We introduce polynomial approximations of the even and odd step functions on the interval $[-1,1]$ with simple Chebyshev coefficients, making their numerical implementation straightforward. We derive rigorous error bounds and demonstrate that these polynomials are nearly optimal in the sense that their error deviates from the theoretically optimal error by a multiplicative factor that grows logarithmically with the polynomial order. From these polynomials, we derive related nearly optimal polynomial approximations that can be used to perform quantum phase estimation, linear amplitude amplification, eigenvalue thresholding, and other quantum algorithms using the quantum singular value transform.
We introduce polynomial approximations of the even and odd step functions on the interval [−1,1] with simple Chebyshev coefficients, making their numerical implementation straightforward. We derive rigorous error bounds and demonstrate that these polynomials are nearly optimal in the sense that their error deviates from the theoretically optimal error by a multiplicative factor that grows logarithmically with the polynomial order. From these polynomials, we derive related nearly optimal polynomial approximations that can be used to perform quantum phase estimation, linear amplitude amplification, eigenvalue thresholding, and other quantum algorithms using the quantum singular value transform.
Quantum matrix inversion with the quantum singular value transformation (QSVT) requires a polynomial approximation to 1/x. Several methods from the literature construct polynomials that achieve the known degree complexity O(κlog(κ/ε)) with condition number κ and uniform error ε. However, the \emph{optimal} polynomial with lowest degree for fixed error ε can only be approximated numerically with the resource-intensive Remez method, leading to impractical preprocessing runtimes. Here, we derive an analytic shortcut to the optimal polynomial. Comparisons with other polynomials from the literature, based on Taylor expansion, Chebyshev iteration, and convex optimization, confirm that our result is optimal. Furthermore, for large κlog(κ/ε), our polynomial has the smallest maximum value on [−1,1] of all approaches considered, leading to reduced circuit depth due to the normalization condition of QSVT. With the Python code provided, this paper will also be useful for practitioners in the field.
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.
Many problems in linear algebra-such as those arising from non-Hermitian physics, transcorrelated quantum chemistry and differential equations-can be solved on a quan-tum computer by processing eigenvalues of the non-normal input matrices. However, the existing Quantum Singular Value Transformation (QSVT) framework is ill-suited to this task, as 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 super-position 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 generatenFourier coefficients with 0 (poly log (n)) gates compared to prior approaches with linear cost.
Solving systems of linear equations is one of the most important primitives in many different areas, including in optimization, simulation, and machine learning. Quantum algorithms for solving linear systems have the potential to provide a quantum advantage for these problems. In this work, we recall the Chebyshev iterative method and the corresponding optimal polynomial approximation of the inverse. We show that the Chebyshev iteration polynomial can be efficiently evaluated both using quantum singular value transformation (QSVT) as well as linear combination of unitaries (LCU). We achieve this by bounding the 1-norm of the coefficients of the polynomial expressed in the Chebyshev basis. This leads to a considerable constant-factor improvement in the runtime of quantum linear system solvers that are based on LCU or QSVT (or, conversely, a several orders of magnitude smaller error with the same runtime/circuit depth).
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) is a powerful framework that unifies many quantum algorithms, including Grover’s search, by enabling polynomial transformations of the singular values of linear operators embedded in unitary matrices. We present an improved quantum search algorithm within this framework. Instead of amplifying the target state amplitude, our method suppresses the non-target amplitude using a modified Chebyshev polynomial. This approach achieves the optimal O(Nlog(1/δ)) query complexity while requiring lower-degree polynomials. For N = 512, our method reduces the polynomial degree from 71 to 43 compared to prior QSVT-based fixed-point search. We also demonstrate how QSVT enables exact search construction.