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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.
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:
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.
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.
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.
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 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 ...
In quantum computing and quantum information theory, the Clifford gates are the elements of the Clifford group, a set of mathematical transformations which normalize the n-qubit Pauli group, i.e., map tensor products of Pauli matrices to tensor products of Pauli matrices through conjugation.