Tight Bound for Estimating Expectation Values from a System of Linear Equations

• URL: http://arxiv.org/abs/2111.10485v3
• Date: Sat, 3 Sep 2022 23:08:59 GMT
• Title: Tight Bound for Estimating Expectation Values from a System of Linear Equations
• Authors: Abhijeet Alase, Robert R. Nerem, Mohsen Bagherimehrab, Peter H{\o}yer, and Barry C. Sanders
• Abstract summary: The System of Linear Equations Problem (SLEP) is specified by a complex invertible matrix $A$. The task is to estimate $xdagger Mx$, where $x$ is the solution vector to the equation $Ax = b$. We show that the quantum query complexity for SLEP in this setting is $Theta(alpha/epsilon)$.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The System of Linear Equations Problem (SLEP) is specified by a complex invertible matrix $A$, the condition number $\kappa$ of $A$, a vector $b$, a Hermitian matrix $M$ and an accuracy $\epsilon$, and the task is to estimate $x^\dagger Mx$, where $x$ is the solution vector to the equation $Ax = b$. We aim to establish a lower bound on the complexity of the end-to-end quantum algorithms for SLEP with respect to $\epsilon$, and devise a quantum algorithm that saturates this bound. To make lower bounds attainable, we consider query complexity in the setting in which a block encoding of $M$ is given, i.e., a unitary black box $U_M$ that contains $M/\alpha$ as a block for some $\alpha \in \mathbb R^+$. We show that the quantum query complexity for SLEP in this setting is $\Theta(\alpha/\epsilon)$. Our lower bound is established by reducing the problem of estimating the mean of a black box function to SLEP. Our $\Theta(\alpha/\epsilon)$ result tightens and proves the common assertion of polynomial accuracy dependence (poly$(1/\epsilon)$) for SLEP, and shows that improvement beyond linear dependence on accuracy is not possible if $M$ is provided via block encoding.

Related papers

• The Communication Complexity of Approximating Matrix Rank [50.6867896228563]
  We show that this problem has randomized communication complexity $Omega(frac1kcdot n2log|mathbbF|)$. As an application, we obtain an $Omega(frac1kcdot n2log|mathbbF|)$ space lower bound for any streaming algorithm with $k$ passes.
  arXiv  Detail & Related papers  (2024-10-26T06:21:42Z)
• Optimal Sketching for Residual Error Estimation for Matrix and Vector Norms [50.15964512954274]
  We study the problem of residual error estimation for matrix and vector norms using a linear sketch. We demonstrate that this gives a substantial advantage empirically, for roughly the same sketch size and accuracy as in previous work. We also show an $Omega(k2/pn1-2/p)$ lower bound for the sparse recovery problem, which is tight up to a $mathrmpoly(log n)$ factor.
  arXiv  Detail & Related papers  (2024-08-16T02:33:07Z)
• Misspecified $Q$-Learning with Sparse Linear Function Approximation: Tight Bounds on Approximation Error [25.777423855881878]
  We show a novel elimination-based algorithm to show one can obtain an $Oleft(Hepsilonright)$-optimal policy. We complement our upper bound with an $widetildeOmegaleft(Hepsilonright)$-optimality lower bound, giving a complete picture of this problem.
  arXiv  Detail & Related papers  (2024-07-18T15:58:04Z)
• Optimal Query Complexities for Dynamic Trace Estimation [59.032228008383484]
  We consider the problem of minimizing the number of matrix-vector queries needed for accurate trace estimation in the dynamic setting where our underlying matrix is changing slowly. We provide a novel binary tree summation procedure that simultaneously estimates all $m$ traces up to $epsilon$ error with $delta$ failure probability. Our lower bounds (1) give the first tight bounds for Hutchinson's estimator in the matrix-vector product model with Frobenius norm error even in the static setting, and (2) are the first unconditional lower bounds for dynamic trace estimation.
  arXiv  Detail & Related papers  (2022-09-30T04:15:44Z)
• Infinite-Horizon Offline Reinforcement Learning with Linear Function Approximation: Curse of Dimensionality and Algorithm [46.36534144138337]
  In this paper, we investigate the sample complexity of policy evaluation in offline reinforcement learning. Under the low distribution shift assumption, we show that there is an algorithm that needs at most $Oleft(maxleft fracleftVert thetapirightVert _24varepsilon4logfracddelta,frac1varepsilon2left(d+logfrac1deltaright)right right)$ samples to approximate the
  arXiv  Detail & Related papers  (2021-03-17T18:18:57Z)
• Model-Free Reinforcement Learning: from Clipped Pseudo-Regret to Sample Complexity [59.34067736545355]
  Given an MDP with $S$ states, $A$ actions, the discount factor $gamma in (0,1)$, and an approximation threshold $epsilon > 0$, we provide a model-free algorithm to learn an $epsilon$-optimal policy. For small enough $epsilon$, we show an improved algorithm with sample complexity.
  arXiv  Detail & Related papers  (2020-06-06T13:34:41Z)
• The Average-Case Time Complexity of Certifying the Restricted Isometry Property [66.65353643599899]
  In compressed sensing, the restricted isometry property (RIP) on $M times N$ sensing matrices guarantees efficient reconstruction of sparse vectors. We investigate the exact average-case time complexity of certifying the RIP property for $Mtimes N$ matrices with i.i.d. $mathcalN(0,1/M)$ entries.
  arXiv  Detail & Related papers  (2020-05-22T16:55:01Z)
• Fast digital methods for adiabatic state preparation [0.0]
  We present a quantum algorithm for adiabatic state preparation on a gate-based quantum computer, with complexity polylogarithmic in the inverse error.
  arXiv  Detail & Related papers  (2020-04-08T18:00:01Z)
• Agnostic Q-learning with Function Approximation in Deterministic Systems: Tight Bounds on Approximation Error and Sample Complexity [94.37110094442136]
  We study the problem of agnostic $Q$-learning with function approximation in deterministic systems. We show that if $delta = Oleft(rho/sqrtdim_Eright)$, then one can find the optimal policy using $Oleft(dim_Eright)$.
  arXiv  Detail & Related papers  (2020-02-17T18:41:49Z)

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.