Qubits are the building blocks of quantum computing, allowing for superposition and enabling powerful quantum algorithms. The Bloch sphere provides a visual way to represent and understand qubit states, making complex quantum concepts more accessible.

Single-qubit gates, like Pauli and Hadamard gates, manipulate qubit states through rotations on the Bloch sphere. These operations are crucial for creating quantum circuits and implementing quantum algorithms, forming the foundation of quantum information processing.

Qubits and Bloch Sphere Representation

Qubits as Quantum Information Units

• A qubit is the fundamental unit of quantum information, analogous to a classical bit but with additional properties derived from quantum mechanics
• The state of a qubit can be represented as a linear combination (superposition) of two orthonormal basis states, typically denoted as |0⟩ and |1⟩, known as computational basis states
• The general state of a qubit is represented as |ψ⟩ = α|0⟩ + β|1⟩, where α and β are complex numbers called probability amplitudes, satisfying the normalization condition $|α|^2 + |β|^2 = 1$
• Qubits enable quantum parallelism and the implementation of quantum algorithms that can solve certain problems more efficiently than classical algorithms (Shor's algorithm for factoring, Grover's algorithm for searching)

Bloch Sphere Representation

• The Bloch sphere is a geometrical representation of the state space of a single qubit, where each point on the surface of the unit sphere corresponds to a unique pure state of the qubit
• In the Bloch sphere representation, the computational basis states |0⟩ and |1⟩ are located at the north and south poles of the sphere, respectively
• The Bloch vector, represented as a point on the surface of the Bloch sphere, characterizes the state of the qubit and is defined by two angles: the polar angle θ (0 ≤ θ ≤ π) and the azimuthal angle φ (0 ≤ φ < 2π)
• The Bloch sphere provides a visual and intuitive way to understand the state of a qubit and the effects of quantum operations on the qubit state

Visualizing Qubit States

Basis States and Superposition

• The north pole of the Bloch sphere corresponds to the computational basis state |0⟩, while the south pole corresponds to the state |1⟩
• The equator of the Bloch sphere represents the equal superposition states, such as |+⟩ = (|0⟩ + |1⟩) / √2 and |-⟩ = (|0⟩ - |1⟩) / √2, which lie on the x-axis and -x-axis, respectively
• The positive and negative y-axis of the Bloch sphere correspond to the states |+i⟩ = (|0⟩ + i|1⟩) / √2 and |-i⟩ = (|0⟩ - i|1⟩) / √2, respectively
• Superposition states allow a qubit to exist in a linear combination of its basis states simultaneously, enabling quantum parallelism

Bloch Vector and Measurement Probabilities

• The Bloch vector, represented by the polar angle θ and azimuthal angle φ, points to the location of the qubit state on the surface of the Bloch sphere
• The probability of measuring the qubit in the |0⟩ state is given by $cos^2(θ/2)$, while the probability of measuring it in the |1⟩ state is $sin^2(θ/2)$
• The Bloch vector representation allows for the calculation of measurement probabilities and the visualization of the effects of quantum operations on the qubit state

Superposition on the Bloch Sphere

Quantum Superposition

• Superposition is a fundamental principle of quantum mechanics, allowing a qubit to exist in a linear combination of its basis states (|0⟩ and |1⟩) simultaneously
• The general state of a qubit in superposition is given by |ψ⟩ = α|0⟩ + β|1⟩, where α and β are complex probability amplitudes satisfying $|α|^2 + |β|^2 = 1$
• The probability amplitudes α and β determine the probabilities of measuring the qubit in the |0⟩ or |1⟩ state, respectively, when a measurement is performed
• Superposition allows qubits to exhibit properties that are not possible with classical bits, such as being in multiple states simultaneously and enabling quantum parallelism in quantum algorithms

Superposition Representation on the Bloch Sphere

• On the Bloch sphere, superposition states are represented by points on the surface of the sphere, excluding the north and south poles (which correspond to the computational basis states)
• The polar angle θ determines the relative amplitudes of the basis states in the superposition, while the azimuthal angle φ determines the relative phase between the basis states
• Equal superposition states, such as |+⟩ and |-⟩, lie on the equator of the Bloch sphere, while states with unequal amplitudes are represented by points at different latitudes
• The Bloch sphere representation provides a clear visualization of the relationship between the probability amplitudes and the location of the qubit state on the sphere

Single-Qubit Gate Operations

Pauli Gates

• Single-qubit gates are unitary operations that transform the state of a qubit, represented by a rotation of the Bloch vector on the Bloch sphere
• The Pauli gates (X, Y, and Z) are fundamental single-qubit gates that perform rotations around the x, y, and z axes of the Bloch sphere, respectively
  • The Pauli X gate (also known as the NOT gate) performs a π rotation around the x-axis, transforming |0⟩ to |1⟩ and vice versa
  • The Pauli Y gate performs a π rotation around the y-axis, transforming |0⟩ to i|1⟩ and |1⟩ to -i|0⟩
  • The Pauli Z gate performs a π rotation around the z-axis, introducing a phase shift of -1 to the |1⟩ state while leaving the |0⟩ state unchanged
• Pauli gates are essential building blocks for constructing more complex quantum circuits and implementing quantum error correction schemes

Hadamard and Rotation Gates

• The Hadamard gate (H) is another important single-qubit gate that creates an equal superposition of the basis states, transforming |0⟩ to (|0⟩ + |1⟩) / √2 and |1⟩ to (|0⟩ - |1⟩) / √2, represented by a 90° rotation around the y-axis followed by a 180° rotation around the x-axis on the Bloch sphere
• Rotation gates (Rx, Ry, and Rz) perform arbitrary rotations around the x, y, and z axes of the Bloch sphere by a specified angle θ
  • The Rx gate performs a rotation around the x-axis by an angle θ, given by the matrix $Rx(θ) = \begin{pmatrix} \cos(θ/2) & -i\sin(θ/2) \ -i\sin(θ/2) & \cos(θ/2) \end{pmatrix}$
  • The Ry gate performs a rotation around the y-axis by an angle θ, given by the matrix $Ry(θ) = \begin{pmatrix} \cos(θ/2) & -\sin(θ/2) \ \sin(θ/2) & \cos(θ/2) \end{pmatrix}$
  • The Rz gate performs a rotation around the z-axis by an angle θ, given by the matrix $Rz(θ) = \begin{pmatrix} e^{-iθ/2} & 0 \ 0 & e^{iθ/2} \end{pmatrix}$
• The phase shift gate (R_φ) introduces a phase shift of φ to the |1⟩ state while leaving the |0⟩ state unchanged, represented by a rotation around the z-axis of the Bloch sphere
• Combining single-qubit gates allows for the creation of any arbitrary single-qubit state on the Bloch sphere, enabling the implementation of quantum algorithms and quantum error correction schemes