Analysis and simulation of QSP beamformer

Shuangning Shi, Yong Shang, Qinglin Liang

Quantum Signal Processing (QSP) beamformer, which is based on Quantum Sig- nal Processing framework, is presented in this paper. This new beamformer can realize the same output gain and signal-to-interference-and-noise ratio (SINR) as conventional beamformers through adjusting impact factors without pre-estimating the covariance matrix of received sig- nals. Finally, the efiectiveness and feasibility of the new beamformer are verifled by simulation results. DOI: 10.2529/PIERS061007112619

Lin Lin

Quantum signal processing (QSP) provides a representation of scalar polynomials of degree $d$ as products of matrices in $\mathrm{SU}(2)$, parameterized by $(d+1)$ real numbers known as phase factors. QSP is the mathematical foundation of quantum singular value transformation (QSVT), which is often regarded as one of the most important quantum algorithms of the past decade, with a wide range of applications in scientific computing, from Hamiltonian simulation to solving linear systems of equations and eigenvalue problems. In this article we survey recent advances in the mathematical and numerical analysis of QSP. In particular, we focus on its generalization beyond polynomials, the computational complexity of algorithms for phase factor evaluation, and the numerical stability of such algorithms. The resolution to some of these problems relies on an unexpected interplay between QSP, nonlinear Fourier analysis on $\mathrm{SU}(2)$, fast polynomial multiplications, and Gaussian elimination for matrices with displacement structure.

V B Gerard

Yulong Dong, Xiang Meng, K. Birgitta Whaley +1 more

Quantum signal processing (QSP) is a powerful quantum algorithm to exactly implement matrix polynomials on quantum computers. Asymptotic analysis of quantum algorithms based on QSP has shown that asymptotically optimal results can in principle be obtained for a range of tasks, such as Hamiltonian simulation and the quantum linear system problem. A further benefit of QSP is that it uses a minimal number of ancilla qubits, which facilitates its implementation on near-to-intermediate term quantum architectures. However, there is so far no classically stable algorithm allowing computation of the phase factors that are needed to build QSP circuits. Existing methods require the use of variable precision arithmetic and can only be applied to polynomials of a relatively low degree. We present here an optimization-based method that can accurately compute the phase factors using standard double precision arithmetic operations. We demonstrate the performance of this approach with applications to Hamiltonian simulation, eigenvalue filtering, and quantum linear system problems. Our numerical results show that the optimization algorithm can find phase factors to accurately approximate polynomials of a degree larger than $10\phantom{\rule{0.16em}{0ex}}000$ with errors below ${10}^{\ensuremath{-}12}$.

Zou Jingbo, Shûichi Ohno