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

• URL: http://arxiv.org/abs/2303.11355v2
• Date: Thu, 29 Feb 2024 16:39:47 GMT
• Title: Here comes the SU(N): multivariate quantum gates and gradients
• Authors: Roeland Wiersema, Dylan Lewis, David Wierichs, Juan Carrasquilla and Nathan Killoran
• Abstract summary: Variational quantum algorithms use non-commuting optimization methods to find optimal parameters for a parametrized quantum circuit. Here, we propose a gate which fully parameterizes the special unitary group $mathrm(N) gate. We show that the proposed gate and its optimization satisfy the quantum limit of the unitary group.
• Score: 1.7809113449965783
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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