Infinite quantum signal processing

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 $d\to \infty$.

32 pages, 7 figures,

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Cited by in corpus (9)

• On multivariate polynomials achievable with quantum signal processing
• Infinite quantum signal processing for arbitrary Szegő functions
• Modular quantum signal processing in many variables
• Quantum phase estimation based filtering: performance analysis and application to low-energy spectral calculation
• Efficient explicit circuit for quantum state preparation of piecewise continuous functions
• Parallel Quantum Signal Processing Via Polynomial Factorization
• Scalable Quantum Computational Science: A Perspective from Block-Encodings and Polynomial Transformations
• Beyond asymptotic reasoning: the practicalities of a quantum ground state projector based on the wall-Chebyshev expansion
• An adversary bound for quantum signal processing