Infinite quantum signal processing

TL;DR

This work generalizes quantum signal processing to infinite, softly decaying phase factors by formulating infinite QSP (iQSP) and proving that, for target Chebyshev coefficients with bounded $\ell^1$ norm, there exists an invertible map in $\ell^1$ linking these coefficients to a convergent infinite sequence of phase factors. It reveals a direct link between the regularity of the target function and the decay of the phase-factor tail, showing algebraic-to-exponential decay aligned with $C^{\alpha}$ smoothness, $C^{\infty}$, or $C^{\omega}$ smoothness. The authors present a simple fixed-point algorithm that provably converges in $\ell^1$ to the phase factors with complexity $\mathcal{O}(d^2\log(1/\epsilon))$, using only double-precision arithmetic and offering numerically stable guarantees as $d\to\infty$. These results provide a rigorous foundation for representing broad classes of non-polynomial functions within QSP/QSVT, with implications for polynomial approximation, circuit synthesis, and Hamiltonian simulation, and suggest directions for relaxing norm constraints and extending to generalized QSP.

Abstract

Quantum signal processing (QSP) represents a real scalar polynomial of degree $d$ using a product of unitary matrices of size $2\times 2$, parameterized by $(d+1)$ real numbers called the phase factors. This innovative representation of polynomials has a wide range of applications in quantum computation. When the polynomial of interest is obtained by truncating an infinite polynomial series, a natural question is whether the phase factors have a well defined limit as the degree $d\to \infty$. While the phase factors are generally not unique, we find that there exists a consistent choice of parameterization so that the limit is well defined in the $\ell^1$ space. This generalization of QSP, called the infinite quantum signal processing, can be used to represent a large class of non-polynomial functions. Our analysis reveals a surprising connection between the regularity of the target function and the decay properties of the phase factors. Our analysis also inspires a very simple and efficient algorithm to approximately compute the phase factors in the $\ell^1$ space. The algorithm uses only double precision arithmetic operations, and provably converges when the $\ell^1$ norm of the Chebyshev coefficients of the target function is upper bounded by a constant that is independent of $d$. This is also the first numerically stable algorithm for finding phase factors with provable performance guarantees in the limit $d\to \infty$.

Figures (4)

• Figure 1: The plot of $\widetilde{C}(\theta)$ and $C(\theta)$ as function of $\theta$.
• Figure 2: The performance of the fixed-point iteration (FPI) algorithm (\ref{['alg:iterative-qsp']}) to find phase factors for $\frac{1}{2} f_{\text{even}}(x)$ and $\frac{1}{2} f_{\text{odd}}(x)$.
• Figure 3: Comparison of the performance of the fixed-point iteration (FPI) algorithm (\ref{['alg:iterative-qsp']}) with the quasi-Newton (QN) method in DongMengWhaleyEtAl2021 to find QSP phase factors for $\frac{1}{2} f_{\text{even}}(x)$ and $\frac{1}{2} f_{\text{odd}}(x)$ with $\tau=50, 100, 150, \cdots, 1000$. The error tolerance is $\epsilon=10^{-12}$.
• Figure 4: Magnitude of the Chebyshev-coefficient vector $c$ and the corresponding reduced phase factors $\Phi$.

Theorems & Definitions (48)

• Definition 1: Target function
• Theorem 3: Invertibility of $\overline{F}$
• Theorem 4: Decay properties of reduced phase factors
• Corollary 5
• Theorem 6: Convergence of the fixed-point iteration algorithm
• Theorem 7: Quantum signal processing GilyenSuLowEtAl2019
• Corollary 8: Quantum signal processing with real target polynomials GilyenSuLowEtAl2019
• Theorem 9: Quantum signal processing with symmetric phase factors WangDongLin2021
• Lemma 10: Phase-factor padding
• proof