QSP phase factors solvers

A toolbox for solving phase factors in quantum signal processing.

Official Website

https://qsppack.gitbook.io/qsppack/ You may find useful tutorials and examples in this website.

Problems and solvers

Given a real polynomial <img src="http://chart.googleapis.com/chart?cht=tx&chl= f" style="border:none;"> of degree <img src="http://chart.googleapis.com/chart?cht=tx&chl= d" style="border:none;"> with definite parity such that <img src="http://chart.googleapis.com/chart?cht=tx&chl= |f(x)| \le 1, x\in[-1,1]" style="border:none;">, the package contains codes for solving phase factors <img src="http://chart.googleapis.com/chart?cht=tx&chl= \Phi=(\phi_0,\dots,\phi_d)" style="border:none;"> such that

<a href="https://www.codecogs.com/eqnedit.php?latex=U_\Phi(x)&space;=&space;e^{\mathrm{i}&space;\phi_0&space;\sigma_z}&space;\prod_{j=1}^{d}&space;\left[&space;e^{\mathrm{i}&space;\arccos(x)&space;\sigma_x}&space;e^{\mathrm{i}&space;\phi_j&space;\sigma_z}&space;\right]&space;=&space;\left(&space;\begin{array}{cc}&space;P(x)&space;&&space;\mathrm{i}&space;Q(x)&space;\sqrt{1&space;-&space;x^2}\\&space;\mathrm{i}&space;Q^*(x)&space;\sqrt{1&space;-&space;x^2}&space;&&space;P^*(x)&space;\end{array}&space;\right)," target="_blank"><img src="https://latex.codecogs.com/gif.latex?U_\Phi(x)&space;=&space;e^{\mathrm{i}&space;\phi_0&space;\sigma_z}&space;\prod_{j=1}^{d}&space;\left[&space;e^{\mathrm{i}&space;\arccos(x)&space;\sigma_x}&space;e^{\mathrm{i}&space;\phi_j&space;\sigma_z}&space;\right]&space;=&space;\left(&space;\begin{array}{cc}&space;P(x)&space;&&space;\mathrm{i}&space;Q(x)&space;\sqrt{1&space;-&space;x^2}\\&space;\mathrm{i}&space;Q^*(x)&space;\sqrt{1&space;-&space;x^2}&space;&&space;P^*(x)&space;\end{array}&space;\right)," title="U_\Phi(x) = e^{\mathrm{i} \phi_0 \sigma_z} \prod_{j=1}^{d} \left[ e^{\mathrm{i} \arccos(x) \sigma_x} e^{\mathrm{i} \phi_j \sigma_z} \right] = \left( \begin{array}{cc} P(x) & \mathrm{i} Q(x) \sqrt{1 - x^2}\\ \mathrm{i} Q^*(x) \sqrt{1 - x^2} & P^*(x) \end{array} \right)," /></a>

where <img src="http://chart.googleapis.com/chart?cht=tx&chl= P_{\mathrm{Re}}=f" style="border:none;">.

The package contains two kinds of solvers:

• Optimization-based solver
• Direct solver (namely the GSLW method and the Haah method)

The package also contains an implementation of the Remez algorithm for finding polynomial approximation.

Citing our work

If you find our work useful or you use our work in your own project, please consider to cite our work.

• Dong, Y., Meng, X., Whaley, K.B. and Lin, L., 2021. Efficient phase-factor evaluation in quantum signal processing. Physical Review A, 103(4), p.042419.
• Wang, J., Dong, Y. and Lin, L., 2021. On the energy landscape of symmetric quantum signal processing. Quantum 6 (2022): 850.
• Dong, Y., Lin, L., Ni, H., & Wang, J. (2024). Infinite quantum signal processing. Quantum, 8, 1558.
• Dong, Y., Lin, L., Ni, H., & Wang, J. (2024). Robust iterative method for symmetric quantum signal processing in all parameter regimes. SIAM Journal on Scientific Computing, 46(5), A2951-A2971.
• Alexis, M., Lin, L., Mnatsakanyan, G., Thiele, C., & Wang, J. (2024). Infinite quantum signal processing for arbitrary Szeg\H {o} functions. arXiv preprint arXiv:2407.05634.
• Ni, H., Sarkar, R., Ying, L., & Lin, L. (2025). Inverse nonlinear fast Fourier transform on SU (2) with applications to quantum signal processing. arXiv preprint arXiv:2505.12615.

Other references:

• A. Gilyén, Y. Su, G. H. Low, and N. Wiebe. Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pages 193–204, 2019
• J. Haah. Product decomposition of periodic functions in quantum signal processing.Quantum, 3:190, 2019

Authors

We hope that the package is useful for your application. If you have any bug reports or comments, please feel free to email one of the software authors:

• Yulong Dong, dongyl@berkeley.edu

• Jiasu Wang, jiasu@berkeley.edu

• Xiang Meng, mengxianglgal@gmail.com

• Hongkang Ni, hongkang@stanford.edu

• Lin Lin, linlin@math.berkeley.edu

Installation

• You can download qsppack and run

  >> startup

  This adds the folder of the solver to MATLAB's path variable.

• Alternatively you can install qsppack to your current directory by pasting the code below to your MATLAB command window:

  unzip('https://github.com/qsppack/qsppack/archive/master.zip')
  movefile('QSPPACK-master', 'qsppack')
  addpath(fullfile(cd,'qsppack','Solvers','Optimization')), savepath
  

Then you can test qsppack by running

>> cd Examples

>> test_HS