# On the energy landscape of symmetric quantum signal processing

Jiasu Wang1, Yulong Dong1, and Lin Lin1,2,3

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

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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.

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### Cited by

[1] 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).

[2] Zane M. Rossi and Isaac L. Chuang, "Multivariable quantum signal processing (M-QSP): prophecies of the two-headed oracle", arXiv:2205.06261.

[3] Di Fang, Lin Lin, and Yu Tong, "Time-marching based quantum solvers for time-dependent linear differential equations", arXiv:2208.06941.

[4] Patrick Rall and Bryce Fuller, "Amplitude Estimation from Quantum Signal Processing", arXiv:2207.08628.

[5] Lexing Ying, "Stable factorization for phase factors of quantum signal processing", arXiv:2202.02671.

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

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

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