Product Decomposition of Periodic Functions in Quantum Signal Processing

Jeongwan Haah

Microsoft Quantum and Microsoft Research, Redmond, Washington, USA

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Abstract

We consider an algorithm to approximate complex-valued periodic functions $f(e^{i\theta})$ as a matrix element of a product of $SU(2)$-valued functions, which underlies so-called quantum signal processing. We prove that the algorithm runs in time $\mathcal O(N^3 \mathrm{polylog}(N/\epsilon))$ under the random-access memory model of computation where $N$ is the degree of the polynomial that approximates $f$ with accuracy $\epsilon$; previous efficiency claim assumed a strong arithmetic model of computation and lacked numerical stability analysis.

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[2] Koen Groenland, Freek Witteveen, Kareljan Schoutens, and Rene Gerritsma, "Signal processing techniques for efficient compilation of controlled rotations in trapped ions", New Journal of Physics 22 6, 063006 (2020).

[3] Guang Hao Low, Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing - STOC 2019 491 (2019) ISBN:9781450367059.

[4] Guang Hao Low and Isaac L. Chuang, "Hamiltonian Simulation by Qubitization", Quantum 3, 163 (2019).

[5] András Gilyén, Yuan Su, Guang Hao Low, and Nathan Wiebe, Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing - STOC 2019 193 (2019) ISBN:9781450367059.

[6] Guang Hao Low and Isaac L. Chuang, "Hamiltonian Simulation by Qubitization", arXiv:1610.06546.

[7] Minh C. Tran, Andrew Y. Guo, Yuan Su, James R. Garrison, Zachary Eldredge, Michael Foss-Feig, Andrew M. Childs, and Alexey V. Gorshkov, "Locality and Digital Quantum Simulation of Power-Law Interactions", Physical Review X 9 3, 031006 (2019).

[8] Jeongwan Haah, Matthew B. Hastings, Robin Kothari, and Guang Hao Low, "Quantum algorithm for simulating real time evolution of lattice Hamiltonians", arXiv:1801.03922.

[9] Guang Hao Low, "Hamiltonian simulation with nearly optimal dependence on spectral norm", arXiv:1807.03967.

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The above citations are from Crossref's cited-by service (last updated successfully 2020-10-22 00:47:34) and SAO/NASA ADS (last updated successfully 2020-10-22 00:47:35). The list may be incomplete as not all publishers provide suitable and complete citation data.