Optimizing a parameterized controlled gate using Free Quaternion Selection

TL;DR

This paper introduces an extension of the Free Quaternion Selection method to optimize parameterized controlled gates in variational quantum algorithms, leading to more efficient, expressive, and shallower quantum circuits for various applications.

Contribution

It extends FQS to controlled gates, enabling local optimization of these gates in variational algorithms, improving circuit depth and expressibility.

Findings

Efficient optimization of controlled gates demonstrated across multiple quantum tasks.

Achieved shallower circuits for molecular time-evolution operators compared to Trotter decomposition.

Enhanced expressibility and circuit depth reduction in variational quantum algorithms.

Abstract

In variational quantum algorithms, parameterization is typically applied to single-qubit gates.In this study, we instead parameterize a generalized controlled gate and propose an algorithm to locally minimize the cost function by maximally optimizing these parameters. This method extends the Free Quaternion Selection (FQS) technique, which was originally developed for single-qubit gate optimization. To evaluate its performance, we apply the proposed method to a variety of quantum optimization tasks, including the Variational Quantum Eigensolver (VQE) for both Ising and molecular Hamiltonians, fidelity maximization in general variational quantum algorithms (VQAs), and unitary compilation of time evolution operators. Across these applications, our method demonstrates efficient optimization, enhanced expressibility, and the ability to construct shallower circuits compared to existing techniques. Moreover, the method can be generalized to optimize particle-number-conserving gates, which are particularly relevant for quantum chemistry. Leveraging this capability, we further demonstrate that the method achieves superior quantum compilation of molecular time-evolution operators by approximating them with shallower circuits than standard Trotter decomposition.