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.
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.
In this work, we introduce a quantum inverse power iteration (QIPI) algorithm based on the quantum singular value transformation (QSVT) to target arbitrary excited states. Given an energy shift ω, QIPI prepares the target excited state by iteratively applying an approximation of the shifted inverse Hamiltonian (H−ωI)−1 to a trial state. Prior quantum inverse power approaches typically relied on Fourier decompositions of the inverse Hamiltonian, with numerical quadrature used to reconstruct the transformation, but such methods are highly sensitive to hyperparameter choices and have been observed to be numerically unstable, effectively restricting their use to ground-state preparation. To enable robust excited-state targeting, we investigate two alternative transformation techniques: a Chebyshev decomposition of the inverse (Cheb-inv) and an eigenstate filtering (EF) approach based on QSVT. We find that EF-based QIPI is substantially more robust than Cheb-inv and other decomposition-based approaches due to the symmetry of the applied filtering polynomial, avoiding divergence with respect to the choice of ω and efficiently suppressing off-target eigenstates even in closely spaced spectra. Numerical simulations for molecular Hamiltonians of H2, LiH, and BeH2 show improved convergence and enhanced access to higher excited states relative to other quantum power methods. Assuming standard oracle access to the Hamiltonian, we further provide logical resource estimates in fault-tolerant settings in terms of T gate counts, and conclude that QIPI can achieve high target state amplification with modest polynomial degrees, thereby making it a promising candidate for scalable excited-state preparation in fault-tolerant quantum chemistry applications.
Modern quantum algorithms are increasingly formulated as coherent procedures for implementing polynomial transformations of operators and singular values. This perspective provides a powerful and unifying language for quantum algorithm design, connecting a wide range of distinct problems through five closely related key tools: block-encoding, qubitization, QSP, QSVT and GQSP. Block-encoding embeds non-unitary matrices into larger unitaries; qubitization converts block-encodings into structured operators; QSP, QSVT and GQSP enable polynomial transformations with near-optimal query complexity. Together, these techniques form a general toolkit for transforming matrix functions into implementable quantum circuits. This paper develops these techniques from first principles as a unified framework for constructing quantum algorithms. We apply this framework to representative applications to highlight design principles and demonstrate how distinct algorithms can be constructed from a unified sequence of operator transformations.
A central contribution is a systematic decision workflow for selecting the appropriate approach according to the operator structure and the desired transformation polynomial. This perspective clarifies when direct GQSP or through qubitization, or Laurent expansion, or QSVT is most appropriate. We organize algorithmic design into an end-to-end pipeline: identifying the target matrix function, constructing an appropriate block-encoding, determining the relevant spectral domain, designing a polynomial or Laurent-polynomial approximation, synthesizing the phase factors, and translating the transformation into an executable quantum circuit. By applying this unified framework to example applications, we showcase a practical methodology for reasoning, designing, and implementing quantum algorithms based on polynomial transformations.
Quantum signal processing (QSP) and quantum singular value transformation (QSVT) currently stand as the most efficient techniques for implementing functions of block-encoded matrices, a central task that lies at the heart of most prominent quantum algorithms. However, current QSP approaches face several challenges, such as the restrictions imposed on the family of achievable polynomials and the difficulty of calculating the required phase angles for specific transformations. In this paper, we present a generalized quantum signal processing (GQSP) approach, employing general SU(2) rotations as our signal-processing operators, rather than relying solely on rotations in a single basis. Our approach lifts all practical restrictions on the family of achievable transformations, with the sole remaining condition being that |P|≤1 , a restriction necessary due to the unitary nature of quantum computation. Furthermore, GQSP provides a straightforward recursive formula for determining the rotation angles needed to construct the polynomials in cases where P and Q are known. In cases where only P is known, we provide an efficient optimization algorithm capable of identifying in under a minute of GPU time, a corresponding Q for polynomials of degree on the order of 107 . We further illustrate GQSP simplifies QSP-based strategies for Hamiltonian simulation, offer an optimal solution to the ϵ -approximate fractional query problem that requires O((1/δ)+log(1/ϵ)) queries to perform where O(1/δ) is a proved lower bound, and introduces novel approaches for implementing bosonic operators. Moreover, we propose a novel framework for the implementation of normal matrices, demonstrating its applicability through synthesis of diagonal matrices, as well as the development of a new algorithm for convolution through synthesis of circulant matrices using only O(dlogN+log2N) 1 and 2-qubit gates for a filter of lengths d . Published by the American Physical Society 2024
Quantum subspace-diagonalization methods, particularly Quantum Krylov Diagonalization (QKD), provide a promising route for computing low-energy spectra of quantum many-body Hamiltonians. However, existing quantum Krylov approaches rely on non-orthogonal Krylov bases, requiring overlap-matrix regularization that limits numerical stability and accuracy. In this work, we introduce an Orthogonal Quantum Krylov Diagonalization (OQKD) framework that reformulates the classical Lanczos recursion at the operator level, enabling an orthogonal quantum implementation of Krylov-subspace diagonalization. By expressing Lanczos vectors as polynomial transformations of the Hamiltonian, OQKD reproduces the orthogonality, tridiagonal structure, and convergence behavior of the classical Lanczos algorithm thus eliminating the need for overlap-matrix regularization. We further show that the required Lanczos polynomials can be implemented using block encoding and Generalized Quantum Signal Processing with the same asymptotic query complexity as Chebyshev-based QKD methods. Numerical simulations of the J1--J2 Heisenberg model confirm the classical Lanczos convergence and numerical stability of the proposed method, while the measurement-complexity scaling is established analytically. Building upon the OQKD framework, we then introduce a restarted state-preparation protocol that replaces a single high-degree polynomial transformation with a sequence of fixed low-degree transformations, maintaining an affordable block encoding success probability while retaining comparable convergence. These results establish OQKD as an orthogonal quantum analog of the classical Lanczos algorithm and identify the restarted protocol as a promising state-preparation strategy for Quantum Phase Estimation.
Implementing polynomial functions of Hermitian matrices on quantum hardware is a foundational task in quantum computing, critical for accurate Hamiltonian simulation, quantum linear system solving, high-fidelity state preparation, machine learning kernels, and other advanced quantum algorithms. Existing state-of-the-art techniques, including Qubitization, Quantum Singular Value Transformation (QSVT), and Quantum Signal Processing (QSP), rely heavily on block-encoding the Hermitian matrix. These methods are often constrained by the complexity of preparing the block-encoded state, the overhead associated with the required ancillary qubits, or the challenging problem of angle synthesis for the polynomial's phase factors, which limits the achievable circuit depth and overall efficiency. In this work, we propose a novel and resource-efficient approach to implement arbitrary polynomials of a Hermitian matrix by leveraging the Generalized Quantum Signal Processing (GQSP) framework. Our method circumvents the need for block-encoding and avoids the compounding post-selection overheads characteristic of LCU-based constructions, achieving a stable, degree-independent success probability. We derive closed-form expressions for symmetric polynomial expansions and demonstrate how linear combinations of GQSP circuits can realize the desired transformation. This approach reduces resource overhead and opens new pathways for quantum algorithm design for functions of Hermitian matrices, particularly in settings where the Hermitian operator arises naturally from symmetric combinations of unitaries.
The deterministic preparation of quantum many-body ground states is essential for advanced quantum simulation, yet optimal algorithms often require prohibitive hardware resources. Here, we propose a highly efficient, non-variational protocol for ground state preparation using a Power-Cosine quantum signal processing (QSP) filter. By eschewing complex block-encoding techniques, our method directly utilizes coherent time-evolution operators controlled by a single ancillary qubit. The integration of mid-circuit measurement and reset (MCMR) drastically minimizes spatial overhead, translating iterative non-unitary filtering into deep temporal coherence. We analytically demonstrate that this approach achieves exponential suppression of excited states with a circuit depth scaling of O(Δ−2log(1/ε)), where Δ denotes the spectral gap, prioritizing implementational simplicity over optimal asymptotic complexity. Numerical simulations on the 1D Heisenberg XYZ model validate the theoretical soundness and shot-noise resilience of our method. Furthermore, an advantage analysis reveals that our protocol exponentially outperforms standard Trotterized Adiabatic State Preparation (TASP) at equivalent circuit depths. This single-ancilla framework provides a highly practical and deterministic pathway for many-body ground state preparation on Early Fault-Tolerant (EFT) quantum architectures.