Quantum singular value transformation (QSVT) provides a unifying framework for quantum algorithms by applying the polynomials of quantum signal processing (QSP) to block-encoded matrices. We connect this framework with spectral analysis, design and topological protection in condensed matter physics. Extending QSVT beyond matrices to possibly infinite-dimensional operators (operator QSVT), we establish the following. First, it transforms every block of the operator by the QSP polynomials; no singular value decomposition is needed. Second, a spectral mapping theorem determines the full spectrum of the QSVT operator, including continuous spectra, from the embedded operator through the QSP polynomial, allowing spectral analysis without translation invariance. Third, the spectral mapping turns QSP into spectral design: the QSP phases can stretch, fold and separate the bands and place the eigenvalues. Finally, the kernel sectors that conventional QSVT usually sets aside carry a topological index, invariant under the choice of QSP phases, local perturbations and gap-preserving deformations, which guarantees topologically protected states. These results unify two quantum walks: Szegedy's walk from quantum algorithms and the split-step walk from condensed matter physics. Their kernel sectors encode the graph's Euler characteristic and the difference of bulk winding numbers, respectively. For the split-step walk, QSP breaks chiral symmetry but not the topological protection, and can move the protected state into a continuous band to form a bound state in the continuum (BIC).
Quantum singular value transformation (QSVT) provides a unifying framework for quantum algorithms by applying the polynomials of quantum signal processing (QSP) to block-encoded matrices. We connect this framework with spectral analysis, design and topological protection in condensed matter physics. Extending QSVT beyond matrices to possibly infinite-dimensional operators (operator QSVT), we establish the following. First, it transforms every block of the operator by the QSP polynomials; no singular value decomposition is needed. Second, a spectral mapping theorem determines the full spectrum of the QSVT operator, including continuous spectra, from the embedded operator through the QSP polynomial, allowing spectral analysis without translation invariance. Third, the spectral mapping turns QSP into spectral design: the QSP phases can stretch, fold and separate the bands and place the eigenvalues. Finally, the kernel sectors that conventional QSVT usually sets aside carry a topological index, invariant under the choice of QSP phases, local perturbations and gap-preserving deformations, which guarantees topologically protected states. These results unify two quantum walks: Szegedy's walk from quantum algorithms and the split-step walk from condensed matter physics. Their kernel sectors encode the graph's Euler characteristic and the difference of bulk winding numbers, respectively. For the split-step walk, QSP breaks chiral symmetry but not the topological protection, and can move the protected state into a continuous band to form a bound state in the continuum (BIC).
Quantum Signal Processing (QSP) and Quantum Singular Value Transformation (QSVT) provide an efficient framework for implementing polynomials of block-encoded matrices, and thus offer a systematic approach to quantum algorithm design. However, despite a number of recent advances, important limitations remain. In particular, QSP can only transform unitary matrices, by applying a polynomial to their eigenvalues, while QSVT is a singular-value transformation and thus one can only obtain the polynomial of Hermitian matrices. As a consequence, these techniques do not directly apply to an arbitrary non-Hermitian matrix that is not diagonalizable. In this work, we propose a simple yet powerful method to extend these ideas to arbitrary square matrices by acting on their eigenvalues. To this end, we introduce the notion of an n-regular block encoding, namely, a block encoding whose k-th power reproduces the k-th power of the encoded matrix for every 0<k<n. We show that applying QSP to any unitary with this property is equivalent to applying a polynomial of degree at most n to the block-encoded matrix, independently of its internal structure. Moreover, we provide a simple construction that transforms any block encoding into an n-regular one using only O(logn) ancillary qubits and operations. Finally, we show that this construction induces the desired transformation on the eigenvalues associated with the Jordan normal form of the matrix; for non-diagonalizable matrices this yields the full matrix function P(A), whose action on each Jordan block involves the derivatives of P, and not merely the map λ↦P(λ).
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 Singular Value Transformation (QSVT) provides a unified framework for applying polynomial functions to the singular values of a block-encoded matrix. QSVT prepares a state proportional to $\bA^{-1}\bb$ with circuit depth O(d⋅polylog(N)), where d is the polynomial degree of the 1/x approximation and N is the size of $\bA$. Current polynomial approximation methods are over the continuous interval [a,1], giving $d = O(\sqrt{\kap}\log(1/\varepsilon))$, and make no use of any properties of $\bA$.
We observe here that QSVT solution accuracy depends only on the polynomial accuracy at the eigenvalues of $\bA$. When all N eigenvalues are known exactly, a pure spectral polynomial pS can interpolate 1/x at these eigenvalues and achieve unit fidelity at reduced degree. But its practical applicability is limited. To address this, we propose a spectral correction that exploits prior knowledge of K eigenvalues of $\bA$. Given any base polynomial p0, such as Remez, of degree d0, a K×K linear system enforces exact interpolation of 1/x only at these K eigenvalues without increasing d0. The spectrally corrected polynomial pSC preserves the continuous error profile between eigenvalues and inherits the parity of p0.
QSVT experiments on the 1D Poisson equation demonstrate up to a 5× reduction in circuit depth relative to the base polynomial, at unit fidelity and improved compliance error. The correction is agnostic to the choice of base polynomial and robust to eigenvalue perturbations up to 10% relative error. Extension to the 2D Poisson equation suggests that correcting a small fraction of the spectrum may suffice to achieve fidelity above 0.999.
Quantum singular value transformation (QSVT) enables the application of polynomial functions to the singular values of near arbitrary linear operators embedded in unitary transforms, and has been used to unify, simplify, and improve most quantum algorithms. QSVT depends on precise results in representation theory, with the desired polynomial functions acting simultaneously within invariant two-dimensional subspaces of a larger Hilbert space. These two-dimensional transformations are largely determined by the related theory of quantum signal processing (QSP). While QSP appears to rely on properties specific to the compact Lie group SU(2), many other Lie groups appear naturally in physical systems relevant to quantum information. This work considers settings in which SU(1,1) describes system dynamics and finds that, surprisingly, despite the non-compactness of SU(1,1), one can recover a QSP-type ansatz, and show its ability to approximate near arbitrary polynomial transformations. We discuss various experimental uses of this construction, as well as prospects for expanded relevance of QSP-like ansätze to other Lie groups.
We present a simplified exposition of some pieces of [GSLW19], which introduces a quantum singular value transformation (QSVT) framework for applying polynomial functions to block-encoded matrices. The QSVT framework has garnered substantial recent interest from the quantum algorithms community, as it was demonstrated by [GSLW19] to encapsulate many existing algorithms naturally phrased as an application of a matrix function. First, we posit that the lifting of quantum singular processing (QSP) to QSVT is better viewed not through Jordan's lemma [Jor75; Reg06] (as was suggested by [GSLW19]) but as an application of the cosine-sine decomposition, which can be thought of as a more explicit and stronger version of Jordan's lemma. Second, we demonstrate that the constructions of bounded polynomial approximations given in [GSLW19], which use a variety of ad hoc approaches drawing from Fourier analysis, Chebyshev series, and Taylor series, can be unified under the framework of truncation of Chebyshev series, and indeed, can in large part be matched via a bounded variant of a standard meta-theorem from [Tre19]. We hope this work finds use to the community as a companion guide for understanding and applying the powerful framework of [GSLW19].
Quantum signal processing (QSP) and the quantum singular value transformation (QSVT) are pivotal tools for simplifying the development of quantum algorithms. These techniques leverage polynomial transformations on the eigenvalues or singular values of block-encoded matrices, achieved with the use of just one control qubit. In contexts where the control qubit is used to extract information about the eigenvalues or singular values, the amount of extractable information is limited to one bit per protocol. In this work, we extend the original QSP ansatz by introducing multiple control qubits. We show that, much like in the single-qubit case, nearly any vector of polynomials can be implemented with a multi-qubit QSP ansatz, and the gate complexity scales polynomially with the dimension of such states. Moreover, assuming that powers of the matrix to transform are easily implementable - as in Shor's algorithm - we can achieve polynomial transformations with degrees that scale exponentially with the number of control qubits. This work aims to provide a partial characterization of the polynomials that can be implemented using this approach, with phase estimation schemes and discrete logarithm serving as illustrative examples.