How is a rotation of an Euler angle represented?
A rotation of Euler angles is represented as a matrix of trigonometric functions of the angles. (2) Quaternions are an algebraic structure that extends the familiar concept of complex numbers.
How to calculate the orientation of an angle?
The first rotation of angle ϕ 1 is around Z, the second rotation of angle Φ is around the new X and the third rotation angle ϕ 2 is around the new Z. The expression of the orientation matrix is obtained by composing the three rotations R ϕ 1 , R Φ and R ϕ 2: Using the given Euler angle yield the following orientation matrix:
How are Euler angles described in Euclidian space?
Rotations in Euclidian space have 3 independent components as shown by Euler. They can be described in a number of ways such as: Euler angles are a common way of defining a rotation by combining 3 successive rotations around different axes.
How to find the expression of the orientation matrix?
The expression of the orientation matrix is obtained by composing the three rotations R ϕ 1 , R Φ and R ϕ 2: Using the given Euler angle yield the following orientation matrix: With the orientation matrix it is the possible to express any vector V c from the cartesian crystal frame to the sample frame by:
When to use HHL to calculate running time?
The HHL is a quantum algorithm to estimate a function of the solution with running time complexity of O(log(N)s2κ2/ϵ) O (log (N) s 2 κ 2 / ϵ) (#hhl) when A A is a Hermitian matrix under the assumptions of efficient oracles for loading the data, Hamiltonian simulation and computing a function of the solution.
How to solve a linear system of equations using HHL?
For the HHL we will use QPE with U = eiAtU = eiAt, where AA is the matrix associated to the system we want to solve. In this case, eiAt = N − 1 ∑ j = 0eiλjt | uj⟩⟨uj |
Is the HHL algorithm suitable for a quantum simulator?
For the quantum simulator, Qiskit already provides an implementation of the HHL algorithm requiring only the matrix A and | b⟩ as inputs in the simplest example. Although we can give the algorithm a general Hermitian matrix and an arbitrary initial state as NumPy arrays, in these cases the quantum algorithm will not achieve an exponential speedup.