On Solving Linear Systems in Sublinear Time

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 x = b , we show that there is a classical algorithm that outputs a data structure for x allowing sampling and querying to the entries, where x is such that ‖ x − A + b ‖ ≤ ϵ‖ A + b ‖. This output can be viewed as a classical analogue to the output of quantum linear solvers. The complexity of our algorithm is \(\widetilde{O}(\kappa _F^6 \kappa ^2/\epsilon ^2) \) , where κ F = ‖ A ‖ F ‖ A + ‖ and κ = ‖ A ‖‖ A + ‖. This improves the previous best algorithm [Gilyén, Song and Tang, arXiv:2009.07268] of complexity \(\widetilde{O}(\kappa _F^6 \kappa ^6/\epsilon ^4) \) . Our algorithm is based on the randomized Kaczmarz method, which is a particular case of stochastic gradient descent. We also find that when A is row sparse, this method already returns an approximate solution x in time \(\widetilde{O}(\kappa _F^2) \) , while the best quantum algorithm known returns | x ⟩ in time \(\widetilde{O}(\kappa _F) \) when A is stored in the QRAM data structure. As a result, assuming access to QRAM and if A is row sparse, the speedup based on current quantum algorithms is quadratic.

show abstract

Quantum and Classical Query Complexities of Functions of Matrices

Montanaro,

Shao

2024

Proceedings of the 56th Annual ACM Symposium on Theory of Computing

View full text Add to dashboard Cite

On Solving Linear Systems in Sublinear Time

Cited by 4 publications

References 0 publications

A Queueing Network-Based Distributed Laplacian Solver

A Queueing Network-Based Distributed Laplacian Solver

Faster Quantum-inspired Algorithms for Solving Linear Systems

Quantum and Classical Query Complexities of Functions of Matrices

Contact Info

Product

Resources

About