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

Roeland Wiersema1,2, Dylan Lewis3, David Wierichs4, Juan Carrasquilla1,2, 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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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.

Our code is freely available on Github:

There is a Demo that illustrates some of the key points of the paper:

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.

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[2] 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).

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

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

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