Christian Arenz (christianarenz.ca@gmail.com), Alexander Pitchford (alex.pitchford@gmail.com)
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. For a $d$ dimensional quantum system in general we represent the Lindbladian as a $d^2 \times d^2$ dimensional matrix by creating the Liouvillian superoperator. Here done for the Lindbladian that describes the amplitude damping channel. Similarly the control generators acting on the qubit are also converted to superoperators. The initial and target maps also need to be in superoperator form.
The user can experiment with the strength of the amplitude damping by changing the gamma variable value. If the rate is sufficiently small then the target fidelity can be achieved within the given tolerence. The drift Hamiltonian and control generators can also be swapped and changed to experiment with controllable and uncontrollable setups.
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
For more background on the pulse optimisation see: QuTiP overview - Optimal Control
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, sigmam, tensor
from qutip.superoperator import liouvillian, sprepost
from qutip.qip import hadamard_transform
import qutip.logging_utils 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()
Sm = sigmam()
Si = identity(2)
#Hadamard gate
had_gate = hadamard_transform(1)
# Hamiltonian
Del = 0.1 # Tunnelling term
wq = 1.0 # Energy of the 2-level system.
H0 = 0.5*wq*sigmaz() + 0.5*Del*sigmax()
#Amplitude damping#
#Damping rate:
gamma = 0.1
L0 = liouvillian(H0, [np.sqrt(gamma)*Sm])
#sigma X control
LC_x = liouvillian(Sx)
#sigma Y control
LC_y = liouvillian(Sy)
#sigma Z control
LC_z = liouvillian(Sz)
#Drift
drift = L0
#Controls - different combinations can be tried
ctrls = [LC_z, LC_x]
# Number of ctrls
n_ctrls = len(ctrls)
# start point for the map evolution
E0 = sprepost(Si, Si)
# target for map evolution
E_targ = sprepost(had_gate, had_gate)
In [4]:
# Number of time slots
n_ts = 10
# Time allowed for the evolution
evo_time = 2
In [5]:
# Fidelity error target
fid_err_targ = 1e-3
# 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, E0, E_targ, 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("Initial fidelity error {}".format(result.initial_fid_err))
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]:
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):
ax1.step(result.time,
np.hstack((result.initial_amps[:, j], result.initial_amps[-1, j])),
where='post')
ax2 = fig1.add_subplot(2, 1, 2)
ax2.set_title("Optimised Control Sequences")
ax2.set_xlabel("Time")
ax2.set_ylabel("Control amplitude")
for j in range(n_ctrls):
ax2.step(result.time,
np.hstack((result.final_amps[:, j], result.final_amps[-1, j])),
where='post')
fig1.tight_layout()
In [11]:
from qutip.ipynbtools import version_table
version_table()
Out[11]:
In [ ]: