The Bloch Sphere: Qubit Representation #quantumcomputing

The Bloch sphere is named after Felix Bloch, a Nobel laureate Swiss physicist who pioneered the study of quantum phenomena in solids. His work inspired the Bloch sphere, a geometric model of a two-level quantum system. Today, it is essential for visualizing qubit states in quantum computing.

A qubit is a fundamental unit of quantum information. Any controllable two-level quantum system such as up and down spin states of an electron is considered to be a qubit. In classical computing a bit can only be either zero or one. However, due to the peculiar properties of quantum mechanics for extremely small physical entities, a qubit can be in superposition of the states 0 and 1. This is a subtle issue in quantum physics which has no analogue in classical physics. By the end of this video, we intend to deliver why the Bloch sphere is central to visualizing and manipulating qubits in quantum computing.

It all starts with the simplest building blocks of quantum mechanics: the basis states, 0 and 1. These two states are like the north and south poles of a sphere. And we call them vectors. Vectors, matrices, and complex numbers are essential mathematical grounds of quantum mechanics. Quantum mechanics allows a qubit to exist in superpositions of these two states, creating a continuum of possibilities.

This is where the Bloch sphere comes in. It's a powerful tool that maps every possible pure state of a qubit onto the surface of a unit sphere. The concept of pure and mixed states is a subtle issue and should not be confused with superposition. Here we explain only the pure states as they are represented on the surface of the Bloch sphere.

It's Important to state that the Bloch sphere is specifically for single-qubit systems. By mapping states onto a three-dimensional sphere, it simplifies understanding of complex superpositions and the effects of quantum operations. The sphere uses spherical coordinates to ensure normalization and capture both amplitudes and phases of complex qubit states. Quantum operations are visualized as rotations on the sphere and utilized in quantum algorithm design. Mixed states are represented by points inside the sphere, reflecting uncertainty or decoherence.