4.1 Single-Qubit Gates and Rotations

Single-qubit gates are the building blocks of quantum computing. They manipulate individual qubits, allowing for state changes, superposition creation, and phase shifts. These gates, including Pauli gates, the Hadamard gate, and rotation gates, form the foundation for more complex quantum operations.

Understanding single-qubit gates is crucial for grasping quantum algorithms and error correction. The Bloch sphere provides a visual representation of qubit states and gate effects, while matrix representations enable precise mathematical descriptions of quantum operations. Mastering these concepts is essential for quantum circuit design and analysis.

Single-Qubit Gates

Pauli gates and qubit states

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Top images from around the web for Pauli gates and qubit states

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  Autonomous calibration of single spin qubit operations | npj Quantum Information View original

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  Cost-optimal single-qubit gate synthesis in the Clifford hierarchy – Quantum View original

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• Pauli X gate (NOT gate) flips qubit state performs 180° rotation around x-axis on Bloch sphere
  • Matrix representation: $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ swaps amplitudes of basis states
  • Effect on basis states: $|0\rangle \rightarrow |1\rangle$, $|1\rangle \rightarrow |0\rangle$ inverts qubit
  • Applications include bit-flip error correction and quantum NOT operations
• Pauli Y gate rotates qubit state by 180° around y-axis introducing complex phase
  • Matrix representation: $\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}$ swaps amplitudes with imaginary unit
  • Effect on basis states: $|0\rangle \rightarrow i|1\rangle$, $|1\rangle \rightarrow -i|0\rangle$ flips and adds phase
  • Used in quantum error correction and certain quantum algorithms (quantum Fourier transform)
• Pauli Z gate (phase flip) changes qubit phase without altering probabilities
  • Matrix representation: $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ negates amplitude of $|1\rangle$ state
  • Effect on basis states: $|0\rangle \rightarrow |0\rangle$, $|1\rangle \rightarrow -|1\rangle$ introduces relative phase
  • Bloch sphere: 180° rotation around z-axis preserving x-y plane projection
  • Crucial in phase error correction and implementing phase shifts

Hadamard gate for superposition

• Hadamard gate (H gate) creates equal superposition of basis states key in many quantum algorithms
  • Matrix representation: $\frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$ combines basis states with equal weight
  • Effect on basis states:
    • $|0\rangle \rightarrow \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$ creates "++" superposition
    • $|1\rangle \rightarrow \frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)$ creates "+-" superposition
  • Bloch sphere: 90° rotation around y-axis followed by 180° rotation around x-axis
• Applications of Hadamard gate span various quantum computing tasks
  • Quantum algorithms (Deutsch-Jozsa, Grover's search) use H gates for initial state preparation
  • Quantum random number generation exploits superposition for true randomness
  • Quantum state tomography employs H gates in measurement basis rotation

Qubit rotations on Bloch sphere

• Bloch sphere provides intuitive 3D representation of qubit states
  • Pure states appear as points on surface, mixed states as points inside sphere
  • Axes correspond to measurement bases (x, y, z) representing different observables
• Rotation angles θ (theta) and φ (phi) define qubit state orientation
  • θ: Angle from z-axis (0 ≤ θ ≤ π) determines superposition between $|0\rangle$ and $|1\rangle$
  • φ: Angle in x-y plane (0 ≤ φ < 2π) represents relative phase between basis states
• General qubit state expressed as $|\psi\rangle = \cos(\frac{\theta}{2})|0\rangle + e^{i\phi}\sin(\frac{\theta}{2})|1\rangle$
  • Coefficients relate to probability amplitudes of measuring $|0\rangle$ or $|1\rangle$
  • Exponential term $e^{i\phi}$ encodes relative phase information
• Rotation operators allow precise control of qubit state
  • Rx(θ): Rotation around x-axis changes superposition without affecting relative phase
  • Ry(θ): Rotation around y-axis alters both superposition and phase
  • Rz(θ): Rotation around z-axis modifies phase while preserving superposition

Single-qubit operations and representations

• Matrix multiplication implements gate operations on qubit states
  • Apply gate U to state $|\psi\rangle$: $U|\psi\rangle$ transforms state vector
  • Compose gates: $(U_2U_1)|\psi\rangle = U_2(U_1|\psi\rangle)$ allows sequential operations
• Rotation gates expressed as matrix exponentials of Pauli operators
  • Rx(θ) = $e^{-i\frac{\theta}{2}X} = \cos(\frac{\theta}{2})I - i\sin(\frac{\theta}{2})X$ rotates around x-axis
  • Ry(θ) = $e^{-i\frac{\theta}{2}Y} = \cos(\frac{\theta}{2})I - i\sin(\frac{\theta}{2})Y$ rotates around y-axis
  • Rz(θ) = $e^{-i\frac{\theta}{2}Z} = \cos(\frac{\theta}{2})I - i\sin(\frac{\theta}{2})Z$ rotates around z-axis
• Bloch sphere visualizations aid in understanding gate effects
  • Pauli gates appear as 180° rotations around respective axes
  • Hadamard gate rotates state from z-axis to x-axis creating superposition
  • Phase gates manifest as rotations around z-axis altering relative phase
• Arbitrary single-qubit unitary decomposed into rotation sequence
  • U = $e^{i\alpha}R_z(\beta)R_y(\gamma)R_z(\delta)$ represents any single-qubit operation
  • Allows implementation of any desired transformation using basic rotations
• Quantum circuits represent gate sequences graphically
  • Read from left to right mirroring time evolution of quantum system
  • Horizontal lines represent qubits, boxes denote gate operations
  • Facilitates design and analysis of quantum algorithms and protocols