Quantum relaxed row and column iteration methods based on block-encoding

TL;DR

This paper introduces quantum algorithms for relaxed row and column iteration methods that accelerate convergence and reduce complexity in solving linear systems on quantum computers, avoiding phase estimation and Hamiltonian simulation.

Contribution

It develops quantum versions of relaxed iteration methods with exponential complexity reduction and linear iteration dependence, enhancing quantum linear system solving techniques.

Findings

Convergence is faster with appropriate parameters.

Complexity is exponentially improved over classical methods.

No phase estimation or Hamiltonian simulation needed.

Abstract

Iteration method is commonly used in solving linear systems of equations. We present quantum algorithms for the relaxed row and column iteration methods by constructing unitary matrices in the iterative processes, which generalize row and column iteration methods to solve linear systems on a quantum computer. Comparing with the conventional row and column iteration methods, the convergence accelerates when appropriate parameters are chosen. Once the quantum states are efficiently prepared, the complexity of our relaxed row and column methods is improved exponentially and is linear with the number of the iteration steps. In addition, phase estimations and Hamiltonian simulations are not required in these algorithms.