Flux Qubit

The flux qubit complements the previous circuits, which have been discussed in this documentation, as another important example since it represents a 3-node circuit. One of the nodes will be declared as a ground node which leads effectively to a 2-dimensional problem that will be discussed here.

import circuitq as cq
import numpy as np
import networkx as nx
import matplotlib.pyplot as plt

Circuit Graph

graph = nx.MultiGraph()
graph.add_edge(0,1, element = 'C')
graph.add_edge(0,1, element = 'J')
graph.add_edge(1,2, element = 'C')
graph.add_edge(1,2, element = 'J')
graph.add_edge(0,2, element = 'C')
graph.add_edge(0,2, element = 'J');

Symbolic Hamiltonian

circuit = cq.CircuitQ(graph)
circuit.h_parameters

[C_{01}, C_{02}, C_{12}, E_{J010}, E_{J020}, E_{J120}, \tilde{\Phi}_{120}]

circuit.h

$\displaystyle - E_{J010} \cos{\left(\frac{\Phi_{1}}{\Phi_{o}} \right)} - E_{J020} \cos{\left(\frac{\Phi_{2}}{\Phi_{o}} \right)} - E_{J120} \cos{\left(\frac{- \Phi_{1} + \Phi_{2} + \tilde{\Phi}_{120}}{\Phi_{o}} \right)} + 0.5 q_{1} \left(\frac{C_{12} q_{2}}{C_{01} C_{02} + C_{01} C_{12} + C_{02} C_{12}} + \frac{q_{1} \left(C_{02} + C_{12}\right)}{C_{01} C_{02} + C_{01} C_{12} + C_{02} C_{12}}\right) + 0.5 q_{2} \left(\frac{C_{12} q_{1}}{C_{01} C_{02} + C_{01} C_{12} + C_{02} C_{12}} + \frac{q_{2} \left(C_{01} + C_{12}\right)}{C_{01} C_{02} + C_{01} C_{12} + C_{02} C_{12}}\right)$

(horizontally scroll the above LaTeX output)

Diagonalization

Here, we will choose two junctions to have the same Josephson energy and the same shunted capacity. The third junction is scaled by alpha, which we choose to be \(0.7\). The parameter dim refers to the dimension for one node which leads to a total matrix dimension of dim $ ^2$.

dim = 50
EJ = 1*circuit.c_v["E"]
alpha = 0.7
C = circuit.c_v["C"]
phi_ext = np.pi*circuit.phi_0
h_num = circuit.get_numerical_hamiltonian(dim,
                    parameter_values=[C,C,alpha*C,EJ,EJ,alpha*EJ,phi_ext])
eigv, eigs = circuit.get_eigensystem(100)

Let’s plot the lowest eigenenergies.

h = 6.62607015e-34
y_scaling = 1/(h *1e9)
plt.plot(np.arange(10), eigv[:10]*y_scaling, 'ro')
plt.xlabel("Eigenvalue No.")
plt.ylabel(r"Energy in GHz$\cdot$h")
plt.show()

The potential of the flux qubit is made up of three cosine terms. This leads to a egg carton shaped potential. As we didn’t specify the grid length above, the flux coordinate grid is set to the default interval \([-4\pi\Phi_0,4\pi\Phi_0 ]\). This will repeat the maximum 4 by 4 times. Let’s plot the potential.

def potential(phi_1, phi_2, phi_ex):
    return (-EJ*np.cos(phi_1/circuit.phi_0) - EJ*np.cos(phi_2/circuit.phi_0) -
             alpha*EJ*np.cos((phi_2-phi_1+phi_ex)/circuit.phi_0) )
phis = np.linspace(-4*np.pi*circuit.phi_0, 4*np.pi*circuit.phi_0, dim)
potential_list = [potential(phi_1,phi_2,phi_ext)*y_scaling
                  for phi_2 in phis for phi_1 in phis]
plt.contourf(phis, phis, np.array(potential_list).reshape(dim,dim))
plt.xticks(np.linspace(-4*np.pi, 4*np.pi, 5)*circuit.phi_0 ,
           [r'$-4\pi$',r'$-2\pi$',r'$0$',r'$2\pi$',r'$4\pi$'])
plt.yticks(np.linspace(-4*np.pi, 4*np.pi, 5)*circuit.phi_0 ,
           [r'$-4\pi$',r'$-2\pi$',r'$0$',r'$2\pi$',r'$4\pi$'])
plt.xlabel(r"$\Phi_1 / \Phi_o$")
plt.ylabel(r"$\Phi_2 / \Phi_o$")
plt.colorbar(label="Potential Energy in GHz$\cdot$h")
plt.show()

Due to the periodicity of the potential, CircuitQ implements the Hamiltonian in the charge basis. To be able to plot the eigenstates as a function of flux, we use the transformation method transform_charge_to_flux().

circuit.transform_charge_to_flux()
eigs = circuit.estates_in_phi_basis

Now, we are able to plot the square of the absolute value of the lowest eigenstates as a function of flux.

plt.figure(figsize=(13,30))
for n in range(10):
    plt.subplot(5,2, n+1)
    plt.contourf(circuit.flux_list, circuit.flux_list,
      abs(np.array(eigs[n].reshape(circuit.n_dim,circuit.n_dim)).transpose())**2)
    plt.colorbar()
    plt.title("Eigenstate " + str(n) )
    plt.xticks(np.linspace(-4*np.pi, 4*np.pi, 5)*circuit.phi_0 ,
               [r'$-4\pi$',r'$-2\pi$',r'$0$',r'$2\pi$',r'$4\pi$'])
    plt.yticks(np.linspace(-4*np.pi, 4*np.pi, 5)*circuit.phi_0 ,
               [r'$-4\pi$',r'$-2\pi$',r'$0$',r'$2\pi$',r'$4\pi$'])
    plt.xlabel(r"$\Phi_1 / \Phi_o$")
    plt.ylabel(r"$\Phi_2 / \Phi_o$")
plt.show()

We find that the lowest two eigenstates are located at the double well between the maxima of the potential.