Abstract
The Bloch sphere is a generalisation of the complex number z with |z|2 = 1 being represented in the complex plane as a point on the unit circle. The goal of the research is to create a simulation that can be used to visualise a Bloch sphere of a single quantum bit, also known as a Qbit. QISKIT (developed by IBM) is an open-source lab for education in the realm of quantum computing, and is used to test and validate this simulator. This study made use of both quantitative and qualitative methods of investigation.
TopIntroduction
A bit is the fundamental unit of a computer. The essential component allows data to be stored in binary numbers, either 0 or 1. In a similar vein, the smallest unit of a quantum computer capable of storing information is referred to as a quantum bit, or in short, a qubit, for short. It has a complicated two-level mechanical system with two states, |1 > and |0 >, divided into two categories. In the same way, that information stored in a classic bit may be updated to meet our requirements, data or data recorded in a quantum bit can likewise be modified within the confines of quantum mechanics. Different gates, such as XOR, OR, and more, are used in digital electronics to change data and transport bits of information, among other things. In the quantum world, we have a variety of gates to choose from. To better understand how these gates function and how the qubit interacts with gates, we must first understand a few mathematical concepts such as complex numbers, matrices, and vectors. This is analogous to a simulation problem in that it is necessary to comprehend and visualize the process. It is possible to tackle this difficulty by modeling and simulating the procedure. Refer to Figure 1 for a visual understanding of how a physical quantum device is simulated on a real quantum simulator.
This is a discrete event simulation, which means that the outputs are fixed for the specified set of input parameters. The work (which is either one of |1> or |0>) is determined by the probability of achieving the desired results. Please keep in mind that this project will only be able to replicate a single qubit at this time. It is possible to simulate several qubits as well.
Figure 1. A simulator of a real quantum computer
TopBackground
It is challenging to visualize a quantum bit’s state consistently visually, and the mathematics required is expensive to solve. As the number of qubits increases, they are simulating the difference in a qubit after a gate is applied gets progressively complex. Matrix multiplication in multiple dimensions is essential in this scenario due to using mathematical techniques and concepts such as live vectors. Visualizing or recreating a Bloch sphere in a graphical environment would be beneficial. The conventional bit contains only two gates: a NOT gate that reverses the bit from 1 to 0 and another that changes the bit from 0 to 1. Four unitary operations or gates are matrix multiplied to form a qubit's final state. After converting this final state to polar form, the changed qubit is acquired. Figure 2 here represents different gates and their corresponding matrixes for unitary operations.
Different gates for a single qubit.
Pasieka (Pasieka et al., 2009) introduces the Bloch sphere interpretation of single qubit quantum channels and operations. This work presents a mathematical technique that begins with an arbitrary mathematical description of a channel and proceeds to identify the geometric aspects of the channel that are relevant for experimentation. The algorithm is implemented in Maple, and the associated code is provided.
Other simulators have also been used for the simulation of the Block sphere. ChangmingHuo (Huo, 2009) a study, a Java simulator and a 3D display based on the Bloch sphere and Bloch ball concepts are used to animate the motion of a qubit or spinor. The simulator may be used to demonstrate a qubit's Larmor precession, the Rabi formula for computing the probability of spin-flip in a time-dependent magnetic field, and the density matrix notion for examining the time evolution of an ensemble of spinors. On the user interface, control panels are developed to le qubit manipulation and change simulation parameters such as external time-independent and time-dependent magnetic fields. Stephen Shary (Shary, 2011) presented simulation software to visually represent a quantum state or qubit based on the Bloch sphere representation. The software uses the Java language and libraries to provide a multi-platform simulator that can be quickly distributed and viewed using the Java web-start technology.
Key Terms in this Chapter
Bloch Sphere: In quantum mechanics and computing, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system (qubit), named after the physicist Felix Bloch. The Bloch sphere is a unit 2-sphere, with antipodal points corresponding to a pair of mutually orthogonal state vectors. The north and south poles of the Bloch sphere are typically chosen to correspond to the standard basis vectors.
Phase Angles: Phase angle refers to a particular point in the time of a cycle whose measurement takes place from some arbitrary zero and its expression is as an angle. Furthermore, a phase angle happens to be one of the most important characteristics of a periodic wave.
Quantum Bit: It is the primary unit of how information is represented in the quantum computing space. It is similar to that of how information is stored in traditional classical computing, which is a binary bit. In a classical system, a bit would have to be in one state or the other. However, quantum mechanics allows the qubit to be in a coherent superposition of both states simultaneously, a property that is fundamental to quantum mechanics and quantum computing.
State Vector: In the mathematical formulation of quantum mechanics, pure quantum states correspond to vectors in a Hilbert space, while each observable quantity (such as the energy or momentum of a particle) is associated with a mathematical operator. The operator serves as a linear function which acts on the states of the system.
Bloch Vector: A Bloch vector is a unit vector (cosfsin?, sinfsin?, cos?) used to represent points on a Bloch sphere.
Quantum Gates: In quantum computing and specifically the quantum circuit model of computation, a quantum logic gate (or simply quantum gate) is a basic quantum circuit operating on a small number of qubits. They are the building blocks of quantum circuits, like classical logic gates are for conventional digital circuits. Unlike many classical logic gates, quantum logic gates are reversible. However, it is possible to perform classical computing using only reversible gates. For example, the reversible Toffoli gate can implement all Boolean functions, often at the cost of having to use ancilla bits. The Toffoli gate has a direct quantum equivalent, showing that quantum circuits can perform all operations performed by classical circuits.
Quantum States: In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in time exhausts all that can be predicted about the system's behavior. A mixture of quantum states is again a quantum state. Quantum states that cannot be written as a mixture of other states are called pure quantum states, while all other states are called mixed quantum states. A pure quantum state can be represented by a ray in a Hilbert space over the complex numbers, while mixed states are represented by density matrices, which are positive semidefinite operators that act on Hilbert spaces.