
An improved quantuminspired algorithm for linear regression
We give a classical algorithm for linear regression analogous to the qua...
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A quantum algorithm for simulating nonsparse Hamiltonians
We present a quantum algorithm for simulating the dynamics of Hamiltonia...
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Spectral sparsification of matrix inputs as a preprocessing step for quantum algorithms
We study the potential utility of classical techniques of spectral spars...
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Fast inversion, preconditioned quantum linear system solvers, and fast evaluation of matrix functions
Preconditioning is the most widely used and effective way for treating i...
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Convergence of Adaptive, Randomized, Iterative Linear Solvers
Deterministic and randomized, rowaction and columnaction linear solver...
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Nearly Linear Row Sampling Algorithm for Quantile Regression
We give a row sampling algorithm for the quantile loss function with sam...
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Fast quantum subroutines for the simplex method
We propose quantum subroutines for the simplex method that avoid classic...
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Faster quantuminspired algorithms for solving linear systems
We establish an improved classical algorithm for solving linear systems in a model analogous to the QRAM that is used by quantum linear solvers. Precisely, for the linear system A = $̱, we show that there is a classical algorithm that outputs a data structure forallowing sampling and querying to the entries, whereis such that A^1≤ϵA^1. This output can be viewed as a classical analogue to the output of quantum linear solvers. The complexity of our algorithm isO(κ_F^6 κ^2/ϵ^2 ), whereκ_F = A_FA^1andκ= AA^1. This improves the previous best algorithm [Gilyén, Song and Tang, arXiv:2009.07268] of complexityO(κ_F^6 κ^6/ϵ^4). Our algorithm is based on the randomized Kaczmarz method, which is a particular case of stochastic gradient descent. We also find that whenAis row sparse, this method already returns an approximate solutionin timeO(κ_F^2), while the best quantum algorithm known returns⟩in timeO(κ_F)whenAis stored in the QRAM data structure. As a result, assuming access to QRAM and ifAis row sparse, the speedup based on current quantum algorithms is quadratic.
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