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arXiv:2406.16556 · Physical Review Physics Education Research

Visualizing quantum states: A pilot study on problem solving in quantum information science education

Jonas Bley, Eva Rexigel, Alda Arias, Lars Krupp, Steffen Steinert, Nikolas Longen, Paul Lukowicz, Stefan Küchemann, Jochen Kuhn, Maximilian Kiefer-Emmanouilidis, Artur Widera

Published Nov 14, 2025• 1 citation
arXivHTMLPDF

Abstract

In the rapidly evolving interdisciplinary field of quantum information science and technology, a major obstacle is the need to understand advanced mathematics to solve complex problems. Current findings in educational research suggest that incorporating visualizations into problem-solving settings can have beneficial effects on students’ performance and cognitive load compared to relying solely on symbolic problem-solving content. Visualizations like the (dimensional) circle notation enable us to represent not only single-qubit but also more complex multiqubit states, entanglement, and quantum algorithms. In this pilot study, we aim to take an initial step toward identifying the contexts in which students benefit from the presentation of visualizations of single- and multiqubit systems in addition to mathematical formalism. For this purpose, we propose a set of test items and a comprehensive methodology to assess students’ performance and cognitive load when solving problems. This is a pilot investigation with a large breadth of questions intended to generate hypotheses and guide larger-scale, more focused studies in the future. Specifically, we compare two approaches: using the mathematical-symbolic Dirac notation alone and using it in combination with the (dimensional) circle notation. In surveys in one-, two- and three-qubit systems, we gather qualitative data from five, five and two think-aloud interviews, identifying problems that students encounter and their problem-solving strategies. In addition, we analyze quantitative data (performance and cognitive load) from 23, 27, and 17 participants in surveys on one-, two-, and three-qubit systems recruited mainly from our quantum computing lectures. We find that most of the test items are appropriate for a heterogeneous target group, as they can differentiate between participants in terms of performance and time taken. In general, the A-B crossover structure of the study is suitable for investigating the benefits of visualization for problem solving with regard to length and feasibility. Future studies should, however, be narrower in scope, given the observable dependence on student characteristics and context, with particular interest lying in the further investigation of the Hadamard gate, the gate, and entanglement in multiqubit systems.

DOI
10.1103/hdpv-frft

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•Jan 1, 2018•0 citations
In the rapidly evolving interdisciplinary field of quantum information science and technology, a major obstacle is the need to understand advanced mathematics to solve complex problems. Current findings in educational research suggest that incorporating visualizations into problem-solving settings can have beneficial effects on students’ performance and cognitive load compared to relying solely on symbolic problem-solving content. Visualizations like the (dimensional) circle notation enable us to represent not only single-qubit but also more complex multiqubit states, entanglement, and quantum algorithms. In this pilot study, we aim to take an initial step toward identifying the contexts in which students benefit from the presentation of visualizations of single- and multiqubit systems in addition to mathematical formalism. For this purpose, we propose a set of test items and a comprehensive methodology to assess students’ performance and cognitive load when solving problems. This is a pilot investigation with a large breadth of questions intended to generate hypotheses and guide larger-scale, more focused studies in the future. Specifically, we compare two approaches: using the mathematical-symbolic Dirac notation alone and using it in combination with the (dimensional) circle notation. In surveys in one-, two- and three-qubit systems, we gather qualitative data from five, five and two think-aloud interviews, identifying problems that students encounter and their problem-solving strategies. In addition, we analyze quantitative data (performance and cognitive load) from 23, 27, and 17 participants in surveys on one-, two-, and three-qubit systems recruited mainly from our quantum computing lectures. We find that most of the test items are appropriate for a heterogeneous target group, as they can differentiate between participants in terms of performance and time taken. In general, the A-B crossover structure of the study is suitable for investigating the benefits of visualization for problem solving with regard to length and feasibility. Future studies should, however, be narrower in scope, given the observable dependence on student characteristics and context, with particular interest lying in the further investigation of the Hadamard gate, the gate, and entanglement in multiqubit systems.
Quantum information science (QIS) is a vast, diverse, and abstract field. In consequence, learners face many challenges. Science, technology, engineering, and mathematics education research has found that visualizations are valuable for aiding learners in complex matters. The conditions under which visualizations pose benefits are largely unexplored in QIS education. In this eye-tracking study, we examine the conditions under which the visualization of multiqubit systems with the dimensional circle notation (DCN) in addition to the mathematical symbolic Dirac notation (DN) is associated with a benefit for solving problems on the ubiquitously used adamard gate operation, in terms of performance, extraneous cognitive load (ECL) and intrinsic cognitive load (ICL). We find that DCN increases performance and reduces cognitive load for participants with little experience in quantum physics. In addition, representational competence predicts reductions in ECL with DCN, but not performance or ICL. Analysis of the eye-tracking results indicates that task solvers with more transitions between DN and DCN benefit less from the visualization. We discuss the generalizability of the results and practical implications.
In the field of quantum information science and technology, the representation and visualization of quantum states and related processes are essential for both research and education. In this context, a focus lies especially on ensembles of few qubits. There exist many powerful representations for single-qubit and multiqubit systems, such as the famous Bloch sphere and generalizations. Here, we utilize the dimensional circle notation as a representation of such ensembles, adapting the so-called circle notation of qubits and the idea of representing the n -particle system in an n -dimensional space. We show that the mathematical conditions for separability lead to symmetry conditions of the quantum state visualized, offering a new perspective on entanglement in few-qubit systems and therefore on various quantum algorithms. In this way, dimensional notations promise significant potential for conveying nontrivial quantum entanglement properties and processes in few-qubit systems to a broader audience, and could enhance understanding of these concepts as a bridge between intuitive quantum insight and formal mathematical descriptions. Published by the American Physical Society 2024
Abstract The revolutionary new field of quantum computing (QC) continues to gain attention in industry, academia, and government in both research and education. At educational institutions, there is a proliferation of introductory courses at various academic levels signaling a growing interest and recognition of the significance of this field. A crucial and often overlooked aspect is the development of research-based materials and pedagogical approaches to effectively teach the complexities of QC to diverse cohorts of learners across multiple disciplines. There is a great need for empirical investigations of the effectiveness of learning materials and pedagogical approaches in this new interdisciplinary field. We present an empirical investigation done at an R1 institution using the multiple case study method. We compare a case study on students in an introductory QC course without research-based mini-tutorials to a study of students taking the QC course with research-based mini-tutorials. We compare the strengths and difficulties of students in the two courses, discuss the general strengths and difficulties of students across both courses, postulate the effectiveness of the mini-tutorials and discuss how they can be revised. Strengths across both classes include the ability to apply single-qubit and two-qubit gates, favoring application of Dirac notation, and a reasonable understanding of normalization, probability and teleportation described qualitatively. Difficulties across both classes included use of matrix representation, use of rotation gates, and an ability to recall and analyze quantitatively a circuit representing teleportation. We postulate that the mini-tutorials on gates, normalization, probability and a qualitative description of teleportation may have been effective in increasing students’ understanding of the concepts. We also identify mini-tutorials that may need revision.
Abstract In quantum circuits, qubits and the quantum gates acting on them have traditionally been analysed using matrix algebra and Dirac notation. While powerful, these can be unintuitive for conceptual understanding and rapid problem solving. In this work, a new schematic representation method is developed that visualizes the effects of quantum gates on qubits without relying on complex mathematical operations. In the new notation, quantum bits (qubits) are represented using black (|0〉) and white (|1〉) circles. When a quantum gate is applied to a qubit, the circle representing the qubit is visually modified. For example, a Hadamard gate transforms a solid black or white circle into a half-black and half-white circle representing superposition. The work shows how this method simplifies the visualization of quantum algorithms, entanglement, and multi- qubit operations. The work proposes how this method can act as a visual aid to support the understanding of quantum algorithms, entanglement, and multi-qubit operations. Thus, the initial conceptual analysis of basic quantum circuits can be facilitated by simple schematic representations, reducing the immediate cognitive load of complex mathematical operations for beginners.
[This paper is part of the Focused Collection in Investigating and Improving Quantum Education through Research.] Quantum information science and engineering (QISE) is a rapidly developing field that leverages the skills of experts from many disciplines to utilize the potential of quantum systems in a variety of applications. It requires talent from a wide variety of traditional fields, including physics, engineering, chemistry, and computer science, to name a few. To prepare students for such opportunities, it is important to give them a strong foundation in the basics of QISE, in which quantum computing plays a central role. In this study, we discuss the development, validation, and evaluation of a Quantum Interactive Learning Tutorial, on the basics and applications of quantum computing. These include an overview of key quantum mechanical concepts relevant to quantum computation (including ways a quantum computer is different from a classical computer), properties of single- and multiqubit systems, and the basics of single-qubit quantum gates. The tutorial uses guided inquiry-based teaching-learning sequences. Its development and validation involved conducting cognitive task analysis from both expert and student perspectives and using common student difficulties as a guide. For example, before engaging with the tutorial, after traditional lecture-based instruction, one reasoning primitive that was common in student responses is that a major difference between an N -bit classical and N -qubit quantum computer is that various things associated with a number N for a classical computer should be replaced with the number 2 N for a quantum computer (e.g., 2 N qubits must be initialized and 2 N bits of information are obtained as the output of the computation on the quantum computer). This type of reasoning primitive also led many students to incorrectly think that there are only N distinctly different states available when computation takes place on a classical computer. Research suggests that this type of reasoning primitive has its origins in students learning that quantum computers can provide exponential advantage for certain problems, e.g., Shor’s algorithm for factoring products of large prime numbers, and that the quantum state during the computation can be in a superposition of 2 N linearly independent states. The inquiry-based learning sequences in the tutorial provide scaffolding support to help students develop a functional understanding. The final version of the validated tutorial was implemented in two distinct courses offered by the physics department with slightly different student populations and broader course goals. Students’ understanding was evaluated after traditional lecture-based instruction on the requisite concepts and again after engaging with the tutorial. We analyze and discuss their improvement in performance on concepts covered in the tutorial. Published by the American Physical Society 2024
My research has three main foci, one which is of interest to the fields of quantum information and quantum optics is to characterize the effects of coupling between quantum systems. Advances in technology made since the conception of quantum mechanics spurred the development of quantum technologies like quantum optics and quantum computing. These applications require a good understanding of the quantum systems that compose them, and the clever ways in which the information in these systems is stored and manipulated. By understanding the ways which quantum systems interact with their environment, their usefulness can be further utilized. The second is to develop methods of calculation which simplify the calculational and computational side of quantum dynamics. Understanding how quantum systems evolve in time is very important, but traditional methods of solving for the dynamics of quantum systems struggle to deal with time dependent parameters. Using results from the theory of Lie algebras, I will show how quantum dynamics problems can be made less computationally intensive by avoiding the partial differential equations typically associated with such problems and reducing the problem to a system of coupled ordinary differential equations. The third focus is to develop methods of visualizing properties of quantum systems. I think this is important as an educational tool and as a way to develop intuition about how quantum systems evolve considering how few opportunities there are to visualize quantum effects on a macroscopic scale. This problem is particularly challenging because quantum mechanics is non-deterministic, meaning that the only nontrivial definitive statements that can be made regarding quantum systems deal with probabilities. This coupled with the fact that many of the systems of interest depend on upwards of six variables and also evolve in time make creating useful visual representations challenging. Despite the challenges, I feel that useful insights can be gained from the visualizations, and will illustrate a few examples.