Calculation of control fields for Lindbladian dynamics using L-BFGS-B algorithm¶

Christian Arenz ([email protected]), Alexander Pitchford ([email protected])

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 :
%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
import datetime
In :
from qutip import Qobj, identity, sigmax, sigmay, sigmaz, sigmam, tensor
from qutip.superoperator import liouvillian, sprepost
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

Defining the physics¶

In :
Sx = sigmax()
Sy = sigmay()
Sz = sigmaz()
Sm = sigmam()
Si = identity(2)

# 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

Defining the time evolution parameters¶

In :
# Number of time slots
n_ts = 10
# Time allowed for the evolution
evo_time = 2

Set the conditions which will cause the pulse optimisation to terminate¶

In :
# 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
# as this tends to 0 -> local minima has been found

Set the initial pulse type¶

In :
# pulse type alternatives: RND|ZERO|LIN|SINE|SQUARE|SAW|TRIANGLE|
p_type = 'RND'

Give an extension for output files¶

In :
#Set to None to suppress output files
f_ext = "{}_n_ts{}_ptype{}.txt".format(example_name, n_ts, p_type)

Run the optimisation¶

In :
# 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,
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)
INFO:qutip.control.dynamics:Setting memory optimisations for level 0
INFO:qutip.control.dynamics:Internal operator data type choosen to be <class 'numpy.ndarray'>
INFO:qutip.control.dynamics:phased dynamics generator caching True
INFO:qutip.control.dynamics:use sparse eigen decomp False
INFO:qutip.control.pulseoptim:System configuration:
Drift dynamics generator:
Quantum object: dims = [[, ], [, ]], shape = (4, 4), type = super, isherm = False
Qobj data =
[[-0.10+0.j    0.00-0.05j  0.00+0.05j  0.00+0.j  ]
[ 0.00-0.05j -0.05+1.j    0.00+0.j    0.00+0.05j]
[ 0.00+0.05j  0.00+0.j   -0.05-1.j    0.00-0.05j]
[ 0.10+0.j    0.00+0.05j  0.00-0.05j  0.00+0.j  ]]
Control 1 dynamics generator:
Quantum object: dims = [[, ], [, ]], shape = (4, 4), type = super, isherm = False
Qobj data =
[[ 0.+0.j  0.+0.j  0.+0.j  0.+0.j]
[ 0.+0.j  0.+2.j  0.+0.j  0.+0.j]
[ 0.+0.j  0.+0.j  0.-2.j  0.+0.j]
[ 0.+0.j  0.+0.j  0.+0.j  0.+0.j]]
Control 2 dynamics generator:
Quantum object: dims = [[, ], [, ]], shape = (4, 4), type = super, isherm = False
Qobj data =
[[ 0.+0.j  0.-1.j  0.+1.j  0.+0.j]
[ 0.-1.j  0.+0.j  0.+0.j  0.+1.j]
[ 0.+1.j  0.+0.j  0.+0.j  0.-1.j]
[ 0.+0.j  0.+1.j  0.-1.j  0.+0.j]]
Initial state / operator:
Quantum object: dims = [[, ], [, ]], shape = (4, 4), type = super, isherm = True
Qobj data =
[[ 1.  0.  0.  0.]
[ 0.  1.  0.  0.]
[ 0.  0.  1.  0.]
[ 0.  0.  0.  1.]]
Target state / operator:
Quantum object: dims = [[, ], [, ]], shape = (4, 4), type = super, isherm = True
Qobj data =
[[ 0.5  0.5  0.5  0.5]
[ 0.5 -0.5  0.5 -0.5]
[ 0.5  0.5 -0.5 -0.5]
[ 0.5 -0.5 -0.5  0.5]]
INFO:qutip.control.pulseoptim:Initial amplitudes output to file: ctrl_amps_initial_Lindblad_n_ts10_ptypeRND.txt
INFO:qutip.control.optimizer:Optimising pulse(s) using GRAPE with 'fmin_l_bfgs_b' method
INFO:qutip.control.pulseoptim:Final amplitudes output to file: ctrl_amps_final_Lindblad_n_ts10_ptypeRND.txt

Report the results¶

In :
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("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)))
------------------------------------
---- Control optimisation stats ----
**** Timings (HH:MM:SS.US) ****
Total wall time elapsed during optimisation: 0:00:01.340385
Wall time computing Hamiltonians: 0:00:00.020057 (1.50%)
Wall time computing propagators: 0:00:01.178558 (87.93%)
Wall time computing forward propagation: 0:00:00.005692 (0.42%)
Wall time computing onward propagation: 0:00:00.004413 (0.33%)
Wall time computing gradient: 0:00:00.091266 (6.81%)

**** Iterations and function calls ****
Number of iterations: 201
Number of fidelity function calls: 265
Number of times fidelity is computed: 265
Number of gradient function calls: 265
Number of times gradients are computed: 265
Number of times timeslot evolution is recomputed: 265

**** Control amplitudes ****
Number of control amplitude updates: 264
Mean number of updates per iteration: 1.3134328358208955
Number of timeslot values changed: 2640
Mean number of timeslot changes per update: 10.0
Number of amplitude values changed: 5280
Mean number of amplitude changes per update: 20.0
------------------------------------
Final evolution
Quantum object: dims = [[, ], [, ]], shape = (4, 4), type = super, isherm = False
Qobj data =
[[ 0.49398354 -9.35884799e-17j  0.43895309 +4.70500365e-04j
0.43895309 -4.70500365e-04j  0.50438676 +1.16197332e-16j]
[ 0.43683752 -2.24436864e-03j -0.44201112 -3.02814020e-03j
0.43197985 +3.42846832e-03j -0.43695905 +4.27534789e-03j]
[ 0.43683752 +2.24436864e-03j  0.43197985 -3.42846832e-03j
-0.44201112 +3.02814020e-03j -0.43695905 -4.27534789e-03j]
[ 0.50601646 +1.18358316e-16j -0.43895309 -4.70500365e-04j
-0.43895309 +4.70500365e-04j  0.49561324 -1.33018845e-16j]]

********* Summary *****************
Initial fidelity error 0.7448841629400895
Final fidelity error 0.005876671350497931
Terminated due to Iteration or fidelity function call limit reached
Number of iterations 201
Completed in 0:00:01.340385 HH:MM:SS.US

Plot the initial and final amplitudes¶

In :
fig1 = plt.figure()
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.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() Versions¶

In :
from qutip.ipynbtools import version_table

version_table()
Out:
SoftwareVersion
QuTiP4.2.0
Numpy1.13.1
SciPy0.19.1
matplotlib2.0.2
Cython0.26
Number of CPUs4
BLAS InfoINTEL MKL
IPython6.1.0
Python3.6.2 |Continuum Analytics, Inc.| (default, Jul 20 2017, 13:51:32) [GCC 4.4.7 20120313 (Red Hat 4.4.7-1)]
OSposix [linux]
Thu Apr 05 11:50:14 2018 BST
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