Can a Hermitian matrix have n different eigenvalues?
In fact, a general Hermitian matrix A of dimension N can have up to N different eigenvalues, therefore calculating them would take O(N) time and the quantum advantage would be lost. Now we are ready to go through the different steps of the HHL algorithm. State preparation in this example is trivial since | b⟩ = | 0⟩.
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.
How to calculate a smooth function of a Hermitian matrix?
Thus, for a real valued smooth function f, we look for a quantum algorithm which, when equipped with a quantum oracle for a Hermitian matrix A and a map that prepares some state , returns a state that is close to in l2 -norm. If measurements are involved, we require the desired output state to be obtained with high probability.
What kind of matrix do you need for HHL?
For the quantum simulator, Qiskit Aqua already provides an implementation of the HHL algorithm requiring the matrix A A and |b⟩ | b ⟩ as basic inputs. The main advantage is that it can take a general Hermitian matrix and an arbitrary initial state as inputs.
Do you need previous knowledge of HHL algorithm?
For the purpose of illustrating the algorithm, we will cheat a bit and calculate the eigenvalues of AA to be able to choose tt to obtain an exact binary representation of the rescaled eigenvalues in the nlnl -register. However, keep in mind that for the HHL algorithm implementation one does not need previous knowledge of the eigenvalues.
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 |