How to implement Hamiltonian $0.01Z$?

Question

I have a task in an assignment that wants me to apply a Hamiltonian to a state. The Hamiltonial is 0.01*sigma_z. I know how to apply a Z gate to a state but I don't know to process the factor 0.01 in front of it.

Context: We want to implement a Trotterized adiabtaic quantum computing algorithim for one qubit and use the Laudau Zener Hamiltonian

Answer 1

If I got it well, to implement what you want in Qiskit you can do the following:

from qiskit import QuantumCircuit
from qiskit.circuit.library import ZGate
from qiskit.quantum_info import Operator
from qiskit.visualization import array_to_latex

k = 0.01
qc1 = QuantumCircuit(1)
h = ZGate().power(k)
qc1.append(h, [0])

array_to_latex(Operator(qc1))

$$ \begin{bmatrix} 1 & 0 \\ 0 & 0.99951 + 0.03141i \\ \end{bmatrix} $$

This will work in general, for any gate and any $k$ value. However, in your specific case, recalling that $Z = U(0, \pi, 0)$, a simple rotation works as well:

from numpy import pi

qc2 = QuantumCircuit(1)
qc2.u(theta=0, phi=pi*k, lam=0, qubit=0)

array_to_latex(Operator(qc2))

$$ \begin{bmatrix} 1 & 0 \\ 0 & 0.99951 + 0.03141i \\ \end{bmatrix} $$

Answer 2

I found the answer. The R_x and R_z gates do exactly that. Decomposing the Pauli Trotter Evolution funktion helped answer that.