From Pulses to Circuits and Back Again: A Quantum Optimal Control Perspective on Variational Quantum Algorithms

The last decade has witnessed remarkable progress in the development of quantum technologies. Although fault-tolerant devices likely remain years away, the noisy intermediate-scale quantum devices of today may be leveraged for other purposes. Leading candidates are variational quantum algorithms (VQAs), which have been developed for applications including chemistry, optimization, and machine learning, but whose implementations on quantum devices have yet to demonstrate improvements over classical capabilities. In this Perspective, we propose a variety of ways that the performance of VQAs could be informed by quantum optimal control theory. A major theme throughout is the need for sufficient control resources in VQA implementations; we discuss different ways this need can manifest, outline a variety of open questions, and look to the future.

Popular Summary

Quantum computers have the potential to enable significant advances spanning numerous scientific disciplines. This fact has led to the development of the first prototype quantum computers in the last few years. However, these early quantum computers remain small and error prone, which significantly restricts the scope of the computations they can perform. As such, a major question is: given their limitations, how can we use near-term quantum computers to solve scientific problems?

This question has motivated the development of variational quantum algorithms (VQAs), which attempt to draw out the full power of near-term quantum computers by linking them to classical computers in a learning control framework. Importantly, such quantum-classical learning control approaches have also proven very successful in another setting: optimizing the controlled dynamics of diverse quantum systems in the context of quantum optimal control (QOC).

In this Perspective, we explore how further progress can be made by considering VQAs, their associated challenges, and potential paths forward, through the lens of QOC. Following a review of the state of the art in VQA theory and experiment and a review of the field of QOC, we explain connections between these two active areas of research. Then, by providing a common framework that captures VQAs and a wide swath of QOC research, we emphasize the need for appropriate control resources to enhance the performance of VQAs, and identify several interesting research directions at the intersection of these two topics that could provide fertile ground for future work.

Figure 1

The quantum-classical optimization loop used in VQA and QOC experiments is shown. The quantum device (green) evaluates the objective functional, whose value is determined via measurements and input into a classical computer (yellow), which updates the optimization parameters. In conventional QOC experiments, these parameters are defined at the pulse level (a), and serve to parametrize a set of control fields $\{f_k(t)\}$ entering in a Hamiltonian describing the physics of the system. In VQAs, the parameters typically enter at the circuit level (b), via a set of gates parametrized through a set of e.g., gate angles $\{{\theta }_i\}$. In this Perspective, we explore how VQAs may be informed by returning to a more physical, pulse-level description inspired by QOC.

Figure 2

A representation of the hierarchy of abstractions used to model quantum hardware, and how variational optimization enters into each level of the hierarchy. Output produced by the hardware is the desired objective function, $J$, which is parametrized in different ways by each level of the hierarchy—by control-field parameters $c_i$ at the pulse level (i), by circuit parameters (usually angles ${\theta }_i$) at the quantum circuit level (ii), and by the structure of the circuit and the encoding and decoding maps (${\mathcal{E}}_{\varphi }$ and ${\mathcal{D}}_{\varphi }$) at the logical circuit level (iii). By understanding the relationship between the natural variational parametrizations at each of the levels, we can develop a rich family of variational ansätze.