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 $d\to \infty$.

翻译：量子信号处理（QSP）利用一个由称为相位因子的 $(d+1)$ 个实数参数化的 $2\times 2$ 酉矩阵的乘积，来表示一个 $d$ 次实标量多项式。这种创新的多项式表示在量子计算中具有广泛的应用。当目标多项式是通过截断一个无穷多项式级数获得时，一个自然的问题是：当次数 $d\to \infty$ 时，相位因子是否具有明确定义的极限？虽然相位因子通常不唯一，但我们发现存在一种一致的参数化选择，使得该极限在 $\ell^1$ 空间中是良定义的。这种QSP的推广，称为无限量子信号处理，可用于表示一大类非多项式函数。我们的分析揭示了目标函数的正则性与相位因子的衰减特性之间的惊人联系。我们的分析还启发了一种非常简单高效的算法，用于在 $\ell^1$ 空间中近似计算相位因子。该算法仅使用双精度算术运算，并且当目标函数的切比雪夫系数的 $\ell^1$ 范数以一个与 $d$ 无关的常数为上界时，可证明收敛。这也是第一个在极限 $d\to \infty$ 下具有可证明性能保证的、数值稳定的寻找相位因子的算法。