Quantum Power Iteration Unified Using Generalized Quantum Signal Processing

TL;DR

This paper introduces a unified quantum framework using generalized quantum signal processing to efficiently implement classical power iteration methods for quantum state preparation, improving scalability and robustness.

Contribution

It develops a comprehensive GQSP-based approach that unifies various quantum power methods, including inverse and folded spectrum, with near-optimal resource scaling and reduced qubit requirements.

Findings

GQSP enables efficient polynomial implementation for quantum power methods.

The proposed methods show improved convergence and lower computational cost.

Numerical validation on molecular Hamiltonians confirms practical advantages.

Abstract

We propose a unifying framework for the state preparation using quantum power method algorithms based on generalized quantum signal processing (GQSP). We apply GQSP to realize quantum analogs of classical power iteration, power Lanczos, inverse iteration, and folded spectrum methods, all within a single coherent framework. GQSP allows efficient realization of methods that require complex polynomials, while avoiding the limitations of approaches based on linear combinations of time-evolution operators. Our constructions, including a Trotter-decomposition-free quantum inverse iteration, achieve near-optimal query scaling, together with reduced qubit requirements. The same formalism yields a quantum folded spectrum method for excited state preparation that avoids explicitly forming powers of the Hamiltonian or performing variational optimization. We provide a theoretical analysis of success probabilities and resource scaling, and we validate the methods numerically using molecular Hamiltonians. The results show that quantum power Lanczos lowers the computational cost and provides robust convergence compared to naive quantum power iteration. Our findings reveal that GQSP-based implementations of power methods combine scalability, flexibility, and robust convergence, paving the way for practical initial state preparations on fault-tolerant quantum devices.