Infinite quantum signal processing
Fuente:
arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866912150091464704 |
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| author | Dong, Yulong Lin, Lin Ni, Hongkang Wang, Jiasu |
| author_facet | Dong, Yulong Lin, Lin Ni, Hongkang Wang, Jiasu |
| contents | 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 $d\to \infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_10162 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Infinite quantum signal processing Dong, Yulong Lin, Lin Ni, Hongkang Wang, Jiasu Quantum Physics Numerical Analysis 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 $d\to \infty$. |
| title | Infinite quantum signal processing |
| topic | Quantum Physics Numerical Analysis |
| url | https://arxiv.org/abs/2209.10162 |