Quantum computing is increasingly becoming part of the advanced undergraduate curriculum in computer science, engineering, mathematics, and physics. But there is still considerable variation in what an introductory university course in the subject actually covers.
Some courses begin from quantum mechanics. Others approach the subject primarily through computer science and algorithms. Still others emphasize programming with current quantum software platforms.
For an interdisciplinary undergraduate audience, I think a different question is useful:
What knowledge should students actually acquire before they move on to more advanced work in quantum algorithms, quantum information, or quantum technologies?
A good introductory course should provide enough mathematical and physical foundation for rigor, while keeping the computational ideas visible throughout. It should also avoid requiring an entire traditional quantum-mechanics sequence before students can begin studying quantum computation.
This article continues from my earlier blog, What Mathematics Is Needed for Quantum Computing?, which focused on the mathematical preparation needed to enter the field. Here, I broaden the question: once that foundation is in place, what should an advanced undergraduate course in quantum computing actually cover?
1. Start with the Necessary Mathematics—but Not Too Much
Linear algebra is the essential mathematical language of quantum computing.
Students should be comfortable with vectors, matrices, inner products, eigenvalues and eigenvectors, complex numbers, and basic probability. Tensor products are especially important once multiple qubits are introduced.
But these ideas do not all have to be prerequisites.
An undergraduate quantum computing course can introduce or review the necessary mathematics as it becomes useful. In fact, students often understand the mathematics better when they immediately see what it represents.
- A vector becomes a quantum state.
- A unitary matrix becomes a quantum operation.
- An inner product becomes connected with overlap and probability.
- A tensor product becomes the mathematical description of a composite quantum system.
This connection between mathematics and physical or computational meaning should continue throughout the course.
2. Quantum Mechanics Should Be Introduced from the Computational Perspective
A previous course in quantum mechanics is certainly useful, particularly for physics students, but I do not think it needs to be a universal prerequisite.
Students from computer science, mathematics, and engineering can learn the quantum principles needed for computation directly.
An introductory course should therefore develop, carefully and progressively:
- quantum states and qubits;
- superposition and phase;
- the computational basis;
- the Bloch-sphere representation;
- measurement and probability;
- unitary evolution; and
- composite quantum systems.
The goal is not to reproduce a traditional quantum-mechanics course in compressed form. It is to give students the quantum framework needed to understand information and computation.
3. Gates and Circuits Should Become a Language for Reasoning
Once students understand quantum states, gates and circuits provide a natural computational language.
Students should learn the standard single-qubit gates, rotations, controlled operations, and multi-qubit circuits. They should also understand reversibility, unitarity, circuit identities, and basic circuit decomposition.
But being able to reproduce a circuit is not enough.
Students should be able to move among several representations:
state → mathematical operation → circuit → measurement outcome
For example, when examining a circuit, students should be able to reason about what happens to the quantum state at each stage rather than treating the circuit as a sequence of software instructions.
That ability becomes increasingly important when they later study quantum algorithms.
4. Multi-Qubit Systems and Entanglement Deserve Special Attention
The transition from one qubit to several qubits is one of the major conceptual transitions in the subject.
This is where tensor products become essential and where students must distinguish product states from entangled states.
Entanglement should not be presented merely as a strange feature of quantum physics. It should be understood as a structural feature of quantum information and, in many contexts, as a computational and informational resource.
Bell states provide a particularly useful entry point because they connect several topics that might otherwise appear unrelated: composite systems, measurement, correlations, entanglement, communication, and later quantum protocols.
5. Measurement Should Be Treated as a Core Topic
Measurement is sometimes introduced quickly and then treated almost as an implementation detail. I think it deserves much more attention.
Students should understand measurement probabilities, measurement in different bases, projective measurement, expectation values, and the state changes associated with measurement.
They should also understand a fundamental feature of quantum computation: the internal state of a quantum system is not simply available for inspection.
A quantum algorithm must therefore be designed not only to transform a state, but also to make useful information accessible through measurement.
This distinction becomes crucial for understanding quantum algorithms.
6. Quantum Information Belongs in an Introductory Quantum Computing Course
Quantum computing and quantum information are too closely connected to be taught as entirely separate subjects.
At the undergraduate level, students should encounter at least the basic ideas of:
- Bell states and Bell inequalities;
- quantum teleportation;
- superdense coding;
- density operators and mixed states;
- quantum channels and noise; and
- introductory quantum error correction.
These topics broaden students' view beyond ideal circuits. They show how quantum states carry information, how that information can be communicated, how noise affects it, and why error correction is necessary for scalable quantum computing.
Density operators are particularly important. Without them, it is difficult to make the transition from ideal pure-state calculations to realistic quantum systems.
7. Teach a Few Algorithms Well Rather Than Many Algorithms Superficially
An introductory course does not need to survey every well-known quantum algorithm.
It is more valuable to choose examples that expose the basic mechanisms that make quantum algorithms work.
Students should gradually encounter concepts such as:
- interference;
- phase manipulation;
- quantum parallelism;
- oracle-based computation;
- Fourier methods; and
- amplitude amplification.
The central pedagogical question should not be:
Can the student reproduce the circuit?
It should be:
Can the student explain why the algorithm works?
That means following amplitudes, phases, interference, transformations, and measurement probabilities through an algorithm.
This prepares students much better for later study of phase estimation, Hamiltonian simulation, quantum linear-algebra methods, optimization, and other advanced topics.
8. Programming Should Reinforce the Theory
Programming is extremely useful in an undergraduate quantum computing course.
Students should construct circuits, simulate them, examine measurement statistics, and compare computational results with their analytical calculations.
But software should support conceptual understanding rather than replace it.
There is a danger in teaching quantum computing mainly through a software framework. Students can learn to create registers, call gates, run circuits, and display histograms without developing a clear model of the quantum state being transformed.
Ideally, students should be able to connect four levels:
mathematical representation → quantum circuit → computation → measurement results
Software then becomes a laboratory for testing and exploring the theory.
9. The Course Should Be Designed as a Progression
For an advanced undergraduate audience, I would organize an introductory quantum computing course approximately along the following path:
- Mathematical foundations as needed
- Qubits and quantum states
- Measurement
- Single-qubit operations
- Multi-qubit systems and gates
- Quantum circuits
- Entanglement and quantum communication
- Density operators, noise, and quantum channels
- Introductory quantum algorithms
- Quantum error correction and selected advanced topics
The details will naturally depend on the institution, the students' backgrounds, and the length of the course.
More important than the exact ordering is the continuity of ideas.
Students should see how quantum states lead to quantum operations; operations lead to circuits; circuits implement information processing; measurement extracts classical information; and these mechanisms eventually combine to produce quantum algorithms.
Preparing Students for What Comes Next
An advanced undergraduate course should not try to make students experts in every part of quantum computing.
It should instead give them a durable conceptual framework from which more advanced topics can be learned.
After such a course, students should be prepared to study subjects such as the quantum Fourier transform and phase estimation, amplitude amplification, Hamiltonian simulation, variational methods, quantum optimization, quantum machine learning, and other modern quantum algorithms and applications.
This is also the perspective that guides Quantum Computing and Information: A Scaffolding Approach. The book is designed around the progression discussed above: developing the mathematical and quantum foundations first, connecting them systematically to circuits and quantum information, and then using that foundation to introduce quantum algorithms.
More broadly, the underlying principle is simple:
Quantum computing is best learned as a connected body of ideas, not as a collection of gates, programming commands, or isolated algorithms.
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