Bloch Sphere, Quantum Gates, and Pauli Matrices

Hiu Yung Wong

Hiu Yung Wong

Mikhail Pavlov

Abstract In the field of quantum computing, the study of single-qubit quantum gates is fundamental to constructing quantum circuits and algorithms. Single-qubit gates, such as the X, Y, and Z gates, enable precise manipulation of qubit states, which can be visualized on the Bloch sphere. The use of matrix representations, including Pauli matrices, provides a rigorous mathematical framework for analyzing gate operations and their effects within quantum circuits. This work also explores the application of the Quantum Fourier Transform (QFT) and introduces the Hadamard test subroutine as a practical method for evaluating quantum states. The presented algorithms were successfully implemented on a 5-qubit IBM quantum computer, demonstrating both the theoretical foundations and practical significance of quantum gates and Pauli matrices in modern quantum computation and their potential role in secure information processing.

Manu Mitra

Irjes, Jianping Li

To enhance the performance of the intelligent optimization algorithm, a new model of performing search on the Bloch sphere is proposed. Then, by integrating the model into the artificial fish swarm optimization, we present a quantum-inspired artificial fish swarm optimization algorithm. In proposed method, the fishes are encoded with the qubits described on the Bloch sphere. The vector product theory is adopted to establish the axis of rotation, and the Pauli matrices are used to construct the rotation matrices. The four fish behaviors, such as moving, tracking, capturing, aggregating, are achieved by rotating the current qubit about the rotation axis to the target qubit on the Bloch sphere. The Bloch coordinates of qubit can be obtained by measuring with the Pauli matrices, and the optimization solutions can be presented through the solution space transformation. The highlight advantages of this method are the ability to simultaneously adjust two parameters of a qubit and automatically achieve the best match between two adjustment quantities, which may accelerate the optimization process. The experimental results show that the proposed method obviously outperforms the classical one in convergence speed and achieves better levels for some benchmark functions.

Ali Al-Bayaty, Marek Perkowski

The conventional design technique of any n-bit quantum gate is mainly achieved using unitary matrices multiplication, where n >= 2 and 1 <= m <= n-1 for m target qubits and n-m control qubits. These matrices represent quantum rotations by an n-bit quantum gate. For a quantum designer, such a conventional technique requires extensive computational time and effort, which may generate an n-bit quantum gate with a too high quantum cost. The Bloch sphere is only utilized as a visualization tool to verify the conventional design correctness for quantum rotations by a quantum gate. In contrast, this paper introduces a new concept of using the Bloch sphere as a "geometrical design tool" to build cost-effective n-bit quantum gates with lower quantum costs. This concept is termed the "Bloch sphere approach (BSA)". In BSA, a cost-effective n-bit quantum gate is built without using any unitary matrices multiplication. Instead, the quantum rotations for such a gate are visually selected using the geometrical planar intersections of the Bloch sphere. The BSA can efficiently map m targets among n-m controls for an n-bit quantum gate, to satisfy the limited layout connectivity for the physical neighboring qubits of a quantum computer. Experimentally, n-bit quantum gates built using the BSA always have lower quantum costs than those for such gates built using the conventional quantum design techniques.