Tight Bound for Estimating Expectation Values from a System of Linear Equations
| Autor: | Alase, Abhijeet, Nerem, Robert R., Bagherimehrab, Mohsen, Høyer, Peter, Sanders, Barry C. |
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| Rok vydání: | 2021 |
| Předmět: | |
| Zdroj: | Phys. Rev. Research, Vol 4, page 023237, 2022 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1103/PhysRevResearch.4.023237 |
| Popis: | 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. Comment: 24 pages |
| Databáze: | arXiv |
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