Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems

TL;DR

This paper introduces a quantum eigenstate filtering algorithm using quantum signal processing and minimax polynomials, enabling efficient preparation of target eigenstates and solving quantum linear systems with near-optimal query complexity.

Contribution

The paper develops a novel quantum eigenstate filtering method that avoids phase estimation and amplitude amplification, improving efficiency in quantum linear system algorithms.

Findings

Achieves near-optimal query complexity of O(deta ext{log}(1/\u03b5)) for sparse matrices

Provides two algorithms based on adiabatic computing and Zeno effect

Prepares solutions as pure states without phase estimation or amplitude amplification

Abstract

We present a quantum eigenstate filtering algorithm based on quantum signal processing (QSP) and minimax polynomials. The algorithm allows us to efficiently prepare a target eigenstate of a given Hamiltonian, if we have access to an initial state with non-trivial overlap with the target eigenstate and have a reasonable lower bound for the spectral gap. We apply this algorithm to the quantum linear system problem (QLSP), and present two algorithms based on quantum adiabatic computing (AQC) and quantum Zeno effect respectively. Both algorithms prepare the final solution as a pure state, and achieves the near optimal $\mathcal{\widetilde{O}}(d\kappa\log(1/\epsilon))$ query complexity for a $d$-sparse matrix, where $\kappa$ is the condition number, and $\epsilon$ is the desired precision. Neither algorithm uses phase estimation or amplitude amplification.