metric and total variation metric between $hatetapi$ and $etapi$ is below $epsilon$ with high probability. Our findings give rise to a unified approach to statistical inference of a wide class of statistical functionals of $etapi$.
  arXiv  Detail & Related papers  (2023-09-29T14:14:53Z)
• High-dimensional Location Estimation via Norm Concentration for Subgamma Vectors [15.802475232604667]
  In location estimation, we are given $n$ samples from a known distribution $f$ shifted by an unknown translation $lambda$. Asymptotically, the maximum likelihood estimate achieves the Cram'er-Rao bound of error $mathcal N(0, frac1nmathcal I)$. We build on the theory using emphsmoothed estimators to bound the error for finite $n$ in terms of $mathcal I_r$, the Fisher information of the $r$-smoothed
  arXiv  Detail & Related papers  (2023-02-05T22:17:04Z)
• Estimating the minimizer and the minimum value of a regression function under passive design [72.85024381807466]
  We propose a new method for estimating the minimizer $boldsymbolx*$ and the minimum value $f*$ of a smooth and strongly convex regression function $f$. We derive non-asymptotic upper bounds for the quadratic risk and optimization error of $boldsymbolz_n$, and for the risk of estimating $f*$.
  arXiv  Detail & Related papers  (2022-11-29T18:38:40Z)
• Polyak-Ruppert Averaged Q-Leaning is Statistically Efficient [90.14768299744792]
  We study synchronous Q-learning with Polyak-Ruppert averaging (a.k.a., averaged Q-leaning) in a $gamma$-discounted MDP. We establish normality for the iteration averaged $barboldsymbolQ_T$. In short, our theoretical analysis shows averaged Q-Leaning is statistically efficient.
  arXiv  Detail & Related papers  (2021-12-29T14:47:56Z)
• Non-Parametric Estimation of Manifolds from Noisy Data [1.0152838128195467]
  We consider the problem of estimating a $d$ dimensional sub-manifold of $mathbbRD$ from a finite set of noisy samples. We show that the estimation yields rates of convergence of $n-frack2k + d$ for the point estimation and $n-frack-12k + d$ for the estimation of tangent space.
  arXiv  Detail & Related papers  (2021-05-11T02:29:33Z)
• Optimal Mean Estimation without a Variance [103.26777953032537]
  We study the problem of heavy-tailed mean estimation in settings where the variance of the data-generating distribution does not exist. We design an estimator which attains the smallest possible confidence interval as a function of $n,d,delta$.
  arXiv  Detail & Related papers  (2020-11-24T22:39:21Z)
• Accelerating Optimization and Reinforcement Learning with Quasi-Stochastic Approximation [2.294014185517203]
  This paper sets out to extend convergence theory to quasi-stochastic approximations. It is illustrated with applications to gradient-free optimization and policy gradient algorithms for reinforcement learning.
  arXiv  Detail & Related papers  (2020-09-30T04:44:45Z)
• Taking a hint: How to leverage loss predictors in contextual bandits? [63.546913998407405]
  We study learning in contextual bandits with the help of loss predictors. We show that the optimal regret is $mathcalO(minsqrtT, sqrtmathcalETfrac13)$ when $mathcalE$ is known.
  arXiv  Detail & Related papers  (2020-03-04T07:36:38Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.