Infinite quantum signal processing

Dong, Yulong [Department of Mathematics, University of California, Berkeley, CA 94720, USA.] (ORCID:0000000305772475); Lin, Lin [Department of Mathematics, University of California, Berkeley, CA 94720, USA.; Applied Mathematics and Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA; Challenge Institute for Quantum Computation, University of California, Berkeley, CA 94720, USA] (ORCID:0000000168609566); Ni, Hongkang [Institute for Computational and Mathematical Engineering (ICME), Stanford University, Stanford, CA 94305, USA.]; Wang, Jiasu [Department of Mathematics, University of California, Berkeley, CA 94720, USA.] (ORCID:0000000213212649)

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

DOE OSTI · 2569021

Infinite quantum signal processing

Dong, Yulong [Department of Mathematics, University of California, Berkeley, CA 94720, USA.] (ORCID:0000000305772475)·Lin, Lin [Department of Mathematics, University of California, Berkeley, CA 94720, USA.; Applied Mathematics and Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA; Challenge Institute for Quantum Computation, University of California, Berkeley, CA 94720, USA] (ORCID:0000000168609566)·Ni, Hongkang [Institute for Computational and Mathematical Engineering (ICME), Stanford University, Stanford, CA 94305, USA.]·Wang, Jiasu [Department of Mathematics, University of California, Berkeley, CA 94720, USA.] (ORCID:0000000213212649)

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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 × 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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Dong, Yulong [Department of Mathematics, University of California, Berkeley, CA 94720, USA.] (ORCID:0000000305772475), Lin, Lin [Department of Mathematics, University of California, Berkeley, CA 94720, USA.; Applied Mathematics and Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA; Challenge Institute for Quantum Computation, University of California, Berkeley, CA 94720, USA] (ORCID:0000000168609566), Ni, Hongkang [Institute for Computational and Mathematical Engineering (ICME), Stanford University, Stanford, CA 94305, USA.], Wang, Jiasu [Department of Mathematics, University of California, Berkeley, CA 94720, USA.] (ORCID:0000000213212649). 2024-12-10. Infinite quantum signal processing. https://doi.org/10.22331/q-2024-12-10-1558

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