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arXiv:2303.11355 · Quantum

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

Roeland Wiersema, Dylan Lewis, David Wierichs, Juan Carrasquilla, Nathan Killoran

Published Mar 7, 2024• 19 citations
arXivHTMLPDF

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 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.

DOI
10.22331/q-2024-03-07-1275

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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 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.
Abstract Currently, variational quantum classification algorithms (VQCAs) generally rely on traditional optimization techniques such as Powell and SLSQP in the parameter optimization session. However, the performance of these methods shows limitations in practical applications. Although the parameter-shift rule can efficiently compute the parameter gradient with quantum circuits, it needs to run the quantum circuit twice repeatedly, which significantly reduces the computation efficiency. In order to overcome this challenge, this paper innovatively integrates the principle of unitary operation in quantum mechanics with the technical characteristics of superconducting quantum chips and elaborately designs some new parameterized quantum gates (PQGs). These PQGs strictly follow the rules of unitary operation, which ensures the stability and accuracy of quantum state evolution while realizing an efficient solution to the parameter gradient. Especially for the gradient calculation of a single qubit and single-angle PQGs, the new method can be completed with only a single quantum circuit run, which greatly improves the computation efficiency. Experimental validation on benchmark datasets such as breast cancer and iris shows that the method proposed in this paper exhibits excellent performance on quantum classification tasks. Compared with the parameter-shift rule, the computation efficiency of the new method is improved by 40%. And the classification accuracy, precision, and other key performance metrics are improved by an average of 5% in comparison with traditional optimization algorithms. This work not only enriches the methodology of quantum machine learning theoretically but also demonstrates its remarkable superiority in practical applications, which indicates that the method has great potential in scientific research and industrial applications.
Abstract Many near-term quantum computing algorithms are conceived as variational quantum algorithms, in which parameterized quantum circuits are optimized in a hybrid quantum–classical setup. Examples are variational quantum eigensolvers, quantum approximate optimization algorithms as well as various algorithms in the context of quantum-assisted machine learning. A common bottleneck of any such algorithm is constituted by the optimization of the variational parameters. A popular set of optimization methods work on the estimate of the gradient, obtained by means of circuit evaluations. We will refer to the way in which one can combine these circuit evaluations as gradient rules. This work provides a comprehensive picture of the family of gradient rules that vary parameters of quantum gates individually. The most prominent known members of this family are the parameter shift rule (PSR) and the finite differences method. To unite this family, we propose a generalized PSR that expresses all members of the aforementioned family as special cases, and discuss how all of these can be seen as providing access to a linear combination of exact first- and second-order derivatives. We further prove that a PSR with one non-shifted evaluation and only one shifted circuit evaluation does not exist, and introduce a novel perspective for approaching new gradient rules.
We propose SUN-VQC, a variational-circuit architecture whose elementary layers are single exponentials of a symmetry-restricted Lie subgroup, SU(2k)⊂SU(2n)\mathrm{SU}(2^{k}) \subset \mathrm{SU}(2^{n})SU(2k)⊂SU(2n) with k≪nk \ll nk≪n. Confining the evolution to this compact subspace reduces the dynamical Lie-algebra dimension from O(4n)\mathcal{O}(4^{n})O(4n) to O(4k)\mathcal{O}(4^{k})O(4k), ensuring only polynomial suppression of gradient variance and circumventing barren plateaus that plague hardware-efficient ansätze. Exact, hardware-compatible gradients are obtained using a generalized parameter-shift rule, avoiding ancillary qubits and finite-difference bias. Numerical experiments on quantum auto-encoding and classification show that SUN-VQCs sustain order-of-magnitude larger gradient signals, converge 2--3×\times× faster, and reach higher final fidelities than depth-matched Pauli-rotation or hardware-efficient circuits. These results demonstrate that Lie-subalgebra engineering provides a principled, scalable route to barren-plateau-resilient VQAs compatible with near-term quantum processors.
Variational quantum algorithms are a promising class of algorithms that can be performed on currently available quantum computers. In most settings, the free parameters of a variational circuit are optimized using a classical optimizer that updates parameters in Euclidean geometry. Since quantum circuits are elements of the special unitary group, we can consider an alternative optimization perspective that depends on the structure of this group. In this work, we investigate a Riemannian optimization scheme over the special unitary group and we discuss its implementation on a quantum computer. We illustrate that the resulting Riemannian gradient-flow algorithm has favorable optimization properties for deep circuits and that an approximate version of this algorithm can be performed on near-term hardware. We highlight the connections of our work with previously proposed heuristics like ADAPT-VQE and show that they can be understood as variants of our algorithm.
Parameterized quantum circuits (PQCs) are pivotal components of variational quantum algorithms (VQAs), which represent a promising pathway to quantum advantage in noisy intermediate-scale quantum (NISQ) devices. PQCs enable flexible encoding of quantum information through tunable quantum gates and have been successfully applied across domains such as quantum chemistry, combinatorial optimization, and quantum machine learning. Despite their potential, PQC performance on NISQ hardware is hindered by noise, decoherence, and the presence of barren plateaus, which can impede gradient-based optimization. To address these limitations, we propose novel methods for improving gradient-free optimizers Rotosolve, Fraxis, and FQS, incorporating information from previous parameter iterations. Our approach conserves computational resources by reallocating optimization efforts toward poorly optimized gates, leading to improved convergence. The experimental results demonstrate that our techniques consistently improve the performance of various optimizers, contributing to more robust and efficient PQC optimization.
Parameterized Quantum Circuits have emerged as a central framework for variational quantum algorithms in the NISQ regime, with major applications in quantum simulation, combinatorial optimization, and, more recently, quantum machine learning (QML). By developing a new approach for computing dynamical Lie algebras associated with quantum circuits combining parametrized gates and non-parametrized entangling gates, we aim to extend the applicability of recent algebraic results to architectures that are actually used in quantum machine learning.