By Kurusch Ebrahimi Fard, Yael Fleischmann, and Alexander Schmeding
NTNU Norwegian University of Science and Technology
Why the Mathematics of Quantum Computing Matters
Quantum computing [4, 6] is an emerging area of research with growing relevance across science and technology. By harnessing quantum mechanical phenomena such as superposition, entanglement, and interference, quantum computers offer fundamentally new ways of processing information – beyond what is feasible with computation on classical machines. As a result, quantum computing is beginning to reshape a wide range of fields, from cryptography and combinatorial optimization to materials science, chemistry, and machine learning.
While large-scale, fault-tolerant quantum computers1are still under development, progress in quantum hardware, algorithms, and hybrid quantum-classical methods has been rapid over the last years [3]. This acceleration has brought with it a growing need for a new generation of scientists and engineers who not only understand how quantum algorithms work, but who also possess the mathematical maturity required to reason about them rigorously. In particular, there is an increasing demand for students who can move fluently between abstract mathematical structures and concrete computational ideas.
At the heart of quantum computing lies a rich and elegant mathematical framework [2, 5, 7, 8]: Hilbert spaces, linear operators, tensor products, unitary dynamics, and probabilistic measurement. These concepts form the language in which quantum algorithms are formulated and analyzed. Yet, despite their central role, they remain challenging to teach within traditional curricula. Often, they are introduced only in fragmented and disconnected ways, spread across courses in linear algebra, functional analysis, probability, or physics. A computation-oriented overview forming a coherent narrative accessible to learners at the tertiary level is often missing.
This is made harder by the fact that competence in quantum computing is relevant far beyond mathematics programs alone. Students in computer science, physics, electrical engineering, and related disciplines increasingly need a solid grasp of the underlying mathematics in order to meaningfully engage with quantum technologies. However, in many study programs, neither the mathematical structures specific to quantum computing nor the necessary foundational topics are taught in an integrated course canon.
Addressing this gap requires rethinking how we teach the mathematics of quantum computing: how concepts are motivated, how learning trajectories are designed, and how abstract theory is connected to algorithms and applications. Developing such an educational framework is not only essential for training future researchers, but also for ensuring that a broader community of scientists and engineers is equipped to participate in the quantum era.
DidactiQC: Rethinking How We Teach the Mathematics of Quantum Computing
As quantum computing and quantum machine learning move to practical relevance, a central challenge becomes clear: how do we teach the underlying mathematics in a way that is both accessible and rigorous? The DidactiQC initiative was launched to address precisely this question.
At its core, DidactiQC aims to develop a coherent and systematic course framework for the mathematics of quantum computing and quantum machine learning. The focus is here on early-career students in mathematics and the mathematical sciences. Rather than treating quantum topics as isolated add-ons, the project seeks to embed them into a carefully designed learning trajectory that connects mathematical ideas to modern quantum algorithms and applications.
Importantly, the initiative looks beyond university courses alone. Mathematical outreach at the secondary-school level could introduce quantum-related concepts early in a student’s learning biography. Further, the education of future mathematics teachers, who play a central role in shaping students’ early encounters with advanced mathematical ideas related to quantum computing, is a key component. These broader perspectives are essential if quantum computing is to become a sustainable part of the mathematical curriculum, rather than a niche topic accessible only to specialists.
Achieving this vision requires more than simply repackaging advanced material. It calls for the development of viable material that respect the specific needs and backgrounds of different target groups, while remaining intellectually honest despite the necessary didactic transformations. Concepts such as linearity, superposition, tensor products, and probabilistic reasoning must be introduced in mathematically meaningful, yet didactically sound ways.
Building a National Learning Space for Quantum Mathematics
DidactiQC builds on concrete teaching experience from recently designed courses in Quantum Computing (2025) and Quantum Machine Learning (2026) at NTNU, as well as comparable courses at other Norwegian universities. The initiative aims to equip students not only with technical competence, but also with the conceptual intuition needed to navigate and contribute to the rapidly evolving quantum landscape.
DidactiQC is supported by funding from the Norwegian Research Council. As a first concrete step, a one-week workshop will serve as a collaborative arena where educators and researchers can share teaching experiences, co-design curricular structures, and identify didactic challenges related to the mathematical foundations of quantum computing, and more generally quantum technologies. By creating space for dialogue, the workshop aims to lay the groundwork for a sustainable educational framework.
This initiative is closely aligned with Norway’s broader efforts to strengthen national competence in quantum technologies1. In recent years, the Norwegian government has committed to coordinated investments in areas such as quantum computing, quantum communication, and quantum sensing, with the aim of pooling Nordic research strengths, building shared infrastructure, promoting talent mobility, and accelerating innovation. Complementing these efforts, the Norsk Kvanteklynge2, launched in 2025, brings together major academic, research, and industry stakeholders — including universities, research institutions such as NTNU and SINTEF, and private-sector actors — to foster a coherent national quantum ecosystem focused on education, competence development, and technology deployment.
By developing a systematic didactic framework for the mathematics of quantum computing and quantum machine learning DidactiQC directly supports the goals of Norsk Kvanteklynge and Norway’s broader quantum strategy. In particular, it addresses a key bottleneck identified across these initiatives: the talent pipeline. Strengthening the mathematical foundations of quantum education is essential if Norway is to develop a base of quantum-literate graduates capable of contributing to research, infrastructure development, and industrial applications. Just as importantly, embedding educational efforts within the national quantum ecosystem helps ensure that investments in hardware and infrastructure are matched by human capacity and conceptual readiness. We believe that this combination will be crucial for long-term competitiveness in the quantum era.
At the local level, DidactiQC is deeply rooted in ongoing initiatives at the Department of Mathematical Sciences (IMF) at NTNU. The project is conceived as a collaboration between the mathematics and the didactics of mathematics groups within the department. The university is currently engaged in a redesign of its basic engineering mathematics courses under the framework of Fremtidens teknologistudier (FTS), with active involvement from the didactics of mathematics group at IMF. This group combines extensive experience in developing mathematics courses for technology students with research expertise in university-level mathematics didactics, and this expertise is directly integrated into the DidactiQC project.
From a didactics perspective, DidactiQC addresses a particularly timely and intellectually rich challenge: how advanced and abstract mathematical structures can be transformed into coherent learning trajectories that remain mathematically meaningful while being accessible to diverse student groups. The project therefore provides a valuable research context for studying didactic transposition, conceptual development, and the role of mathematical representations in an emerging and rapidly evolving domain.
In parallel, researchers at IMF are involved in the national Centre for Sustainable, Risk-averse and Ethical AI (SURE-AI), where quantum computing plays a thematic role due to its proximity to artificial intelligence research [1]. While these initiatives are distinct from DidactiQC, the project is expected to yield strong local synergies at the intersection of mathematics, quantum technology, and AI.
References
[1] Yuri Alexeev, Marwa H. Farag, Taylor L. Patti, Mark E. Wolf, Natalia Ares, Alán Aspuru Guzik, Simon C. Benjamin, Zhenyu Cai, Shuxiang Cao, Christopher Chamberland, Zohim Chandani, Federico Fedele, Ikko Hamamura, Nicholas Harrigan, Jin-Sung Kim, Elica Kyoseva, Justin G. Lietz, Tom Lubowe, Alexander McCaskey, Roger G. Melko, Kouhei Nakaji, Alberto Peruzzo, Pooja Rao, Bruno Schmitt, Sam Stanwyck, Norm M. Tubman, Hanrui Wang, and Timothy Costa. Artificial intelligence for quantum computing. Nature Communications , 16(1), December 2025.
[2] Johannes A. Buchmann. Introduction to quantum algorithms , volume 64 of Pure and Applied Undergraduate Texts . American Mathematical Society, Providence, RI, [2024] ©2024.
[3] Davide Castelvecchi. Quantum computers will finally be useful: what’s behind the revolution. Nature, 650(8100):24–26, February 2026.
[4] Editorial. 40 years of quantum computing. Nature Reviews Physics , 4(1):1–1, January 2022.
[5] Michael A. Nielsen and Isaac L. Chuang. Quantum computation and quantum information. Cambridge University Press, Cambridge, 2000.
[6] JohnPreskill. Quantumcomputing40yearslater. arXiv:2106.10522v3 , in”Feynman Lectures on Computation”, 2nd Edition, 2023, CRC Press.
[7] Wolfgang Scherer. Mathematics of Quantum Computing: An Introduction . Springer International Publishing, 2019.
[8] Maria Schuld and Francesco Petruccione. Machine Learning with Quantum Computers. Springer International Publishing, 2021.
- See Harnessing our quantum potential – towards a Norwegian quantum strategy (https://www.regjeringen.no/no/aktuelt/harnessing-our-quantum-potential-towards-a-norwegian-quantum-strategy/id3103651/) and the report on quantum strategies and research cooperation in the Nordic countries, Nordic Quantum Technology Research Co-operation (https://www.nordforsk.org/2025/nordic-quantum-technology-research-co-operation). ↩︎
- See Norsk kvanteklynge (https://www.norskkvanteklynge.no/) ↩︎