High schoolers excel at Oxford quantum course using pictorial mathematics
Bob Coecke, Aleks Kissinger, Stefano Gogioso, Selma Dündar‐Coecke, Caterina Puca, Lia Yeh, Muhammad Waseem, Emmanuel M. Pothos, Sieglinde M.-L. Pfaendler, Vincent Wang-Maścianica, Thomas Cervoni, Ferdi Tomassini, Vincent Anandraj, Peter Sigrist, Ilyas Khan
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.
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.
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.
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.
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 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.
Nothing endures but change. – Heraclitus of Ephesus, 535–475 BC Who Are You and What Do You Want? While there is already a plethora of textbooks on quantum theory and its features, this one is unique because it is based on quantum picturalism. Prerequisites. There are hardly any prerequisites to this textbook. We do not expect our readers to have a background in physics or computer science or to have any profound background in mathematics. In principle, some basic secondary-school mathematics should be sufficient. For example, linear algebra (and of course quantum theory) is presented from scratch in a diagrammatic manner. However, this does not mean that the first half of this book will be a boring read for the specialist, since these presentations from scratch are radically different from the usual ones. Target audience. Given its low entrance fee, as well as its unique form and content, this textbook should appeal to a broad audience, ranging from students to experts from a wide range of disciplines including physicists, computer scientists, mathematicians, logicians, philosophers of science, and researchers from other areas with a multidisciplinary interest, such as biologists, engineers, cognitive scientists, and educational scientists. One particular target audience for this book consists of students and researchers in quantum computation and quantum information, as we will apply the tools from quantum picturalism directly to these areas. A practising quantum computing researcher may discover a new set of tools to attack open problems where traditional methods have failed, and a student may find some subjects explained in a manner that is much easier to grasp. Another target audience consists of students and experts with an interest in foundations and/or philosophy of physics, who can read in this book about a process-oriented approach to physics that takes composition of systems as a first-class citizen, rather than a derived notion. In particular, this is the first book that uses diagrammatic language to capture the idea of a process theory and puts such process theories forward as a new foundation for quantum theory, in which all standard quantum theoretical notions can be expressed.
Quantum computing is an emerging field with growing implications across science and industry, making early educational exposure increasingly important. This paper examines how quantum computing concepts can be introduced into high-school STEM curricula within existing structures to enhance foundational learning in mathematics, computer science, and physics. We outline a modular integration strategy introducing key quantum ideas into standard courses, leveraging open-source educational resources to ensure global accessibility. Emphasis is placed on educational opportunity and equity: the approach is designed to be inclusive and to bridge current curricular gaps so that students worldwide can develop basic quantum literacy. Our analysis demonstrates that integrating quantum topics at the secondary level is feasible and can enrich STEM learning.