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International Conference on Engineering and Emerging Technologies (ICEET)

Matrix representation of basic quantum circuits

Pedro Juan Roig, Salvador Alcaraz, Katja Gilly, Cristina Bernad, Blas Trigueros

Published Oct 22, 2025• 0 citations
Publisher

Abstract

Quantum computing fundamentals are based on Hilbert spaces and linear algebra, which allows to describe quantum gates as matrix representations. In this paper, a matrix approach is taken in order to analyze basic quantum circuits, composed of basic quantum gates of 1, 2, and 3 qubits. The most common cases of all those gates are analyzed. One-qubit gates are expressed as 2x2 matrices, accounting for a rotation around a determined axis. Two-qubit gates are described as 4x4 matrices, standing for controlled gates. Three-quibit gates are defined as 8x8 matrices, extending the concept of controlled gates and rotations.

DOI
10.1109/iceet67911.2025.11424136

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Quantum computing fundamentals are based on Hilbert spaces and linear algebra, which allows to describe quantum gates as matrix representations. In this paper, a matrix approach is taken in order to analyze basic quantum circuits, composed of basic quantum gates of 1, 2, and 3 qubits. The most common cases of all those gates are analyzed. One-qubit gates are expressed as 2x2 matrices, accounting for a rotation around a determined axis. Two-qubit gates are described as 4x4 matrices, standing for controlled gates. Three-quibit gates are defined as 8x8 matrices, extending the concept of controlled gates and rotations.
Quantum gates typically represent either rotations of controlled gates. Rotation gates usually involve one qubit, although Ising coupling gates involve two qubits. On the other hand, controlled gates can have any number of control qubits with at least a target qubit. In this paper, the decomposition of controlled gates in terms of rotation gates around a given axis has been carried out. The focus herein has been set on theX\mathbf{X}Xaxis, such that AND-CmX gates and OR-CmX gates have been decomposed and proved form={1,2,3,4}\mathbf{m}=\{\mathbf{1, 2, 3, 4}\}m={1,2,3,4}and different polarities.
The field of quantum computing is growing rapidly and there is a surprisingly large literature.Research in this area includes the design of quantum reversible circuits and developing quantum algorithms for the models of quantum computing.This paper is focused on representing quantum reversible gates in matrix form.In turn these matrices can be used to develop quantum circuits with help of K-Map.Also this paper gives the historical development of quantum algorithms and basics concepts in quantum compuation.
In this article, the problem of synthesizing a general Hermitian quantum gate into a set of primary quantum gates is addressed. To this end, an extended version of the Jacobi approach for calculating the eigenvalues of Hermitian matrices in linear algebra is considered as the basis of the proposed synthesis method. The quantum circuit synthesis method derived from the Jacobi approach and its optimization challenges are described. It is shown that the proposed method results in multiple-control rotation gates around the y axis, multiple-control phase shift gates, multiple-control NOT gates, and a middle diagonal Hermitian matrix, which can be synthesized to multiple-control Pauli Z gates. Using the proposed approach, it is shown how multiple-control U gates, where U is a single-qubit Hermitian quantum gate, can be implemented using a linear number of elementary gates in terms of circuit lines with the aid of one auxiliary qubit in an arbitrary state.
This chapter introduces Boolean quantum gates that allow us to map any Boolean circuit in such a way that the circuit becomes reversible. It describes quantum gates for one qubit, like for example Clifford gates that are the elements of the Clifford group and can be efficiently simulated with a classical computer. The chapter introduces the controlled-U gates and the unitary decomposition and formulate the process of transpilation. A Boolean quantum circuit represents a permutation in Hilbert space and can be represented by a unitary permutation matrix composed of the unitary matrices representing the quantum gates. Such a mapping does not alter the distribution of the amplitudes; the distribution remains unchanged during the execution of the quantum Boolean gates.
Quantum mechanics has proven its position as the primary theory for accurately describing microscopic domains. Numerous fields in physics utilize quantum mechanics to great effects, with one prominent application being quantum computing. Quantum computing extends the field of quantum mechanics to information theory, striving to perform computational tasks utilizing quantum phenomena with notably greater efficiency compared to classical alternatives. In this thesis, I will go through the basics required in the field of quantum computing. To understand quantum computing demands adapting a new type of matrix mechanical view. This involves considering Hilbert spaces, which are linear vector space having a well defined inner product. Hilbert spaces contain state vectors describing quantum states, as well as operators for altering and measuring these states. The dynamics of quantum systems are governed by three quantum postulates. These specify Hilbert spaces and state vectors as the fundamental components of quantum systems, Hermitian operators as providing the physical manifestation of these systems, and finally, the Schrödinger's equation to describe the time evolution of such systems. These notions are required in producing quantum computing. Quantum computing uses qubits as the computational basis, analogous to classical bits but with the added capability of quantum superposition. Qubits are represented as two-state vectors, and can be physically realized with any quantum-behaving binary-valued system, like the electron spins. Qubits can be combined into higher dimensional states with the use of tensor products, causing the system's Hilbert space to expand as 2^n with n being the number of qubits, allowing for exponential computational speedup. Computational logic can be constructed by acting on the qubits using unitary operators, known as quantum gates. The combinations of qubits and quantum gates can be visualized as quantum circuits, depicted using specialized circuit diagram notations. An essential part of quantum computing, and the phenomena enabling exponential computational speedup, is quantum entanglement. Entanglement arises in interacting qubits, intertwining them so that the individual states become correlated. Entanglement can be characterized by being unable to reduce a combined state into its pure constituents. The amount of entanglement in a system can be measured using von Neumann entropy, analogous to Shannon entropy in classical information theory. Entanglement serves various purposes in quantum computing and is a defining feature distinguishing it from the classical domain. In this thesis, the entanglement properties of a periodic four-qubit circuit with alternating quantum gates were simulated. The simulation was performed computationally, and measured entanglement entropy as a function of the amount of circuit periods. The results demonstrated different characteristics of entanglement depending on the parameters used for the quantum gates, with the entanglement either being periodic, non-occurring or chaotic in nature.
This lecture continues Lectures One and Two, where we introduced the basic concepts of quantum computing and reviewed the essential linear algebra background for undergraduate students. In this lecture, we focus on the basic computational units of quantum computation, known as quantum gates. Quantum gates play a central role in quantum algorithms, as each quantum algorithm can be implemented as a sequence of quantum gate operations. This idea has a close analogy with classical computing, where computations can be represented by Boolean circuits constructed from elementary logic gates. In principle, classical algorithms can be implemented through combinations of such logic gates. To establish this connection, we first review several fundamental logic gates in classical computing, including the AND, OR, and NOT gates. These elementary Boolean gates provide a useful foundation for understanding quantum gates and how classical computational concepts are extended into the quantum domain. In general, a quantum processing unit (QPU), which executes quantum gate operations, can be viewed as the quantum counterpart of a classical central processing unit (CPU), where computations are carried out through logical operations and circuits. This lecture establishes the foundation for understanding the quantum circuit model and for designing and analyzing quantum algorithms in subsequent lectures. To provide a smooth transition to the topics that follow, we also introduce several tiny quantum algorithms and examples in this lecture. These examples illustrate how quantum gates can be combined to perform meaningful computational tasks and prepare students for the more advanced quantum algorithms discussed later.