In my previous article, “What Should an Advanced Undergraduate Course in Quantum Computing Cover?”, I discussed a possible progression from mathematical foundations through qubits, circuits, entanglement, quantum information, and introductory algorithms. That article, in turn, continued from an earlier discussion of “What Mathematics Is Needed for Quantum Computing?”
The natural next question is:
What should students study once they are ready to move from introductory quantum computing to quantum algorithms and applications in greater depth?
Quantum algorithms are often introduced through a handful of famous examples—Deutsch-Jozsa, Grover's search, and Shor's factoring algorithm. These remain important. But an advanced curriculum should go beyond a catalog of individual algorithms.
The field is increasingly better understood through a collection of reusable algorithmic primitives and frameworks that appear repeatedly across different problems. At the same time, the application side of quantum computing has expanded substantially, encompassing quantum simulation, chemistry, optimization, quantum machine learning, scientific computing, and other areas.
There is therefore a practical curricular consideration. Both quantum algorithms and quantum applications now encompass a large and rapidly expanding range of topics. Trying to cover both adequately in a single course can easily lead to a broad but shallow survey. For a more substantial curriculum, it may be better to separate them into two advanced courses—one focused primarily on algorithmic principles and reusable computational primitives, and another focused on applications such as quantum simulation, chemistry, optimization, quantum machine learning, and scientific computing.
Both courses should also be supplemented by regular programming practice. Students should implement important ideas, simulate circuits, analyze numerical results, and compare theory with computational behavior. Programming is most useful when it reinforces the mathematical and conceptual structure rather than replacing it.
An advanced course—or sequence of courses—should therefore help students understand not only how an algorithm works, but also:
- what computational primitive it uses;
- what assumptions it makes about the input;
- what information it actually returns;
- how precision and success probability affect its cost;
- how it compares with classical methods; and
- when a theoretical speedup might—or might not—translate into practical quantum advantage.
1. Begin with the Computational Framework
Before studying more sophisticated algorithms, students need a framework for evaluating what an algorithm actually accomplishes.
They should already be comfortable with quantum states, gates, circuits, measurement, and basic quantum information. At the algorithmic level, however, several additional ideas become essential:
- circuit complexity;
- query complexity;
- oracle models;
- success probability;
- approximation and precision;
- state preparation;
- classical and quantum input models;
- classical and quantum output; and
- overall resource requirements.
These details are not merely technical qualifications.
A claim that an algorithm requires $O(\log N)$ operations, for example, means relatively little until we ask how an $N$-dimensional data set enters the quantum computer, what assumptions are made about accessing it, and what form of output is required.
This habit of examining assumptions should begin early and continue throughout an advanced curriculum.
2. Quantum Fourier Transform and Phase Estimation
The quantum Fourier transform (QFT) and quantum phase estimation (QPE) belong near the center of an advanced quantum algorithms course.
They introduce ideas that reappear throughout the subject:
- phase kickback;
- interference in the Fourier basis;
- eigenvalue estimation;
- controlled unitary evolution;
- the relationship between precision and circuit resources;
- success probability;
- state overlap; and
- the use of one algorithm as a primitive inside another.
Period finding and Shor's factoring algorithm then become important applications of these ideas rather than isolated constructions.
This is an important pedagogical transition. Students begin to see that the central subject is not simply “Shor's algorithm,” but a collection of mechanisms—Fourier transformation, controlled evolution, phase estimation, and classical post-processing—that can be reused elsewhere.
QPE also prepares students for later applications in quantum simulation, chemistry, spectral analysis, and linear algebra.
3. From Grover Search to Amplitude Amplification
A similar shift should occur when teaching Grover's algorithm.
Grover search is historically important, but amplitude amplification is the more general idea.
Students should understand both geometrically and algebraically how amplitudes associated with desirable states can be increased through repeated reflections. They should see how the familiar $O(\sqrt{N})$ search complexity emerges and under what assumptions that improvement applies.
This framework then leads naturally to amplitude estimation, which turns amplitude information into estimates of probabilities and expectation values.
Applications can include:
- numerical integration;
- Monte Carlo-type estimation;
- probability estimation;
- risk analysis; and
- other statistical computational tasks.
Again, the educational goal is to move from a particular named algorithm to a reusable computational mechanism.
4. Hamiltonian Simulation as Both Application and Primitive
One of the original motivations for quantum computing was the simulation of quantum systems.
But Hamiltonian simulation is more than an application area. It is also a fundamental algorithmic primitive.
Students should encounter several approaches, such as:
- product formulas;
- linear combinations of unitaries;
- block-encoding-based methods;
- qubitization; and
- related modern simulation techniques.
Equally important is understanding how the computational cost depends on:
- the representation of the Hamiltonian;
- simulation time;
- desired accuracy;
- sparsity or other structural properties;
- state preparation; and
- the observable that ultimately needs to be measured.
This is also an excellent place to introduce resource analysis as a normal part of studying an algorithm rather than something performed only after the theoretical derivation is complete.
5. Block Encoding, Quantum Signal Processing, and QSVT
For a modern advanced course, I think block encoding, quantum signal processing (QSP), and quantum singular value transformation (QSVT) deserve substantial attention.
These ideas provide one of the most important unifying viewpoints in contemporary quantum algorithms.
The basic progression is conceptually powerful:
Represent an operator → Transform its spectrum → Use the transformed operator to solve a computational problem
Students can then see how polynomial approximations allow quantum circuits to implement carefully chosen functions of eigenvalues or singular values.
From this perspective, algorithms for:
- Hamiltonian simulation;
- matrix inversion;
- linear systems;
- spectral filtering;
- threshold functions; and
- other operator transformations
no longer appear as unrelated constructions.
For advanced students, this is one of the places where quantum algorithms begin to look like a coherent discipline rather than a collection of clever examples.
6. Quantum Linear Systems—and the Importance of Asking What the Output Means
Quantum algorithms for solving linear systems provide a particularly useful case study.
The headline result can appear remarkable: under appropriate assumptions, a quantum algorithm may obtain information about the solution to a very large linear system with complexity exponentially smaller in the matrix dimension than straightforward classical methods.
But students should immediately learn to ask:
- How is the matrix provided?
- How is the vector on the right-hand side prepared?
- What assumptions are made about sparsity or block encoding?
- How does the condition number affect complexity?
- What precision is required?
- What does the quantum computer actually output?
- Do we need the full classical solution vector?
The last question is especially important.
If the solution is encoded as a quantum state, extracting every component classically may eliminate much of the apparent advantage.
Linear-system algorithms therefore provide an excellent setting for teaching students how to evaluate quantum speedup claims critically.
They also provide a bridge to broader applications in numerical analysis, differential equations, optimization, and scientific computing.
7. Variational and Hybrid Quantum Algorithms
An advanced course should also address the major class of hybrid quantum-classical algorithms that emerged partly in response to the limitations of near-term hardware.
These include:
- the variational quantum eigensolver (VQE);
- the quantum approximate optimization algorithm (QAOA);
- parameterized quantum circuits;
- quantum-classical optimization loops; and
- related variational methods.
Students should understand both why these approaches are attractive and why their performance can be difficult to assess.
Important issues include:
- optimization landscapes;
- barren plateaus;
- measurement cost;
- noise;
- circuit depth;
- ansatz design;
- classical optimization;
- parameter initialization; and
- the absence, in many cases, of demonstrated quantum advantage.
Hybrid algorithms should therefore not be presented simply as “algorithms that work on today's quantum computers.”
They are better treated as an important algorithmic paradigm whose capabilities and limitations remain active areas of research.
8. Measurement Is Part of the Algorithm
Measurement is sometimes treated as the final step of an algorithm: run the circuit and read the answer.
For many practical quantum algorithms, this picture is inadequate.
Useful quantities often need to be estimated statistically from repeated measurements.
Students should therefore understand topics such as:
- expectation-value estimation;
- sampling complexity;
- observable grouping;
- repeated circuit execution;
- overlap and success probability;
- measurement in different bases; and
- tradeoffs between circuit depth and measurement count.
In realistic quantum workflows, the cost of extracting useful classical information can be comparable to—or even greater than—the cost of preparing and evolving the quantum state.
Measurement should therefore be treated as an algorithmic resource.
This also reinforces an important general lesson: the complexity of a quantum algorithm should be assessed from input preparation through final usable output, not merely by counting gates in its central circuit.
9. Quantum Simulation and Many-Body Physics
Once students have developed the major algorithmic primitives, a second advanced course can turn more fully toward applications.
Quantum simulation is a natural place to begin because it connects directly with many of the algorithms already studied.
Topics can include:
- ground-state problems;
- excited states and spectra;
- real-time dynamics;
- correlation functions;
- Ising and Heisenberg models;
- Hubbard-type models;
- state preparation; and
- observable estimation.
A particularly important lesson is that the computational problem begins before the quantum circuit.
A physical problem must first be modeled, discretized, encoded, and mapped into a representation suitable for quantum computation.
This broader workflow helps students see how algorithmic primitives are actually used in scientific problems.
10. Quantum Chemistry
Quantum chemistry is one of the most developed application areas for quantum computing and provides an excellent interdisciplinary case study.
Students should understand the progression from the underlying physical problem to a quantum-computational representation.
Topics can include:
- the Born-Oppenheimer approximation;
- electronic structure;
- basis functions;
- first and second quantization;
- molecular Hamiltonians;
- fermion-to-qubit mappings;
- Hartree-Fock initial states;
- active-space methods;
- VQE;
- phase-estimation-based approaches; and
- measurement of molecular observables.
This application area is especially useful pedagogically because it demonstrates that the cost of quantum computation depends not only on the algorithm, but also on modeling choices, encoding choices, basis size, state preparation, and measurement strategy.
11. Optimization: Formulate the Problem Before Solving It
Quantum optimization provides another important application domain.
Students should learn concepts such as:
- QUBO formulations;
- Ising representations;
- constraint penalties;
- slack variables;
- QAOA;
- variational optimization;
- adiabatic approaches; and
- hybrid workflows.
But the course should distinguish carefully between two problems:
How do we formulate the optimization problem?
and
How do we attempt to solve it quantum mechanically?
Mapping scheduling, routing, portfolio selection, resource allocation, or other practical problems into an Ising Hamiltonian is itself a nontrivial modeling task.
Students should also compare quantum approaches with strong classical methods. For many optimization problems, the relevant classical baseline is not brute-force search but highly developed heuristics, approximation algorithms, mixed-integer programming, or problem-specific solvers.
This comparison is essential when discussing possible quantum advantage.
12. Quantum Machine Learning—with Careful Attention to Claims of Advantage
Quantum machine learning deserves attention because of its rapid development and broad interest.
An advanced applications course might introduce:
- data encoding;
- parameterized quantum models;
- quantum neural networks;
- parameter-shift methods;
- quantum kernels;
- generative quantum models;
- Born machines;
- hybrid learning workflows; and
- measurement and training costs.
But QML also provides an excellent opportunity to teach critical reading of research claims.
A mathematical speedup based on efficient quantum access to a data set is different from experimentally demonstrating an advantage on realistic classically stored data.
Students should learn to distinguish carefully among:
- theoretical complexity advantage;
- advantage under a particular data-access model;
- empirical performance on a small problem;
- scaling evidence; and
- demonstrated practical quantum advantage.
These distinctions are useful far beyond quantum machine learning.
13. Quantum Algorithms for Scientific Computing
Quantum algorithms increasingly intersect with traditional numerical scientific computing.
Linear systems naturally lead to topics such as:
- ordinary differential equations;
- partial differential equations;
- finite-difference discretization;
- spectral methods;
- numerical linear algebra;
- eigenvalue problems;
- operator transformations; and
- scientific simulation.
This connection is educationally valuable because students can compare quantum methods directly with techniques already used in physics, engineering, applied mathematics, and computational science.
Quantum computation then appears not as a replacement for classical scientific computing, but as a possible component of a larger computational workflow.
That perspective is likely to become increasingly important as practical quantum applications develop.
14. Make Quantum Advantage a Topic in Its Own Right
Perhaps the most important addition to a modern quantum algorithms and applications curriculum is a serious discussion of quantum advantage.
Students should not finish an advanced course believing that a better asymptotic expression automatically implies a useful quantum algorithm.
For every claimed advantage, they should learn to ask:
- What is the best relevant classical baseline?
- What input-access model is assumed?
- What output is required?
- What precision is needed?
- What is the state-preparation cost?
- What are the data-loading costs?
- How many logical qubits are needed?
- How many gates are needed?
- How many circuit repetitions and measurements are required?
- What happens when fault tolerance and error correction are included?
- Can the result be efficiently verified?
- Is the claimed advantage asymptotic, experimental, or practically useful?
These questions connect theoretical computer science with the practical realities of quantum computation.
They also give students a more durable way to evaluate new algorithms as the field continues to develop.
A Possible Two-Course Progression
Given the breadth of the field, I would organize the advanced material approximately as two connected courses.
Course I: Quantum Algorithms
A possible progression is:
Computational Foundations → QFT and QPE → Amplitude Amplification and Estimation → Hamiltonian Simulation → Block Encoding → QSP and QSVT → Quantum Linear Systems → Variational and Hybrid Algorithms → Resource Analysis
The emphasis would be on reusable algorithmic structures.
Students would learn not simply to reproduce named algorithms, but to recognize the mathematical and computational ideas that recur across many algorithms.
Course II: Quantum Applications
A possible progression is:
Quantum Simulation → Quantum Chemistry → Many-Body Systems → Optimization → Quantum Machine Learning → Scientific Computing and Differential Equations → End-to-End Workflows → Quantum Advantage
The emphasis would shift from algorithmic primitives to problem formulation and complete computational workflows.
Students would ask how a scientific, engineering, or data problem becomes a quantum-computational problem in the first place—and what must happen before and after the quantum circuit.
Programming Practice Across Both Courses
Programming should run through both courses rather than being isolated into a short software unit.
Students might:
- construct and simulate circuits;
- implement QFT, QPE, and amplitude amplification;
- study approximation and finite-precision effects;
- implement small variational algorithms;
- construct simple Hamiltonians;
- simulate quantum dynamics;
- formulate QUBO problems;
- estimate observables from measurement samples; and
- compare theoretical resource estimates with actual circuit implementations.
The goal is not merely to learn a particular software package.
Students should continually connect:
Mathematical formulation → Algorithm → Circuit → Computation → Measurement → Classical interpretation
This connection is especially important as quantum algorithms become more sophisticated.
From Quantum Computing to Quantum Algorithms and Applications
Taken together, the curriculum can be viewed as a progression:
Mathematical Foundations → Quantum Computing and Information → Quantum Algorithms → Quantum Applications
The first stage supplies the mathematical language.
The second develops qubits, quantum states, circuits, measurement, entanglement, and quantum information.
The third develops the major algorithmic primitives and frameworks.
The fourth asks how these methods can be used in realistic problems—and whether they can provide meaningful computational advantage.
This is also the perspective that guides Quantum Algorithms and Applications: A Scaffolding Approach. The book develops the subject from foundational algorithmic primitives through simulation, optimization, quantum machine learning, linear systems, differential equations, and the broader question of quantum advantage.
But the larger pedagogical principle is independent of any particular textbook:
Advanced quantum algorithms and applications are best learned not as a catalog of celebrated algorithms and disconnected use cases, but as a connected set of reusable computational ideas, modeling methods, and end-to-end workflows.
That perspective gives students something more valuable than familiarity with today's algorithms. It gives them a framework for understanding the algorithms and applications that have yet to be developed.
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