Contents
How are the Pauli matrices used in quantum gates?
The Pauli Gates You should be familiar with the Pauli matrices from the linear algebra section. If any of the maths here is new to you, you should use the linear algebra section to bring yourself up to speed. We will see here that the Pauli matrices can represent some very commonly used quantum gates.
How to represent quantum states in density matrix?
In Qiskit, we can use the quantum_info module to represent quantum states either in state vector notation, or in the density matrix representation. For convenience, we will import this module as qi: from qiskit import QuantumCircuit import qiskit.quantum_info as qi Let’s once again consider the entangled pure state
Which is the correct equation for Qiskit U 3 gate?
Qiskit provides U 2 and U 1 -gates, which are specific cases of the U 3 gate in which θ = π 2θ= π 2, and θ = ϕ = 0 θ = ϕ= 0 respectively. You will notice that the U 1 -gate is equivalent to the R ϕ -gate. U 3(π 2,ϕ,λ) = U 2 = 1 √2 [ 1 −eiλ eiϕ eiλ+iϕ] U 3(0,0,λ) = U 1 =[10
How to create a short circuit in Qiskit?
In Qiskit, we can create a short circuit to verify this: Let’s see the result of the above circuit. Note: Here we use plot_bloch_multivector () which takes a qubit’s statevector instead of the Bloch vector. We can indeed see the state of the qubit is | 1⟩|1⟩ as expected.
What are the commutation relations of the Pauli matrix?
The Pauli matrices obey the following commutation relations: and anticommutation relations: where the structure constant ε abc is the Levi-Civita symbol, Einstein summation notation is used, δ ab is the Kronecker delta, and I is the 2 × 2 identity matrix.
What kind of vector space is the Pauli matrix?
Each Pauli matrix is Hermitian, and together with the identity matrix I (sometimes considered as the zeroth Pauli matrix σ0 ), the Pauli matrices form a basis for the real vector space of 2 × 2 Hermitian matrices.
Which is the computational basis of the X gate?
In fact, the computational basis (the basis formed by the states | 0⟩ |0⟩ and | 1⟩|1⟩) is often called the Z-basis. This is not the only basis we can use, a popular basis is the X-basis, formed by the eigenstates of the X-gate. We call these two vectors | + ⟩|+⟩ and | − ⟩|−⟩: