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arXiv:2512.00141

High schoolers excel at Oxford quantum course using pictorial mathematics

Bob Coecke, Aleks Kissinger, Stefano Gogioso, Selma Dündar-Coecke, Caterina Puca, Lia Yeh, Muhammad Hamza Waseem, Emmanuel M. Pothos, Sieglinde Pfaendler, Vincent Wang-Mascianica, Thomas Cervoni, Ferdi Tomassini, Vincent Anandraj, Peter Sigrist, Ilyas Khan

physics.ed-phcs.LOmath.CTquant-phSubmitted Nov 28, 2025• Announced Dec 1, 2025• 0 citations
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Abstract

We are at the dawn of the second quantum revolution, where our ability to create and control individual quantum systems is poised to drive transformative advancements in basic science, computation, and everyday life. However, quantum theory has long been conceived as notoriously hard to learn, creating a significant barrier to workforce development, informed decision-making by stakeholders and policymakers, and broader public understanding. This paper is concerned with Quantum Picturalism, a novel visual mathematical language for quantum physics. Originally developed over two decades ago to explore the foundational structure of quantum theory, this rigorous diagrammatic framework has since been adopted in both academia and industry as a powerful tool for quantum computing research and software development. Here, we demonstrate its potential as a transformative educational methodology. We report the findings from a pilot study involving 54 UK high school students, randomly selected from a pool of 734 volunteers across the UK. Despite the absence of advanced mathematical prerequisites, these students demonstrated a strong conceptual grasp of key quantum principles and operations. On an assessment comprising university graduate-level exam questions, participants achieved an 82% pass rate, with 48% obtaining a distinction-level grade. These results pave the way for making quantum more inclusive, lowering traditional cognitive and demographic barriers to quantum learning. This approach has the potential to broaden participation in the field and provide a promising new entry point for stakeholders, future experts, and the general public.

Comments
9 pages, many pictures

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High schoolers excel at Oxford quantum course using pictorial mathematics

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Arnab Buragohain

Quantum computing has emerged as one of the most influential technological developments of the twenty-first century. While considerable attention has been devoted to hardware development, algorithm design, and industrial applications, comparatively less emphasis has been placed on the educational challenges associated with introducing quantum concepts to engineering students. Most available learning resources are designed primarily for physics students and often require prior knowledge of advanced quantum mechanics. Consequently, engineering students frequently encounter significant conceptual barriers when attempting to understand quantum computation. This chapter presents a mathematics-first educational framework aimed at simplifying the learning process for engineering students. Instead of introducing quantum theory through wave mechanics and advanced physical interpretations, the framework begins with familiar mathematical concepts such as vectors, matrices, probability, and linear transformations. The chapter demonstrates how core quantum concepts can be taught using accessible mathematical tools already encountered in undergraduate engineering curricula. A structured educational model is proposed and illustrated through practical examples, matrix operations, simple quantum gates, and engineering-oriented learning activities. The framework is intended to reduce cognitive barriers, improve conceptual understanding, and encourage interdisciplinary learning among future engineers. The chapter further discusses potential applications of quantum mathematics in thermal engineering, computational fluid dynamics, optimization, artificial intelligence, and advanced manufacturing.

•Aug 1, 2026•0 citations
We are at the dawn of the second quantum revolution, where our ability to create and control individual quantum systems is poised to drive transformative advancements in basic science, computation, and everyday life. However, quantum theory has long been conceived as notoriously hard to learn, creating a significant barrier to workforce development, informed decision-making by stakeholders and policymakers, and broader public understanding.
This paper is concerned with Quantum Picturalism, a novel visual mathematical language for quantum physics. Originally developed over two decades ago to explore the foundational structure of quantum theory, this rigorous diagrammatic framework has since been adopted in both academia and industry as a powerful tool for quantum computing research and software development. Here, we demonstrate its potential as a transformative educational methodology.
We report the findings from a pilot study involving 54 UK high school students, randomly selected from a pool of 734 volunteers across the UK. Despite the absence of advanced mathematical prerequisites, these students demonstrated a strong conceptual grasp of key quantum principles and operations. On an assessment comprising university graduate-level exam questions, participants achieved an 82% pass rate, with 48% obtaining a distinction-level grade.
These results pave the way for making quantum more inclusive, lowering traditional cognitive and demographic barriers to quantum learning. This approach has the potential to broaden participation in the field and provide a promising new entry point for stakeholders, future experts, and the general public.
Abstract We are at the dawn of the second quantum revolution, where our ability to create and control individual quantum systems is poised to drive transformative advancements in basic science, computation, and everyday life. However, quantum theory has long been conceived as notoriously hard to learn, creating a significant barrier to workforce development, informed decision-making by stakeholders and policymakers, and broader public understanding. This paper is concerned with Quantum Picturalism [15], a novel visual mathematical language for quantum physics. Originally developed over two decades ago to explore the foundational structure of quantum theory [2], this rigorous diagrammatic framework has since been adopted in both academia and industry as a powerful tool for quantum computing research and software development. Here, we demonstrate its potential as a transformative educational methodology. We report the findings from a pilot study involving 54 UK high school students [27], randomly selected from a pool of 734 volunteers across the UK. Despite the absence of advanced mathematical prerequisites, these students demonstrated a strong conceptual grasp of key quantum principles and operations. On an assessment comprising university graduate-level exam questions, participants achieved an 82% pass rate, with 48% obtaining a distinction-level grade. These results pave the way for making quantum more inclusive, lowering traditional cognitive and demographic barriers to quantum learning. This approach has the potential to broaden participation in the field and provide a promising new entry point for stakeholders, future experts, and the general public.
Quantum theory is often regarded as challenging to learn and teach, with advanced mathematical prerequisites ranging from complex numbers and probability theory to matrix multiplication, vector space algebra and symbolic manipulation within the Hilbert space formalism. It is traditionally considered an advanced undergraduate or graduate-level subject. In this work, we challenge the conventional view by proposing “Quantum Picturalism” as a new approach to teaching the fundamental concepts of quantum theory and computation. We establish the foundations and methodology for an ongoing educational experiment to investigate the question “From what age can students learn quantum theory if taught using a dia-grammatic approach?”. We anticipate that the primary benefit of leveraging such a diagrammatic approach, which is conceptually intuitive yet mathematically rigorous, will be eliminating some of the most daunting barriers to teaching and learning this subject while enabling young learners to reason proficiently about high-level problems. We posit that transitioning from symbolic presentations to pictorial ones will increase the appeal of STEM education, attracting more diverse audience.
The educational value of a fully diagrammatic approach in a scientific field has never been explored. We present Quantum Picturalism (QPic), an entirely diagrammatic formalism for all of qubit quantum mechanics. This framework is particularly advantageous for young learners as a novel way to teach key concepts such as entanglement, measurement, and mixed-state quantum mechanics in a math-intensive subject. This eliminates traditional obstacles without compromising mathematical correctness - removing the need for matrices, vectors, tensors, complex numbers, and trigonometry as prerequisites to learning. Its significance lies in that a field as complex as Quantum Information Science and Technology (QIST), for which educational opportunities are typically exclusive to the university level and higher, can be introduced at high school level. In this study, we tested this hypothesis, examining whether QPic reduces cognitive load by lowering complex mathematical barriers while enhancing mental computation and conceptual understanding. The data was collected from an experiment conducted in 2023, whereby 54 high school students (aged 16-18) underwent 16 hours of training spread over eight weeks. The post-assessments illustrated promising outcomes in all three specific areas of focus: (1) whether QPic can alleviate technical barriers in learning QIST, (2) ensures that the content and teaching method are age appropriate, (3) increases confidence and motivation in science and STEM fields. There was a notable success rate in terms of teaching outcomes, with 82% of participants successfully passing an end-of-training exam and 48% achieving a distinction, indicating a high level of performance. The unique testing and training regime effectively reduced the technical barriers typically associated with traditional approaches, as hypothesized.
The more advanced the symbolic mathematics, the more impenetrable the meaning. This is particularly evident in quantum theory, where symbolic formalisms foreground states and objects. An alternative diagrammatic approach, Quantum Picturalism (QPic), privileges structure, relation, and transformation. As a category theoretic formalism, it supports reasoning through the composition of visually represented quantum processes, a distinction that not only concerns representation but also shapes how quantum phenomena are conceptualized, opening new educational possibilities and ways of knowing. The choice of formalism—algebraic or diagrammatic—is not neutral, but embodies underlying ontological commitments that shape which structural features of quantum phenomena—such as states vs. relations; objects vs. transformations—are made accessible. This paper examines the epistemological, cognitive, and educational significance of non-symbolic thought in mathematics, and in quantum mechanics in particular, using QPic as a guiding example. We raise a central question: can quantum processes be represented in ways that align more naturally with human cognition? What is new here is showing how the epistemological and cognitive aspects of non-symbolic formalism are brought into pedagogical practice. Such paradigm largely benefits from the language of thought hypothesis alongside theories of embodied cognition, dual coding, and conceptual metaphor theory to articulate a holistic framework in which multiple, interacting conceptual systems underpin understanding. Drawing on historical precedents—from Euclidean geometry to Leibniz's universal calculus of thought—we contend that human intuition works more naturally within relational and visual frameworks, particularly those that help complex ideas to be seen and manipulated as structured wholes. The broader impact of this paradigm lies in democratizing access to quantum innovation and rethinking what it means to understand the quantum world, offering an inclusive entry point into it.
Quantum computing was once regarded as a mere theoretical possibility, but recent advances in engineering and materials science have brought practical quantum computers closer to reality. Currently, representatives from industry, academia, and governments across the world are working to build the educational structures needed to produce the quantum workforce of the future. Less attention has been paid to growing quantum computing capacity at the high school level. This article details work at The University of Texas at Austin to develop and pilot the first full-year high school quantum computing class. Over the course of two years, researchers and practitioners involved with the project learned several pedagogical and practical lessons that can be helpful for quantum computing course design and implementation at the secondary level. In particular, we find that the use of classical optics provides a clear and accessible avenue for representing quantum states and gate operators and facilitates both learning and the transfer of knowledge to other Science, Technology, and Engineering (STEM) skills. Furthermore, students found that exploring quantum optical phenomena prior to the introduction of mathematical models helped in the understanding and mastery of the material.
Quantum computing has emerged as one of the most influential technological developments of the twenty-first century. While considerable attention has been devoted to hardware development, algorithm design, and industrial applications, comparatively less emphasis has been placed on the educational challenges associated with introducing quantum concepts to engineering students. Most available learning resources are designed primarily for physics students and often require prior knowledge of advanced quantum mechanics. Consequently, engineering students frequently encounter significant conceptual barriers when attempting to understand quantum computation. This chapter presents a mathematics-first educational framework aimed at simplifying the learning process for engineering students. Instead of introducing quantum theory through wave mechanics and advanced physical interpretations, the framework begins with familiar mathematical concepts such as vectors, matrices, probability, and linear transformations. The chapter demonstrates how core quantum concepts can be taught using accessible mathematical tools already encountered in undergraduate engineering curricula. A structured educational model is proposed and illustrated through practical examples, matrix operations, simple quantum gates, and engineering-oriented learning activities. The framework is intended to reduce cognitive barriers, improve conceptual understanding, and encourage interdisciplinary learning among future engineers. The chapter further discusses potential applications of quantum mathematics in thermal engineering, computational fluid dynamics, optimization, artificial intelligence, and advanced manufacturing.