To find the matrix representation of a two-qubit gate composed of two single-qubit gates one needs to calculate the tensor product of the mentioned two single-qubit gates matrices?
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 consider 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.
Other recent questions and answers regarding EITC/QI/QIF Quantum Information Fundamentals:
- Are amplitudes of quantum states always real numbers?
- How the quantum negation gate (quantum NOT or Pauli-X gate) operates?
- Why is the Hadamard gate self-reversible?
- If measure the 1st qubit of the Bell state in a certain basis and then measure the 2nd qubit in a basis rotated by a certain angle theta, the probability that you will obtain projection to the corresponding vector is equal to the square of sine of theta?
- How many bits of classical information would be required to describe the state of an arbitrary qubit superposition?
- How many dimensions has a space of 3 qubits?
- Will the measurement of a qubit destroy its quantum superposition?
- Can quantum gates have more inputs than outputs similarily as classical gates?
- Does the universal family of quantum gates include the CNOT gate and the Hadamard gate?
- What is a double-slit experiment?
View more questions and answers in EITC/QI/QIF Quantum Information Fundamentals