Single-qubit gates are the building blocks of quantum circuits, allowing us to manipulate individual qubits. These gates, represented by 2x2 matrices, transform a qubit's state through operations like flipping, phase shifting, or creating superpositions.

Common gates include the Pauli-X (NOT) gate, Hadamard gate, and rotation gates. By combining these gates, we can perform complex transformations on qubits, visualized on the Bloch sphere. Understanding single-qubit gates is crucial for designing quantum algorithms and circuits.

Single-qubit gates and circuits

Introduction to single-qubit gates

• Single-qubit gates are unitary operations that act on a single qubit, manipulating its quantum state
• Single-qubit gates are represented by 2x2 unitary matrices in the computational basis (|0⟩ and |1⟩)
• Single-qubit gates are fundamental building blocks of quantum circuits, allowing for the manipulation and control of individual qubits
• Single-qubit gates, along with multi-qubit gates, enable the implementation of quantum algorithms and computations

Matrix representation and action on qubits

• The action of a single-qubit gate on a qubit's state vector is described by matrix multiplication, where the gate matrix is multiplied with the state vector
• The resulting state vector represents the transformed quantum state of the qubit after the gate is applied
• The unitary nature of single-qubit gates ensures that the quantum state remains normalized and preserves the quantum information
• The matrix representation of single-qubit gates allows for the mathematical analysis and simulation of quantum circuits

Applying single-qubit gates

Common single-qubit gates

• The Pauli-X gate (also known as the NOT gate) is represented by the matrix $[0 1; 1 0]$ and flips the state of a qubit from |0⟩ to |1⟩ and vice versa
• The Hadamard gate (H) is represented by the matrix $[1/√2 1/√2; 1/√2 -1/√2]$ and creates an equal superposition of the computational basis states (|0⟩ and |1⟩)
• The Phase Shift gate (also known as the S gate) applies a phase shift of π/2 to the |1⟩ state, leaving the |0⟩ state unchanged
• The T gate (also known as the π/8 gate) applies a phase shift of π/4 to the |1⟩ state, leaving the |0⟩ state unchanged

Rotation gates

• Rotation gates, such as the rotation around the X-axis (Rx), Y-axis (Ry), and Z-axis (Rz), apply a rotation to the qubit's state vector by a specified angle
  • The Rx(θ) gate rotates the qubit's state vector around the X-axis by an angle θ
  • The Ry(θ) gate rotates the qubit's state vector around the Y-axis by an angle θ
  • The Rz(θ) gate rotates the qubit's state vector around the Z-axis by an angle θ
• Rotation gates provide a continuous range of transformations, allowing for fine-grained control over the qubit's state
• The angle of rotation determines the extent to which the qubit's state is modified (45°, 90°, 180°)

Effects of single-qubit gates

Transforming the quantum state

• Single-qubit gates transform the quantum state of a qubit by modifying the amplitudes and/or phases of the computational basis states
• The Pauli-X gate flips the amplitudes of the |0⟩ and |1⟩ states, effectively inverting the qubit's state
• The Hadamard gate transforms a qubit's state into an equal superposition of the computational basis states, with amplitudes of 1/√2 for both |0⟩ and |1⟩
• Rotation gates (Rx, Ry, Rz) modify the amplitudes and phases of the computational basis states, resulting in a rotation of the qubit's state vector on the Bloch sphere
• The Phase Shift gate and T gate introduce a phase difference between the |0⟩ and |1⟩ states, affecting the relative phase of the qubit's state

Bloch sphere representation

• The Bloch sphere is a geometrical representation of a qubit's state, where single-qubit gates correspond to rotations on the Bloch sphere
• The north and south poles of the Bloch sphere represent the computational basis states |0⟩ and |1⟩, respectively
• Points on the surface of the Bloch sphere represent pure quantum states, while points inside the sphere represent mixed states
• Single-qubit gates can be visualized as rotations of the qubit's state vector around the X, Y, or Z axes of the Bloch sphere (Rx(π/2), Ry(π/4), Rz(π))

Composing single-qubit gate sequences

Combining single-qubit gates

• Single-qubit gates can be combined in a sequence to perform complex transformations on a qubit's state
• The order of gates in a sequence matters, as quantum gates do not generally commute with each other
• Composing single-qubit gates allows for the implementation of specific quantum operations and algorithms
• Gate decomposition techniques can be used to express a desired single-qubit operation as a sequence of basic single-qubit gates (Rz(π/2) = Rx(π/2) + Rz(π) + Rx(-π/2))

Optimization and compilation

• Quantum compilers optimize sequences of single-qubit gates to minimize the number of gates and improve circuit efficiency
• Optimization techniques, such as gate fusion and cancellation, are applied to reduce the gate count and depth of the circuit
• Compilation maps the high-level quantum operations to the available gate set of the target quantum hardware
• Efficient compilation is crucial for executing quantum circuits on real quantum devices with limited coherence times and gate fidelities

Integration with multi-qubit gates

• Composing single-qubit gates with multi-qubit gates enables the creation of entangled states and the realization of quantum algorithms
• Single-qubit gates are often used in conjunction with multi-qubit gates (CNOT, CZ) to perform controlled operations and create entanglement between qubits
• The interplay between single-qubit and multi-qubit gates forms the basis for implementing quantum circuits and algorithms (Quantum Fourier Transform, Grover's search)
• Understanding the effects and composition of single-qubit gates is essential for designing and analyzing complex quantum circuits and algorithms