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arXiv:2509.19195

Quantum Krylov Algorithm for Szegö Quadrature

Kirby, William, Yizhi Shen, Daan Camps, Anirban Chowdhury, Katherine Klymko, Roel Van Beeumen

Published Sep 23, 2025• 0 citations
arXivHTMLPDF

Abstract

We present a quantum algorithm to evaluate matrix elements of functions of unitary operators. The method is based on calculating quadrature nodes and weights using data collected from a quantum processor. Given a unitary $U$ and quantum states $|ψ_0\rangle$, $|ψ_1\rangle$, the resulting quadrature rules form a functional that can then be used to classically approximate $\langleψ_1|f(U)|ψ_0\rangle$ for any function $f$. In particular, the algorithm calculates Szegö quadrature rules, which, when $f$ is a Laurent polynomial, have the optimal relation between degree of $f$ and number of distinct quantum circuits required. The unitary operator $U$ could approximate a time evolution, opening the door to applications like estimating properties of Hamiltonian spectra and Gibbs states, but more generally could be any operator implementable via a quantum circuit. We expect this algorithm to be useful as a subroutine in other quantum algorithms, much like quantum signal processing or the quantum eigenvalue transformation of unitaries. Key advantages of our algorithm are that it does not require approximating $f$ directly, via a series expansion or in any other way, and once the output functional has been constructed using the quantum algorithm, it can be applied to any $f$ classically after the fact.

DOI
10.48550/arxiv.2509.19195

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We present a quantum algorithm to evaluate matrix elements of functions of unitary operators. The method is based on calculating quadrature nodes and weights using data collected from a quantum processor. Given a unitary UUU and quantum states ∣ψ0⟩|ψ_0\rangle∣ψ0​⟩, ∣ψ1⟩|ψ_1\rangle∣ψ1​⟩, the resulting quadrature rules form a functional that can then be used to classically approximate ⟨ψ1∣f(U)∣ψ0⟩\langleψ_1|f(U)|ψ_0\rangle⟨ψ1​∣f(U)∣ψ0​⟩ for any function fff. In particular, the algorithm calculates Szegö quadrature rules, which, when fff is a Laurent polynomial, have the optimal relation between degree of fff and number of distinct quantum circuits required. The unitary operator UUU could approximate a time evolution, opening the door to applications like estimating properties of Hamiltonian spectra and Gibbs states, but more generally could be any operator implementable via a quantum circuit. We expect this algorithm to be useful as a subroutine in other quantum algorithms, much like quantum signal processing or the quantum eigenvalue transformation of unitaries. Key advantages of our algorithm are that it does not require approximating fff directly, via a series expansion or in any other way, and once the output functional has been constructed using the quantum algorithm, it can be applied to any fff classically after the fact.
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Quantum signal processing (QSP) is a powerful quantum algorithm to exactly implement matrix polynomials on quantum computers. Asymptotic analysis of quantum algorithms based on QSP has shown that asymptotically optimal results can in principle be obtained for a range of tasks, such as Hamiltonian simulation and the quantum linear system problem. A further benefit of QSP is that it uses a minimal number of ancilla qubits, which facilitates its implementation on near-to-intermediate term quantum architectures. However, there is so far no classically stable algorithm allowing computation of the phase factors that are needed to build QSP circuits. Existing methods require the use of variable precision arithmetic and can only be applied to polynomials of a relatively low degree. We present here an optimization-based method that can accurately compute the phase factors using standard double precision arithmetic operations. We demonstrate the performance of this approach with applications to Hamiltonian simulation, eigenvalue filtering, and quantum linear system problems. Our numerical results show that the optimization algorithm can find phase factors to accurately approximate polynomials of a degree larger than 1000010\phantom{\rule{0.16em}{0ex}}00010000 with errors below ${10}^{\ensuremath{-}12}$.
Understanding the capacity of quantum circuits through the lens of approximation theory is essential for evaluating the complexity of quantum circuits required to solve various problems in scientific computation. We design quantum circuits capable of approximating d-dimensional functions within the Korobov function space. This is achieved by leveraging the quantum signal processing (QSP) and the linear combination of unitaries (LCU) algorithms to build quantum circuits that output Chebyshev polynomials. We also present a quantitative analysis of the approximation error rates and evaluates the computational complexity of implementing the proposed circuits. Since the Korobov function space is a subspace of the certain Sobolev spaces, our work develops a theoretical foundation for implementing a large class of functions suitable for applications on a quantum computer.
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 J1J_1J1​--J2J_2J2​ 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.
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.