Quantum computing through the lens of control: A tutorial introduction

TL;DR

The paper addresses the challenge of understanding and developing quantum computing from a control-theoretic perspective, emphasizing robustness, scalability, and feedback concepts in the NISQ era. It builds a cohesive mathematical framework—from qubits and gates to density matrices and error models—and connects these to control-inspired methods, notably variational quantum algorithms (VQAs) and error mitigation/correction. Key contributions include a structured tutorial on building quantum algorithms with control-theory intuition, a detailed treatment of VQAs (including VQE, QAOA, and QML), and an accessible survey of quantum error correction and mitigation techniques (Shor code, 3-qubit codes, ZNE). The work aims to equip readers with the mathematical tools to analyze, design, and evaluate quantum algorithms under noise, highlighting practical pathways and research challenges at the interface of quantum computing and control. This synthesis fosters cross-disciplinary insight, enabling control theorists to contribute to quantum algorithm design and hardware-aware implementations toward more robust and scalable quantum computation.

Abstract

Quantum computing is a fascinating interdisciplinary research field that promises to revolutionize computing by efficiently solving previously intractable problems. Recent years have seen tremendous progress on both the experimental realization of quantum computing devices as well as the development and implementation of quantum algorithms. Yet, realizing computational advantages of quantum computers in practice remains a widely open problem due to numerous fundamental challenges. Interestingly, many of these challenges are connected to performance, robustness, scalability, optimization, or feedback, all of which are central concepts in control theory. This paper provides a tutorial introduction to quantum computing from the perspective of control theory. We introduce the mathematical framework of quantum algorithms ranging from basic elements including quantum bits and quantum gates to more advanced concepts such as variational quantum algorithms and quantum errors. The tutorial only requires basic knowledge of linear algebra and, in particular, no prior exposure to quantum physics. Our main goal is to equip readers with the mathematical basics required to understand and possibly solve (control-related) problems in quantum computing. In particular, beyond the tutorial introduction, we provide a list of research challenges in the field of quantum computing and discuss their connections to control.

Figures (10)

• Figure 1: Bloch sphere representation of a qubit $\ket{\psi}=\cos\frac{\vartheta}{2}\ket0+e^{i\varphi}\sin\frac{\vartheta}{2}\ket1$ for angles $\varphi$ and $\vartheta$. The Bloch sphere provides a useful illustration of a qubit, in particular how it generalizes a classical bit. While qubits can take values at any point on the sphere, classical bits can only take two values corresponding to $\ket0$ and $\ket1$.
• Figure 2: Circuit representation of a quantum gate acting on the input state $\ket{\psi_{\mathrm{in}}}$ and producing the output state $\ket{\psi_{\mathrm{out}}}$. Mathematically, the quantum gate is characterized via multiplication by the unitary matrix $U$, that is, $\ket{\psi_{\mathrm{out}}}=U\ket{\psi_{\mathrm{in}}}$.
• Figure 3: A generic quantum algorithm including an input state $\ket{\psi_0}$, a unitary matrix $U$, and a projective measurement with respect to the observable $\mathcal{M}$. The unitary matrix $U$ usually consists of parallel and series interconnections of smaller quantum gates (typically single- or two-qubit gates).
• Figure 4: Circuit representation of a projective measurement in the computational basis. Performing this measurement
• Figure 5: Basic scheme of a variational quantum algorithm (VQA). VQAs are a feedback interconnection of a parameterized quantum circuit $f(\theta)$ with an update rule $\theta^+=g(\theta,v)$ for the parameter vector $\theta$. Evaluating $f(\theta)$ requires executing the quantum algorithm with the unitary matrix $U(\theta)$ from \ref{['eq:vqa_U_def']} and the observable $\mathcal{M}$, compare \ref{['eq:vqa_f_definition']}. On the contrary, the update rule for $\theta$ is commonly implemented on a classical computer. In many VQAs, $f(\theta)$ plays the role of a cost function, in which case the update rule $g(\theta,v)$ is chosen such that $\theta$ iteratively converges to a (local) minimum of $f$.