Quantum signal processing (QSP) and its extensions are increasingly popular frameworks for developing quantum algorithms. Yet QSP implementations still struggle to complete a classical pre-processing step (‘QSP-processing’) that determines the set of SU (2) rotation matrices defining the QSP circuit. We introduce a method of QSP-processing for complex polynomials that identifies a solution without optimization or root-finding and verify the success of our methods with polynomials characterized by floating point precision coefficients. We demonstrate the success of our technique for relevant target polynomials and precision regimes, including the Jacobi-Anger expansion used in QSP Hamiltonian Simulation. For popular choices of sign and inverse function approximations, we characterize regimes where all known QSP-processing methods should be expected to struggle without arbitrary precision arithmetic.
Quantum signal processing (QSP) and its extensions are increasingly popular frameworks for developing quantum algorithms. Yet QSP implementations still struggle to complete a classical pre-processing step (‘QSP-processing’) that determines the set of SU (2) rotation matrices defining the QSP circuit. We introduce a method of QSP-processing for complex polynomials that identifies a solution without optimization or root-finding and verify the success of our methods with polynomials characterized by floating point precision coefficients. We demonstrate the success of our technique for relevant target polynomials and precision regimes, including the Jacobi-Anger expansion used in QSP Hamiltonian Simulation. For popular choices of sign and inverse function approximations, we characterize regimes where all known QSP-processing methods should be expected to struggle without arbitrary precision arithmetic.
Quantum signal processing (QSP) relies on a historically costly pre-processing step, "QSP-processing/phase-factor finding." QSP-processing is now a developed topic within quantum algorithms literature, and a beginner accessible review of QSP-processing is overdue. This work provides a whirlwind tour through QSP conventions and pre-processing methods, beginning from a pedagogically accessible QSP convention. We then review QSP conventions associated with three common polynomial types: real polynomials with definite parity, sums of reciprocal/anti-reciprocal Chebyshev polynomials, and complex polynomials. We demonstrate how the conventions perform with respect to three criteria: circuit length, polynomial conditions, and pre-processing methods. We then review the recently introduced Wilson method for QSP-processing and give conditions where it can succeed with bound error. Although the resulting bound is not computationally efficient, we demonstrate that the method succeeds with linear error propagation for relevant target polynomials and precision regimes, including the Jacobi-Anger expansion used in Hamiltonian simulation algorithms. We then apply our benchmarks to three QSP-processing methods for QSP circuits and show that a method introduced by Berntson and Sünderhauf outperforms both the Wilson method and the standard optimization strategy for complex polynomials.
Quantum signal processing (QSP) offers a framework for implementing polynomial transformations of operators that are block-encoded within unitary matrices. It has been widely adopted as a grand unification framework for quantum algorithms. In this work, we demonstrate that the coefficients of the induced polynomials P(x) and Q(x) of a QSP sequence in the Chebyshev basis can be uniquely determined by the coefficients of a Laurent polynomial. Based on this correspondence, we develop an efficient method for computing QSP matrix multiplications by evaluating this Laurent polynomial to obtain the QSP sequence. Finally, by applying this framework to the fixed-point iterative method for phase factor recovery, we achieve highly efficient computation of phase factors.
Quantum signal processing is a powerful framework in quantum algorithms, playing a central role in Hamiltonian simulation and related applications. The sequence of polynomials implemented at each step of this protocol provides a polynomial basis for block-encoding any polynomial of a unitary. We characterize the achievable polynomial bases in terms of their orthogonality or biorthogonality with respect to a linear functional admitting an integral representation. Explicit expressions for the quantum signal processing angles are derived for families of polynomial sequences, including Hermite, Jacobi, and Rogers-Szegő polynomials. We show that 2n+2 rotation angles are required to encode a sequence of polynomials in these classes up to degree n. We use this result to show that an ε-approximation of a smooth function f can be block-encoded using O(log(1/ε)) gates via its Hermite series expansion. The connections established with the theory of orthogonal and biorthogonal polynomials lead to a new method for solving the quantum signal processing angle-finding problem, yielding explicit expressions for the angles. They also provide a complete characterization of the polynomials achievable by SU(1,1)-QSP in terms of their roots. Biorthogonality properties are shown to hold in the bivariate QSP setting, yielding a set of necessary conditions for achievable polynomials.
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 10000 with errors below ${10}^{\ensuremath{-}12}$.
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 signal processing (QSP) serves as the asymptotically optimal technique for Hamiltonian simulation on a quantum computer. By approximating the time evolution operator via the Jacobi-Anger expansion, the Hamiltonian simulation problem reduces to a problem in polynomial approximation theory: find a sufficient degree-d polynomial series to approximate e−iτx on [−1,1] within error ε. While d∈O~(τ) is known to be asymptotically optimal, there exists a gap between state-of-the-art bounds and the optimal constant multiplicative factor, which is approximately equal to 1. Here, we close this gap almost entirely, to the point where possible future improvements will not be of practical significance. Our improvement resides in a careful treatment of the Bessel tail in the Jacobi-Anger series using Kapteyn's and Watson's inequalities, thereby reducing the overhead estimates for all Hamiltonian simulation tasks on quantum computers by a factor of ≈e/2.