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Selected Quantum Algorithms in High School Level

Matěj Maroušek

Published Jan 1, 2026• 0 citations
OpenAlex

Abstract

This bachelor thesis deals with the creation of a Czech didactic text for teaching quantum algorithms at high schools. The thesis describes five basic quantum algorithms, namely Deutsch's, Deutsch-Jozsa, Bernstein-Vazirani, Simon's, and Grover's, which uti- lize a quantum oracle and form a theoretical foundation for more complex quantum algorithms. The theoretical explanation is supplemented with motivational and illustra- tive examples. The individual procedures are elaborated in such a way that the text is fully comprehensible to students at an upper secondary school level of knowledge. To facilitate orientation in the subject matter, an introductory chapter was created, summa- rizing the reader's expected knowledge and the necessary mathematical apparatus. The thesis concludes with a chapter providing a comprehensive summary of all the described algorithms.

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This bachelor thesis deals with the creation of a Czech didactic text for teaching quantum algorithms at high schools. The thesis describes five basic quantum algorithms, namely Deutsch's, Deutsch-Jozsa, Bernstein-Vazirani, Simon's, and Grover's, which uti- lize a quantum oracle and form a theoretical foundation for more complex quantum algorithms. The theoretical explanation is supplemented with motivational and illustra- tive examples. The individual procedures are elaborated in such a way that the text is fully comprehensible to students at an upper secondary school level of knowledge. To facilitate orientation in the subject matter, an introductory chapter was created, summa- rizing the reader's expected knowledge and the necessary mathematical apparatus. The thesis concludes with a chapter providing a comprehensive summary of all the described algorithms.
These notes begin in Chapter 1 with a review of linear algebra and the postulates of quantum mechanics, leading to an explanation of single- and multi-qubit gates. Chapter 2 explores the challenge of constructing arbitrary quantum states from given initial states, and introduces circuits for building oracles. Chapter 3 presents foundational algorithms such as entanglement creation, quantum teleportation, Deutsch-Jozsa, Bernstein-Vazirani, and Simon's algorithm. Chapters 4 and 5 cover algorithms based on the quantum Fourier transform, including phase estimation, period finding, factoring, and logarithm computation. These chapters also include complexity analysis and detailed quantum circuits suitable for implementation in code. Chapter 6 introduces Grover's algorithm for quantum search and amplitude amplification, including its realization via Hamiltonian simulation and a method for derandomization. Chapter 7 discusses basic techniques for Hamiltonian simulation, such as Lie-Trotter decomposition, sparse Hamiltonians, and the linear combination of unitaries. It also provides example circuits for simulating Hamiltonians expressed as linear combinations of Pauli operators. Chapter 8 introduces variational quantum algorithms, and Chapter 9 presents an algorithm for simulating fermionic many-particle systems, with an emphasis on molecular Hamiltonians. It also outlines the key transformations needed to map a molecular Hamiltonian to a form suitable for simulation on a quantum computer.
The main aim of this thesis was to create a set of problems in Quantum Mechanics, which would be included in a collection of solved problems in Physics available online on the website http://www.fyzikalniulohy.cz. As a result, thirty nine problems were completed, covering seven thematic areas partially of high school and mostly of introductory college Quantum Mechanics course. In general, the collection is mainly designed for college students as an exercise in basic quantum-mechanical problems and also for high school students, who have a desire to deal with some special problems. A detailed description of content, structure, and technical execution of this collection can be found in chapter 1 of this thesis. Another task was to review existing online collections of applets dealing with Quantum Mechanics. A selection of them is introduced in chapter 2 of this thesis.
The quantum algorithms covered in this chapter, i.e., the Deutsch, Deutsch-Jozsa and Bernstein-Vazirani algorithms, have been chosen for their relative simplicity and theoretical interest. They do not address problems of real practical value, their purpose being to demonstrate, using simple examples, the great potential power of quantum computation relative to the digital case. The most significant breakthrough in quantum computing is Shor's integer factorisation algorithm. Shor's algorithm is beyond the scope of this introductory text, but its significance lies in the fact that it demonstrates the potential to factor large integers efficiently. No efficient digital algorithms for factoring large integers exist, and many current encryption algorithms rely on this being the case. Shor's algorithm therefore threatens current encryption methods, and is responsible for much of the recent accelerated interest in quantum computation.
The Deutsch and Deutsch-Jozsa algorithms are foundational in the study of quantum computing, providing early and elegant demonstrations of quantum speedup over classical computation. This paper offers a pedagogical exposition of both algorithms, beginning with the necessary background in quantum computation and culminating in detailed, step-by-step derivations of the algorithms, complete with mathematical formulations and circuit diagrams. Simulations using quantum gates are also discussed to enhance understanding. The aim is to make these seminal algorithms accessible to readers with a basic background in linear algebra and quantum mechanics.
This chapter expounds on quantum algorithms, outlining fundamental concepts and their practical implementations. It starts with an explanation of core quantum principles such as functions and oracles. Then, it introduces Qiskit, an IBM-developed open-source quantum computing software framework. The discussion integrates theoretical underpinnings with empirical demonstrations, providing readers with an understanding of quantum programming intricacies. Through the utilisation of practical examples implemented in Qiskit, readers are given firsthand exposure, thereby reinforcing their comprehension of the subject matter. The overarching objective is to equip readers with a comprehensive knowledge base, thereby fostering proficiency in navigating the theoretical underpinnings and practical applications of quantum computing. To this end, the chapter ends with an examination of three seminal quantum algorithms: Deutsch, Deutsch-Jozsa, and Bernstein-Vazirani. These algorithms serve as keystones in the quantum computing domain, illustrating the efficacy and adaptability of quantum computation in resolving diverse computational problems.