Summary
Highlights
This module delves into quantum gates, the foundational elements of quantum computing, and how they manipulate quantum states. It begins with a quick recap of quantum states, including the representation of a single qubit as a linear combination of basis states (0 and 1) with probability amplitudes (alpha and beta). The concepts of bra and ket notation, inner and outer products, and the probabilistic collapse of a qubit upon measurement are revisited. The Bloch sphere is introduced as a visualization tool for single qubits, where a qubit's state is defined by two angles, theta and phi, representing its position on the sphere.
A single qubit gate is the most basic computation, designed to change the state of a qubit through a linear transformation represented by a 2x2 unitary matrix. The Pauli X, Y, and Z gates are discussed as examples. The Pauli X gate acts as a bit-flip, transforming 0 to 1 and 1 to 0, which visualizes as a 180-degree rotation around the X-axis on the Bloch sphere. The Pauli Z gate performs a phase flip, rotating a qubit around the Z-axis by 180 degrees. The Pauli Y gate, built from a combination of X and Z gates, performs both a bit and phase flip, rotating around the Y-axis by 180 degrees. Combinations of these Pauli gates can generate any 2x2 matrix, allowing for arbitrary rotations of a single qubit on the Bloch sphere.
The Hadamard gate (H) is a crucial gate that transforms a qubit between the X and Z bases. Applied to a 0 state, it creates a superposition state (plus state), and applying it again reverts the state to 0. This ability to change bases is fundamental to quantum computing. The S gate adds a 90-degree phase shift (pi/2) to the relative phase of a qubit, transforming states like the plus state to the plus-i state. These gates are important for creating superposition and manipulating the phase of qubits.
A quantum circuit is a sequence of quantum gates, where each gate performs an elementary computation. The combined effect of multiple gates in a circuit is determined by matrix multiplication of their respective matrices, applied in reverse order of operation due to the convention that time flows from left to right. Any sequence of gates, when combined, also results in a unitary matrix. The measurement of a qubit at the end of a circuit causes its state to probabilistically collapse to one of its basis states (0 or 1), providing a classical output. This measured value is the interpretable result of the quantum computation.
To increase the computational power beyond a single qubit, multiple qubits are combined to form multi-qubit states. For 'n' qubits, the system can represent 2^n entries, indicating an exponential increase in state space. A two-qubit state is formed by the tensor product of two single-qubit states, resulting in a 4x1 matrix. This bipartite state encompasses all four possible basis states (00, 01, 10, 11). For 'n' qubits, the tensor product results in a 2^n by 1 matrix, with 2^n probability amplitudes. These multi-qubit states can be classified as 'product states' if they can be expressed as a tensor product of individual qubit states, or 'entangled states' if they cannot, possessing unique quantum correlations.
While the Bloch sphere visualizes single qubits, multi-qubit states are visualized using a Q-sphere. A Q-sphere represents multi-qubit states by marking specific points (small spheres) on its surface, where the north pole typically represents the state with all qubits in the 0 state (e.g., 00 for two qubits), and the south pole represents all qubits in the 1 state (e.g., 11). Intermediate states are located at various latitudes and longitudes. The color of the sphere indicates the relative phase, and the size of the sphere represents the relative weight or probability amplitude of that particular state. The Q-sphere extends its complexity for three or more qubits, requiring more latitudes to represent the increasing number of possible basis states.