Alexander Pitchford (agp1@aber.ac.uk)
Example to demonstrate using the control library to determine control pulses using the ctrlpulseoptim.optimize_pulse function. The (default) L-BFGS-B algorithm is used to optimise the pulse to minimise the fidelity error, which in this case is given by the 'Trace difference' norm.
This in an open quantum system example, with a single qubit subject to an amplitude damping channel. The target evolution is the Hadamard gate. The user can experiment with the strength of the amplitude damping by changing the gamma variable value
The user can experiment with the timeslicing, by means of changing the number of timeslots and/or total time for the evolution. Different initial (starting) pulse types can be tried. The initial and final pulses are displayed in a plot
This example assumes that the example-control-pulseoptim-Hadamard has already been tried, and hence explanations in that notebook are not repeated here.
In [1]:
%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
import datetime
In [2]:
from qutip import Qobj, identity, sigmax, sigmay, sigmaz, tensor
from qutip.qip import hadamard_transform
import qutip.logging as logging
logger = logging.get_logger()
#Set this to None or logging.WARN for 'quiet' execution
log_level = logging.INFO
#QuTiP control modules
import qutip.control.pulseoptim as cpo
example_name = 'Lindblad'
In [3]:
Sx = sigmax()
Sy = sigmay()
Sz = sigmaz()
Si = identity(2)
Sd = Qobj(np.array([[0, 1],
[0, 0]]))
Sm = Qobj(np.array([[0, 0],
[1, 0]]))
Sd_m = Qobj(np.array([[1, 0],
[0, 0]]))
Sm_d = Qobj(np.array([[0, 0],
[0, 1]]))
#Amplitude damping#
#Damping rate:
gamma = 0.1
L0_Ad = gamma*(2*tensor(Sm, Sd.trans()) -
(tensor(Sd_m, Si) + tensor(Si, Sd_m.trans())))
#sigma X control
LC_x = -1j*(tensor(Sx, Si) - tensor(Si, Sx))
#sigma Y control
LC_y = -1j*(tensor(Sy, Si) - tensor(Si, Sy.trans()))
#sigma Z control
LC_z = -1j*(tensor(Sz, Si) - tensor(Si, Sz))
#Drift
drift = L0_Ad
#Controls
ctrls = [LC_z, LC_x]
# Number of ctrls
n_ctrls = len(ctrls)
initial = identity(4)
#Target
#Hadamard gate
had_gate = hadamard_transform(1)
target_DP = tensor(had_gate, had_gate)
In [4]:
# Number of time slots
n_ts = 200
# Time allowed for the evolution
evo_time = 2
In [5]:
# Fidelity error target
fid_err_targ = 1e-10
# Maximum iterations for the optisation algorithm
max_iter = 200
# Maximum (elapsed) time allowed in seconds
max_wall_time = 30
# Minimum gradient (sum of gradients squared)
# as this tends to 0 -> local minima has been found
min_grad = 1e-20
In [6]:
# pulse type alternatives: RND|ZERO|LIN|SINE|SQUARE|SAW|TRIANGLE|
p_type = 'RND'
In [7]:
#Set to None to suppress output files
f_ext = "{}_n_ts{}_ptype{}.txt".format(example_name, n_ts, p_type)
In [8]:
# Note that this call will take the defaults
# dyn_type='GEN_MAT'
# This means that matrices that describe the dynamics are assumed to be
# general, i.e. the propagator can be calculated using:
# expm(combined_dynamics*dt)
# prop_type='FRECHET'
# and the propagators and their gradients will be calculated using the
# Frechet method, i.e. an exact gradent
# fid_type='TRACEDIFF'
# and that the fidelity error, i.e. distance from the target, is give
# by the trace of the difference between the target and evolved operators
result = cpo.optimize_pulse(drift, ctrls, initial, target_DP, n_ts, evo_time,
fid_err_targ=fid_err_targ, min_grad=min_grad,
max_iter=max_iter, max_wall_time=max_wall_time,
out_file_ext=f_ext, init_pulse_type=p_type,
log_level=log_level, gen_stats=True)
In [9]:
result.stats.report()
print("Final evolution\n{}\n".format(result.evo_full_final))
print("********* Summary *****************")
print("Final fidelity error {}".format(result.fid_err))
print("Final gradient normal {}".format(result.grad_norm_final))
print("Terminated due to {}".format(result.termination_reason))
print("Number of iterations {}".format(result.num_iter))
print("Completed in {} HH:MM:SS.US".format(datetime.timedelta(seconds=result.wall_time)))
In [10]:
t = result.time[:n_ts]
fig1 = plt.figure()
ax1 = fig1.add_subplot(2, 1, 1)
ax1.set_title("Initial Control amps")
ax1.set_xlabel("Time")
ax1.set_ylabel("Control amplitude")
for j in range(n_ctrls):
amps = result.initial_amps[:, j]
ax1.plot(t, amps)
ax2 = fig1.add_subplot(2, 1, 2)
ax2.set_title("Optimised Control Amplitudes")
ax2.set_xlabel("Time")
ax2.set_ylabel("Control amplitude")
for j in range(n_ctrls):
amps = result.final_amps[:, j]
ax2.plot(t, amps)
plt.show()
In [11]:
from qutip.ipynbtools import version_table
version_table()
Out[11]: