On the energy landscape of symmetric quantum signal processing

Jiasu Wang; Yulong Dong; Lin Lin

doi:10.22331/q-2022-11-03-850

On the energy landscape of symmetric quantum signal processing

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

¹Department of Mathematics, University of California, Berkeley, CA 94720, USA.
²Challenge Institute for Quantum Computation, University of California, Berkeley, CA 94720, USA
³Applied Mathematics and Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA

Published:	2022-11-03, volume 6, page 850
Eprint:	arXiv:2110.04993v2
Doi:	https://doi.org/10.22331/q-2022-11-03-850
Citation:	Quantum 6, 850 (2022).

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Abstract

Symmetric quantum signal processing provides a parameterized representation of a real polynomial, which can be translated into an efficient quantum circuit for performing a wide range of computational tasks on quantum computers. For a given polynomial $f$, the parameters (called phase factors) can be obtained by solving an optimization problem. However, the cost function is non-convex, and has a very complex energy landscape with numerous global and local minima. It is therefore surprising that the solution can be robustly obtained in practice, starting from a fixed initial guess $\Phi^0$ that contains no information of the input polynomial. To investigate this phenomenon, we first explicitly characterize all the global minima of the cost function. We then prove that one particular global minimum (called the maximal solution) belongs to a neighborhood of $\Phi^0$, on which the cost function is strongly convex under the condition ${\left\lVert f\right\rVert}_{\infty}=\mathcal{O}(d^{-1})$ with $d=\mathrm{deg}(f)$. Our result provides a partial explanation of the aforementioned success of optimization algorithms.

► BibTeX data

@article{Wang2022energylandscapeof,
  doi = {10.22331/q-2022-11-03-850},
  url = {https://doi.org/10.22331/q-2022-11-03-850},
  title = {On the energy landscape of symmetric quantum signal processing},
  author = {Wang, Jiasu and Dong, Yulong and Lin, Lin},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {6},
  pages = {850},
  month = nov,
  year = {2022}
}

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[3] Jasmine Sinanan-Singh, Gabriel L. Mintzer, Isaac L. Chuang, and Yuan Liu, "Single-shot Quantum Signal Processing Interferometry", Quantum 8, 1427 (2024).

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[7] Michel Alexis, Gevorg Mnatsakanyan, and Christoph Thiele, "Quantum signal processing and nonlinear Fourier analysis", Revista Matemática Complutense (2024).

[8] Sho Sugiura, Arkopal Dutt, William J. Munro, Sina Zeytinoğlu, and Isaac L. Chuang, "Power of Sequential Protocols in Hidden Quantum Channel Discrimination", Physical Review Letters 132 24, 240805 (2024).

[9] Zane M. Rossi and Isaac L. Chuang, "Semantic embedding for quantum algorithms", Journal of Mathematical Physics 64 12, 122202 (2023).

[10] Daan Camps, Lin Lin, Roel Van Beeumen, and Chao Yang, "Explicit Quantum Circuits for Block Encodings of Certain Sparse Matrices", SIAM Journal on Matrix Analysis and Applications 45 1, 801 (2024).

[11] Yulong Dong, Lin Lin, and Yu Tong, "Ground-State Preparation and Energy Estimation on Early Fault-Tolerant Quantum Computers via Quantum Eigenvalue Transformation of Unitary Matrices", PRX Quantum 3 4, 040305 (2022).

[12] Zane M. Rossi, Jack L. Ceroni, and Isaac L. Chuang, "Modular quantum signal processing in many variables", arXiv:2309.16665, (2023).

[13] Yulong Dong, Lin Lin, Hongkang Ni, and Jiasu Wang, "Infinite quantum signal processing", arXiv:2209.10162, (2022).

[14] Zane M. Rossi and Isaac L. Chuang, "Multivariable quantum signal processing (M-QSP): prophecies of the two-headed oracle", Quantum 6, 811 (2022).

[15] Michel Alexis, Lin Lin, Gevorg Mnatsakanyan, Christoph Thiele, and Jiasu Wang, "Infinite quantum signal processing for arbitrary Szegő functions", arXiv:2407.05634, (2024).

[16] Zane M. Rossi, Victor M. Bastidas, William J. Munro, and Isaac L. Chuang, "Quantum signal processing with continuous variables", arXiv:2304.14383, (2023).

[17] Lexing Ying, "Stable factorization for phase factors of quantum signal processing", Quantum 6, 842 (2022).

[18] Yulong Dong, Jonathan Gross, and Murphy Yuezhen Niu, "Beyond Heisenberg Limit Quantum Metrology through Quantum Signal Processing", arXiv:2209.11207, (2022).

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