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].
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 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 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).
Abstract. Many problems in linear algebra—such as those arising from non-Hermitian physics, transcorrelated quantum chemistry, and differential equations—can be solved on a quantum computer by processing eigenvalues of the non-normal input matrices. However, the existing Quantum Singular Value Transformation (QSVT) framework is ill-suited for this task because eigenvalues and singular values are different in general. We present a Quantum EigenValue Transformation (QEVT) framework for applying arbitrary polynomial transformations on eigenvalues of block-encoded non-normal operators and a related Quantum EigenValue Estimation (QEVE) algorithm for operators with real spectra. QEVT has query complexity to the block encoding nearly recovering that of the QSVT for a Hermitian input, and QEVE achieves the Heisenberg-limited scaling for diagonalizable input matrices. As applications, we develop a linear differential equation solver with strictly linear time query complexity for average-case diagonalizable operators, as well as a ground state preparation algorithm that upgrades previous nearly optimal results for Hermitian Hamiltonians to diagonalizable matrices with real spectra. Underpinning our algorithms is an efficient method to prepare a quantum superposition of Faber polynomials, which generalize the nearly-best uniform approximation properties of Chebyshev polynomials to the complex plane. Of independent interest, we also develop techniques to generate [Formula: see text] Fourier coefficients with [Formula: see text] gates compared to prior approaches with linear cost.
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 signal processing and quantum singular value transformation are powerful tools to implement polynomial transformations of block-encoded matrices on quantum computers, and has achieved asymptotically optimal complexity in many prominent quantum algorithms. We propose a framework of quantum signal processing and quantum singular value transformation on U ( N ) , which realizes multiple polynomials simultaneously from a block-encoded input, as a generalization of those on U ( 2 ) in the original frameworks. We provide a comprehensive characterization of achievable polynomial matrices and give recursive algorithms to construct the quantum circuits that realize desired polynomial transformations. As three example applications, we propose a framework to realize bi-variate polynomial functions, demonstrate N -interval decision achieving O ( d ) query complexity with a log 2 N improvement over iterative U ( 2 ) -QSP requiring O ( d log 2 N ) queries, and present a quantum amplitude estimation algorithm achieving the Heisenberg limit without adaptive measurements.