GRAPE calculation of control fields for single-qubit rotation¶

Robert Johansson (robert@riken.jp)

In [1]:

%matplotlib inline
import matplotlib.pyplot as plt
import time
import numpy as np
from numpy import pi

In [2]:

from qutip import *
from qutip.control import *

In [3]:

T = 1
times = np.linspace(0, T, 100)

In [4]:

theta, phi = np.random.rand(2)

In [5]:

# target unitary transformation (random single qubit rotation)
U = rz(phi) * rx(theta); U

Out[5]:

Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = False\begin{equation*}\left(\begin{array}{*{11}c}(0.986-0.054j) & (-0.009-0.157j)\\(0.009-0.157j) & (0.986+0.054j)\\\end{array}\right)\end{equation*}

In [6]:

R = 150
H_ops = [sigmax(), sigmay(), sigmaz()]

H_labels = [r'$u_{x}$',
            r'$u_{y}$',
            r'$u_{z}$',
        ]

In [7]:

H0 = 0 * pi * sigmaz()

GRAPE¶

In [8]:

from qutip.control.grape import plot_grape_control_fields, _overlap
from qutip.control.cy_grape import cy_overlap
from qutip.control.grape import cy_grape_unitary, grape_unitary_adaptive

In [9]:

from scipy.interpolate import interp1d
from qutip.ui.progressbar import TextProgressBar

In [10]:

u0 = np.array([np.random.rand(len(times)) * 2 * pi * 0.005 for _ in range(len(H_ops))])

u0 = [np.convolve(np.ones(10)/10, u0[idx,:], mode='same') for idx in range(len(H_ops))]

In [11]:

result = cy_grape_unitary(U, H0, H_ops, R, times, u_start=u0, eps=2*pi/T, phase_sensitive=False,
                          progress_bar=TextProgressBar())

10.0%. Run time:   7.45s. Est. time left: 00:00:01:07
20.0%. Run time:  14.05s. Est. time left: 00:00:00:56
30.0%. Run time:  21.91s. Est. time left: 00:00:00:51
40.0%. Run time:  27.99s. Est. time left: 00:00:00:41
50.0%. Run time:  34.34s. Est. time left: 00:00:00:34
60.0%. Run time:  40.20s. Est. time left: 00:00:00:26
70.0%. Run time:  46.05s. Est. time left: 00:00:00:19
80.0%. Run time:  52.30s. Est. time left: 00:00:00:13
90.0%. Run time:  58.29s. Est. time left: 00:00:00:06
Total run time:  63.70s

Plot control fields for iSWAP gate in the presense of single-qubit tunnelling¶

In [12]:

plot_grape_control_fields(times, result.u[:,:,:] / (2 * pi), H_labels, uniform_axes=True);

No description has been provided for this image

In [13]:

# target unitary
U

Out[13]:

Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = False\begin{equation*}\left(\begin{array}{*{11}c}(0.986-0.054j) & (-0.009-0.157j)\\(0.009-0.157j) & (0.986+0.054j)\\\end{array}\right)\end{equation*}

In [14]:

# unitary from grape pulse
result.U_f

Out[14]:

Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = False\begin{equation*}\left(\begin{array}{*{11}c}(0.986-0.054j) & (-0.009-0.157j)\\(0.009-0.157j) & (0.986+0.054j)\\\end{array}\right)\end{equation*}

In [15]:

# target / result overlap
_overlap(U, result.U_f).real, abs(_overlap(U, result.U_f))**2

Out[15]:

(0.9999999999999987, 0.9999999999999973)

Verify correctness of the Hamiltonian pulses by integration¶

In [16]:

c_ops = []

In [17]:

U_f_numerical = propagator(result.H_t, times[-1], c_ops, args={})

In [18]:

U_f_numerical

Out[18]:

Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = False\begin{equation*}\left(\begin{array}{*{11}c}(0.987-0.049j) & (-3.399\times10^{-04}-0.151j)\\(3.399\times10^{-04}-0.151j) & (0.987+0.049j)\\\end{array}\right)\end{equation*}

In [19]:

_overlap(U, U_f_numerical)

Out[19]:

(0.9999289415682254+0j)

Bloch sphere dynamics¶

In [20]:

psi0 = basis(2, 0)
e_ops = [sigmax(), sigmay(), sigmaz()]

In [21]:

me_result = mesolve(result.H_t, psi0, times, c_ops, e_ops)

In [22]:

b = Bloch()

b.add_points(me_result.expect)

b.add_states(psi0)
b.add_states(U * psi0)
b.render()

Process tomography¶

Ideal gate¶

In [23]:

op_basis = [[qeye(2), sigmax(), sigmay(), sigmaz()]]
op_label = [["i", "x", "y", "z"]]

In [24]:

fig = plt.figure(figsize=(8,6))

U_ideal = spre(U) * spost(U.dag())

chi = qpt(U_ideal, op_basis)

fig = qpt_plot_combined(chi, op_label, fig=fig, threshold=0.001)

Gate calculated using GRAPE¶

In [25]:

fig = plt.figure(figsize=(8,6))

U_ideal = spre(result.U_f) * spost(result.U_f.dag())

chi = qpt(U_ideal, op_basis)

fig = qpt_plot_combined(chi, op_label, fig=fig, threshold=0.001)

Versions¶

In [26]:

from qutip.ipynbtools import version_table

version_table()

Out[26]:

Software	Version
QuTiP	4.2.0
Numpy	1.13.1
SciPy	0.19.1
matplotlib	2.0.2
Cython	0.25.2
Number of CPUs	2
BLAS Info	INTEL MKL
IPython	6.1.0
Python	3.6.1 \|Anaconda custom (x86_64)\| (default, May 11 2017, 13:04:09) [GCC 4.2.1 Compatible Apple LLVM 6.0 (clang-600.0.57)]
OS	posix [darwin]
Wed Jul 19 22:14:32 2017 MDT