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The fact that the Pauli matrices, along with the identity matrix I, form an orthogonal basis for the Hilbert space of all 2 × 2 complex matrices , over , means that we can express any 2 × 2 complex matrix M as = + where c is a complex number, and a is a 3-component, complex vector.
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.
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:
Common quantum logic gates by name (including abbreviation), circuit form(s) and the corresponding unitary matrices. In quantum computing and specifically the quantum circuit model of computation, a quantum logic gate (or simply quantum gate) is a basic quantum circuit operating on a small number of qubits.
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 Pauli matrices abide by the physicists' convention for Lie algebras. In that convention, Lie algebra elements are multiplied by i, the exponential map (below) is defined with an extra factor of i in the exponent and the structure constants remain the same, but the definition of them acquires a factor of i.
The most obvious relation to the Pauli matrices (from the definitions of the matrices in this article, and using their commutation relations) would be to have u i = −i σ i. However, as is apparent at the other article, u 1 = i σ 1 , u 2 = − i σ 2 and u 3 = i σ 3 works as well, with an unexpected minus sign on the second matrix (the ...
The matrix exponential then gives us a map : (,) from the space of all n×n matrices to the general linear group of degree n, i.e. the group of all n×n invertible matrices. In fact, this map is surjective which means that every invertible matrix can be written as the exponential of some other matrix [ 9 ] (for this, it is essential to consider ...