In the realm of quantum information processing, the manipulation of quantum states is fundamental to the design and implementation of quantum algorithms and protocols. Two-qubit gates are essential building blocks in quantum circuits, allowing for the entanglement and interaction of qubits. When constructing a two-qubit gate from two single-qubit gates, the matrix representation of the composite gate can indeed be obtained by calculating the tensor product of the individual single-qubit gate matrices.
To delve into this concept further, let's consider the general form of a single-qubit gate represented by a 2×2 unitary matrix. For instance, let's denote two single-qubit gates as (U) and (V), with their respective matrix representations as (U) and (V). The composite two-qubit gate, denoted as (UV), can be constructed by taking the tensor product of (U) and (V). The tensor product operation is denoted by the symbol (otimes), and for two matrices A and B, the tensor product A (otimes) B results in a block matrix of size (m*n) x (p*q), where A is m x n and B is p x q.
Mathematically, the matrix representation of the two-qubit gate (UV) is given by the tensor product of the matrices (U) and (V), as follows:
[ UV = U otimes V = begin{bmatrix} u_{11}V & u_{12}V \ u_{21}V & u_{22}V end{bmatrix} ]Here, (u_{ij}) represents the elements of matrix (U) and (V) is the single-qubit gate matrix. The resulting matrix is a 4×4 unitary matrix that operates on a composite system of two qubits.
To illustrate this with an example, let's consider two well-known single-qubit gates, the Pauli-X gate denoted as (X) and the Hadamard gate denoted as (H). The matrix representations of these gates are:
[ X = begin{bmatrix} 0 & 1 \ 1 & 0 end{bmatrix} ] [ H = frac{1}{sqrt{2}} begin{bmatrix} 1 & 1 \ 1 & -1 end{bmatrix} ]To find the matrix representation of the two-qubit gate formed by (X) and (H), we calculate the tensor product of (X) and (H):
[ XH = X otimes H = begin{bmatrix} 0 cdot H & 1 cdot H \ 1 cdot H & 0 cdot H end{bmatrix} = begin{bmatrix} 0 & 0 & frac{1}{sqrt{2}} & frac{1}{sqrt{2}} \ 0 & 0 & frac{1}{sqrt{2}} & -frac{1}{sqrt{2}} \ frac{1}{sqrt{2}} & frac{1}{sqrt{2}} & 0 & 0 \ frac{1}{sqrt{2}} & -frac{1}{sqrt{2}} & 0 & 0 end{bmatrix} ]This resulting 4×4 matrix represents the two-qubit gate obtained by composing the Pauli-X gate and the Hadamard gate.
When constructing a two-qubit gate from two single-qubit gates, the matrix representation of the composite gate can be derived by calculating the tensor product of the individual single-qubit gate matrices. This mathematical operation allows for the representation of entangling operations on composite quantum systems, enabling the manipulation of quantum states in quantum information processing tasks.
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