Infinite quantum signal processing

Yulong Dong; Lin Lin; Hongkang Ni; Jiasu Wang

doi:10.22331/q-2024-12-10-1558

Infinite quantum signal processing

Yulong Dong¹, Lin Lin^1,2,3, Hongkang Ni⁴, and Jiasu Wang¹

¹Department of Mathematics, University of California, Berkeley, CA 94720, USA.
²Applied Mathematics and Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
³Challenge Institute for Quantum Computation, University of California, Berkeley, CA 94720, USA
⁴Institute for Computational and Mathematical Engineering (ICME), Stanford University, Stanford, CA 94305, USA.

Published:	2024-12-10, volume 8, page 1558
Eprint:	arXiv:2209.10162v3
Doi:	https://doi.org/10.22331/q-2024-12-10-1558
Citation:	Quantum 8, 1558 (2024).

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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 $d\to \infty$.

► BibTeX data

@article{Dong2024infinitequantum,
  doi = {10.22331/q-2024-12-10-1558},
  url = {https://doi.org/10.22331/q-2024-12-10-1558},
  title = {Infinite quantum signal processing},
  author = {Dong, Yulong and Lin, Lin and Ni, Hongkang and Wang, Jiasu},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1558},
  month = dec,
  year = {2024}
}

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