Quantum Power Iteration Unified Using Generalized Quantum Signal Processing

TL;DR

The paper introduces generalized quantum signal processing (GQSP) as a unifying framework to implement polynomial transforms of a block-encoded Hamiltonian, enabling four quantum power methods for state preparation: QPI, QPL, QII, and QFSM. It provides rigorous resource-scaling analyses and demonstrates, through numerical simulations on molecular Hamiltonians, that QPI and QPL offer near-optimal ground-state preparation with or without energy estimates, while QII accelerates convergence when a rough energy is known, and QFSM enables excited-state preparation without variational optimization. The results show that GQSP-based filters can achieve robust convergence with favorable qubit and query counts, offering a practical toolbox for initializing ground and excited states on fault-tolerant quantum devices. The work also discusses limitations, such as low success probabilities requiring amplitude amplification, and identifies future directions like perturbation-based coefficient derivation, improved energy-shift strategies, Green's function extensions, and multi-excitation block variants to tackle larger systems.

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.

Figures (12)

• Figure 1: Generalized Quantum Signal Processing (GQSP) circuit. The circuit comprises three registers: the first one qubit register controls the GQSP procedure, the second $n$ qubits register performs the block encoding, and the third $m$ qubits register holds the state on which the Hamiltonian acts.
• Figure 2: (a) Potential energy curve for the hydrogen molecule obtained with the QPI method ($\mathrm{QPL}0$). The basis set is cc-pVDZ, and the active space is $(2,2)$. Star markers indicate the corresponding $\mathrm{QPL}1$ energies immediately after the variational optimization step.
• Figure 3: Bond length dependence of the Lanczos polynomial coefficients for the hydrogen molecule. Coefficients are obtained after the variational optimization of the Lanczos polynomial (see Eq. \ref{['eq:H2_QPL']}).
• Figure 4: (a) Potential energy curves for singlet (S) and triplet (T) CH$_2$ obtained using QPL at various polynomial degrees ($k=0,1,2$). All $\mathrm{QPL}k$ curves correspond to energies after $50$ iterations. The basis set is cc-pVDZ with an active space of $(6,6)$. (b) The right panels show deviations from CASCI$(6,6)$ for the first $50$ iterations of the power method using QPL with $k=0,1,2,3,4$. The dashed black lines indicate chemical accuracy ($1\,\text{kcal/mol}$). Solid lines correspond to the triplet state and dashed lines to the singlet state.
• Figure 5: (a) Potential energy curves for the nitrogen molecule obtained using QPL with different polynomial degrees ($k=0,1,2,3$) and initial guesses (HF in dashed, and partially converged UCCD in solid). All $\mathrm{QPL}k$ curves correspond to energies after $50$ iterations. The basis set is cc-pVDZ, and the active space is $(6,6)$. (b) The right panels display energy differences from CASCI$(6,6)$ over the first $50$ iterations of the power method. The black dashed line marks chemical accuracy ($1\,\text{kcal/mol}$), and the red dot-dashed line is the HF energy.

Theorems & Definitions (1)

• Theorem 1: Generalized Quantum Signal Processing Motlagh2024