Infinite quantum signal processing
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Quantum signal processing (QSP) represents a real scalar polynomial of degree d using a product of unitary matrices of size 2× 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→∞. 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 ℓ^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 ℓ^1 space. The algorithm uses only double precision arithmetic operations, and provably converges when the ℓ^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→∞.
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