Quantum Power Iteration Unified Using Generalized Quantum Signal Processing

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.