Here Comes The SU(N): Multivariate Quantum Gates And Gradients

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Roeland Wiersema^1,2, Dylan Lewis³, David Wierichs⁴, Juan Carrasquilla^1,2, and Nathan Killoran⁴

¹Vector Institute, MaRS Centre, Toronto, Ontario, M5G 1M1, Canada
²Department of Physics and Astronomy, University of Waterloo, Ontario, N2L 3G1, Canada
³Department of Physics and Astronomy, University College London, London WC1E 6BT, United Kingdom
⁴Xanadu, Toronto, ON, M5G 2C8, Canada

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Abstract

Variational quantum algorithms use non-convex optimization methods to find the optimal parameters for a parametrized quantum circuit in order to solve a computational problem. The choice of the circuit ansatz, which consists of parameterized gates, is crucial to the success of these algorithms. Here, we propose a gate which fully parameterizes the special unitary group $mathrm{SU}(N)$. This gate is generated by a sum of non-commuting operators, and we provide a method for calculating its gradient on quantum hardware. In addition, we provide a theorem for the computational complexity of calculating these gradients by using results from Lie algebra theory. In doing so, we further generalize previous parameter-shift methods. We show that the proposed gate and its optimization satisfy the quantum speed limit, resulting in geodesics on the unitary group. Finally, we give numerical evidence to support the feasibility of our approach and show the advantage of our gate over a standard gate decomposition scheme. In doing so, we show that not only the expressibility of an ansatz matters, but also how it’s explicitly parameterized.

Featured image: Rethinking the parameterization of general unitary gates.

Our code is freely available on Github:
https://github.com/dwierichs/Here-comes-the-SUN

There is a Demo that illustrates some of the key points of the paper:
https://pennylane.ai/qml/demos/tutorial_here_comes_the_sun/

Popular summary

In the realm of variational quantum computing, numerous circuit ansätze exist, yet the quest for a time-efficient circuit with optimal trainability remains a challenge. We introduce a new type of multivariate quantum gate, called an $mathrm{SU}(N)$ gate and show how to differentiate it on quantum hardware. We explore gate speed limits, biases in gradient-based training as well as trainability in practice. We argue that our proposed SU(N) gate has advantages over other general unitary gates with both qualitative and quantitative arguments, which illustrates how important it is to choose the right parameterization for a variational quantum gate.

► BibTeX data

@article{Wiersema2024herecomessun, doi = {10.22331/q-2024-03-07-1275}, url = {https://doi.org/10.22331/q-2024-03-07-1275}, title = {Here comes the {SU}({N}): multivariate quantum gates and gradients}, author = {Wiersema, Roeland and Lewis, Dylan and Wierichs, David and Carrasquilla, Juan and Killoran, Nathan}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{“{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {8}, pages = {1275}, month = mar, year = {2024} }

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Cited by

[1] Ronghang Chen, Zhou Guang, Cong Guo, Guanru Feng, and Shi-Yao Hou, “Pure quantum gradient descent algorithm and full quantum variational eigensolver”, Frontiers of Physics 19 2, 21202 (2024).

[2] David Wierichs, Richard D. P. East, Martín Larocca, M. Cerezo, and Nathan Killoran, “Symmetric derivatives of parametrized quantum circuits”, arXiv:2312.06752, (2023).

[3] Yaswitha Gujju, Atsushi Matsuo, and Rudy Raymond, “Quantum Machine Learning on Near-Term Quantum Devices: Current State of Supervised and Unsupervised Techniques for Real-World Applications”, arXiv:2307.00908, (2023).

[4] Korbinian Kottmann and Nathan Killoran, “Evaluating analytic gradients of pulse programs on quantum computers”, arXiv:2309.16756, (2023).

The above citations are from SAO/NASA ADS (last updated successfully 2024-03-07 16:43:15). The list may be incomplete as not all publishers provide suitable and complete citation data.

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Source: https://quantum-journal.org/papers/q-2024-03-07-1275/

Time Stamp: March 7, 2024

Time Stamp: Nov 6, 2023

Here comes the SU(N): multivariate quantum gates and gradients

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Quantum Optimal Control via Semi-Automatic Differentiation

Finite-time Landauer principle beyond weak coupling

On the role of entanglement in qudit-based circuit compression

Minimum Trotterization Formulas for a Time-Dependent Hamiltonian

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