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2.8 Exponential of a Pauli vector. ... the Pauli matrices are a set of three 2 × 2 complex matrices that are traceless, ... For example, it can be shown that ...
Multi-qubit Pauli matrices can be written as products of single-qubit Paulis on disjoint qubits. Alternatively, when it is clear from context, the tensor product symbol can be omitted, i.e. unsubscripted Pauli matrices written consecutively represents tensor product rather than matrix product. For example:
The Pauli matrices also anti-commute, for example = =. The matrix exponential of a Pauli matrix is a rotation operator, often ...
As Pauli matrices are related to the generator of rotations, these rotation operators can be written as matrix exponentials with Pauli matrices in the argument. Any 2 × 2 {\displaystyle 2\times 2} unitary matrix in SU(2) can be written as a product (i.e. series circuit) of three rotation gates or less.
the matrix exponential reduces to a plain product of the exponentials of the two respective pieces. This is a formula often used in physics, as it amounts to the analog of Euler's formula for Pauli spin matrices, that is rotations of the doublet representation of the group SU(2).
Given a unit vector in 3 dimensions, for example (a, b, c), one takes a dot product with the Pauli spin matrices to obtain a spin matrix for spin in the direction of the unit vector. The eigenvectors of that spin matrix are the spinors for spin-1/2 oriented in the direction given by the vector. Example: u = (0.8, -0.6, 0) is a unit vector ...
Example: Spinor in a magnetic field [ edit ] The Hamiltonian of a spin-1/2 particle in a magnetic field can be written as [ 3 ] H = μ σ ⋅ B , {\displaystyle H=\mu \mathbf {\sigma } \cdot \mathbf {B} ,} where σ {\displaystyle \mathbf {\sigma } } denote the Pauli matrices , μ {\displaystyle \mu } is the magnetic moment , and B is the ...
The classical analog of the CNOT gate is a reversible XOR gate. How the CNOT gate can be used (with Hadamard gates) in a computation.. In computer science, the controlled NOT gate (also C-NOT or CNOT), controlled-X gate, controlled-bit-flip gate, Feynman gate or controlled Pauli-X is a quantum logic gate that is an essential component in the construction of a gate-based quantum computer.