# Quantum algorithm for linear systems of equations.

@article{Harrow2009QuantumAF, title={Quantum algorithm for linear systems of equations.}, author={Aram Wettroth Harrow and Avinatan Hassidim and Seth Lloyd}, journal={Physical review letters}, year={2009}, volume={103 15}, pages={ 150502 } }

Solving linear systems of equations is a common problem that arises both on its own and as a subroutine in more complex problems: given a matrix A and a vector b(-->), find a vector x(-->) such that Ax(-->) = b(-->). We consider the case where one does not need to know the solution x(-->) itself, but rather an approximation of the expectation value of some operator associated with x(-->), e.g., x(-->)(dagger) Mx(-->) for some matrix M. In this case, when A is sparse, N x N and has condition…

## 1,443 Citations

### Quantum Linear System Algorithm for Dense Matrices.

- Computer SciencePhysical review letters
- 2018

A quantum algorithm is described that achieves a sparsity-independent runtime scaling of O(κ^{2}sqrt[n]polylog(n)/ε) for an n×n dimensional A with bounded spectral norm, which amounts to a polynomial improvement over known quantum linear system algorithms when applied to dense matrices.

### Quantum Linear System Algorithm for General Matrices in System Identification

- Computer ScienceEntropy
- 2022

A modified quantum scheme to obtain the quantum state x corresponding to the solution of the linear system of equations in O(κ2rpolylog(mn)/ϵ) time for a general m×n dimensional A, which is superior to existing quantum algorithms, where κ is the condition number, r is the rank of matrix A and ϵ is the precision parameter.

### Quantum algorithm for solving matrix equations of the form AX = B

- Computer ScienceLaser Physics Letters
- 2022

This paper presents a quantum algorithm to solve matrix equations of the form AX = B, and demonstrates that in this algorithm the scale of complexity is O(lognN) .

### QUBO Formulations for a System of Linear Equations

- Computer Science
- 2022

This work forms unconstrained binary optimization (QUBO) models for a linear system solving problem using a linear least-square problem with binary representation of the solution and validate those QUBO models on the D-Wave system and discuss the results.

### A near-term quantum algorithm for solving linear systems of equations based on the Woodbury identity

- Computer Science
- 2022

A quantum algorithm for solving linear systems of equations that is based on the Woodbury identity, which analytically describes the inverse of a matrix that is a low-rank modiﬁcation of another (easily-invertible) matrix that can produce accurate results on current hardware.

### Quantum linear systems algorithms: a primer

- Computer ScienceArXiv
- 2018

The Harrow-Hassidim-Lloyd quantum algorithm for sampling from the solution of a linear system provides an exponential speed-up over its classical counterpart, and a linear solver based on the quantum singular value estimation subroutine is discussed.

### Proving the BQP-Completeness of the Quantum Linear Systems Problem Using a Clock Construction

- Mathematics, Computer Science
- 2022

In virtually any technical field, the classical problem of solving a linear system of equations has a wide range of applications, and the best classical algorithms for solving linear systems of equations scale with time polynomial in N.

### Quantum algorithms for the polynomial eigenvalue problems

- Computer ScienceArXiv
- 2020

This work attempts to solve polynomial eigenvalue problems (PEPs) in a quantum computer using the Fourier spectral method to solve ordinary differential equations (ODEs) and provides two quantum algorithms to solve PEPs by extending the quantum algorithm for GEPs.

### QUBO formulations for numerical quantum computing

- Computer Science
- 2021

To formulate QUBO models for a linear system solving problem, a linear leastsquare problem with binary representation of the solution is made use of and those models are validated on the D-Wave system.

### A quantum interior-point predictor–corrector algorithm for linear programming

- Computer ScienceJournal of Physics A: Mathematical and Theoretical
- 2020

We introduce a new quantum optimization algorithm for dense linear programming problems, which can be seen as the quantization of the interior point predictor–corrector algorithm [] using a quantum…

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