Entangling gate on two coupled qubits¶
Imports¶
!pip install -q -U pip
!pip install -q matplotlib
# System imports
import copy
import numpy as np
import time
import itertools
import matplotlib.pyplot as plt
import tensorflow as tf
import tensorflow_probability as tfp
from typing import List
# Main C3 objects
from c3.c3objs import Quantity as Qty
from c3.parametermap import ParameterMap as PMap
from c3.experiment import Experiment as Exp
from c3.model import Model as Mdl
from c3.generator.generator import Generator as Gnr
# Building blocks
import c3.generator.devices as devices
import c3.signal.gates as gates
import c3.libraries.chip as chip
import c3.signal.pulse as pulse
import c3.libraries.tasks as tasks
# Libs and helpers
import c3.libraries.algorithms as algorithms
import c3.libraries.hamiltonians as hamiltonians
import c3.libraries.fidelities as fidelities
import c3.libraries.envelopes as envelopes
import c3.utils.qt_utils as qt_utils
import c3.utils.tf_utils as tf_utils
Model components¶
The model consists of two qubits with 3 levels each and slightly different parameters:
qubit_lvls = 3
freq_q1 = 5e9
anhar_q1 = -210e6
t1_q1 = 27e-6
t2star_q1 = 39e-6
qubit_temp = 50e-3
q1 = chip.Qubit(
name="Q1",
desc="Qubit 1",
freq=Qty(value=freq_q1, min_val=4.995e9, max_val=5.005e9, unit='Hz 2pi'),
anhar=Qty(value=anhar_q1, min_val=-380e6, max_val=-120e6, unit='Hz 2pi'),
hilbert_dim=qubit_lvls,
t1=Qty(value=t1_q1, min_val=1e-6, max_val=90e-6, unit='s'),
t2star=Qty(value=t2star_q1, min_val=10e-6, max_val=90e-3, unit='s'),
temp=Qty(value=qubit_temp, min_val=0.0, max_val=0.12, unit='K')
)
freq_q2 = 5.6e9
anhar_q2 = -240e6
t1_q2 = 23e-6
t2star_q2 = 31e-6
q2 = chip.Qubit(
name="Q2",
desc="Qubit 2",
freq=Qty(value=freq_q2, min_val=5.595e9, max_val=5.605e9, unit='Hz 2pi'),
anhar=Qty(value=anhar_q2, min_val=-380e6, max_val=-120e6, unit='Hz 2pi'),
hilbert_dim=qubit_lvls,
t1=Qty(value=t1_q2, min_val=1e-6, max_val=90e-6,unit='s'),
t2star=Qty(value=t2star_q2, min_val=10e-6, max_val=90e-6, unit='s'),
temp=Qty(value=qubit_temp, min_val=0.0, max_val=0.12, unit='K')
)
There is a static coupling in x-direction between them: \((b_1+b_1^\dagger)(b_2+b_2^\dagger)\)
coupling_strength = 50e6
q1q2 = chip.Coupling(
name="Q1-Q2",
desc="coupling",
comment="Coupling qubit 1 to qubit 2",
connected=["Q1", "Q2"],
strength=Qty(
value=coupling_strength,
min_val=-1 * 1e3 ,
max_val=200e6 ,
unit='Hz 2pi'
),
hamiltonian_func=hamiltonians.int_XX
)
and each qubit has a drive line
drive1 = chip.Drive(
name="d1",
desc="Drive 1",
comment="Drive line 1 on qubit 1",
connected=["Q1"],
hamiltonian_func=hamiltonians.x_drive
)
drive2 = chip.Drive(
name="d2",
desc="Drive 2",
comment="Drive line 2 on qubit 2",
connected=["Q2"],
hamiltonian_func=hamiltonians.x_drive
)
All parts are collected in the model. The initial state will be thermal at a non-vanishing temperature.
init_temp = 50e-3
init_ground = tasks.InitialiseGround(
init_temp=Qty(value=init_temp, min_val=-0.001, max_val=0.22, unit='K')
)
model = Mdl(
[q1, q2], # Individual, self-contained components
[drive1, drive2, q1q2], # Interactions between components
[init_ground] # SPAM processing
)
model.set_lindbladian(False)
model.set_dressed(True)
Control signals¶
The devices for the control line are set up
sim_res = 100e9 # Resolution for numerical simulation
awg_res = 2e9 # Realistic, limited resolution of an AWG
v2hz = 1e9
lo = devices.LO(name='lo', resolution=sim_res)
awg = devices.AWG(name='awg', resolution=awg_res)
mixer = devices.Mixer(name='mixer')
resp = devices.Response(
name='resp',
rise_time=Qty(value=0.3e-9, min_val=0.05e-9, max_val=0.6e-9, unit='s'),
resolution=sim_res
)
dig_to_an = devices.DigitalToAnalog(name="dac", resolution=sim_res)
v_to_hz = devices.VoltsToHertz(
name='v_to_hz',
V_to_Hz=Qty(value=v2hz, min_val=0.9e9, max_val=1.1e9, unit='Hz/V')
)
The generator combines the parts of the signal generation and assignes a signal chain to each control line.
generator = Gnr(
devices={
"LO": lo,
"AWG": awg,
"DigitalToAnalog": dig_to_an,
"Response": resp,
"Mixer": mixer,
"VoltsToHertz": v_to_hz
},
chains={
"d1": ["LO", "AWG", "DigitalToAnalog", "Response", "Mixer", "VoltsToHertz"],
"d2": ["LO", "AWG", "DigitalToAnalog", "Response", "Mixer", "VoltsToHertz"]
}
)
Gates-set and Parameter map¶
Following a general cross resonance scheme, both qubits will be resonantly driven at the frequency of qubit 2 with a Gaussian envelope. We drive qubit 1 (the control) at the frequency of qubit 2 (the target) with a higher amplitude to compensate for the reduced Rabi frequency.
t_final = 45e-9
sideband = 50e6
gauss_params_single_1 = {
'amp': Qty(value=0.8, min_val=0.2, max_val=3, unit="V"),
't_final': Qty(value=t_final, min_val=0.5 * t_final, max_val=1.5 * t_final, unit="s"),
'sigma': Qty(value=t_final / 4, min_val=t_final / 8, max_val=t_final / 2, unit="s"),
'xy_angle': Qty(value=0.0, min_val=-0.5 * np.pi, max_val=2.5 * np.pi, unit='rad'),
'freq_offset': Qty(value=-sideband - 3e6, min_val=-56 * 1e6, max_val=-52 * 1e6, unit='Hz 2pi'),
'delta': Qty(value=-1, min_val=-5, max_val=3, unit="")
}
gauss_params_single_2 = {
'amp': Qty(value=0.03, min_val=0.02, max_val=0.6, unit="V"),
't_final': Qty(value=t_final, min_val=0.5 * t_final, max_val=1.5 * t_final, unit="s"),
'sigma': Qty(value=t_final / 4, min_val=t_final / 8, max_val=t_final / 2, unit="s"),
'xy_angle': Qty(value=0.0, min_val=-0.5 * np.pi, max_val=2.5 * np.pi, unit='rad'),
'freq_offset': Qty(value=-sideband - 3e6, min_val=-56 * 1e6, max_val=-52 * 1e6, unit='Hz 2pi'),
'delta': Qty(value=-1, min_val=-5, max_val=3, unit="")
}
gauss_env_single_1 = pulse.Envelope(
name="gauss1",
desc="Gaussian envelope on drive 1",
params=gauss_params_single_1,
shape=envelopes.gaussian_nonorm
)
gauss_env_single_2 = pulse.Envelope(
name="gauss2",
desc="Gaussian envelope on drive 2",
params=gauss_params_single_2,
shape=envelopes.gaussian_nonorm
)
The carrier signal of each drive is set to the resonance frequency of the target qubit.
lo_freq_q1 = freq_q1 + sideband
lo_freq_q2 = freq_q2 + sideband
carr_1 = pulse.Carrier(
name="carrier",
desc="Carrier on drive 1",
params={
'freq': Qty(value=lo_freq_q2, min_val=0.9 * lo_freq_q2, max_val=1.1 * lo_freq_q2, unit='Hz 2pi'),
'framechange': Qty(value=0.0, min_val=-np.pi, max_val=3 * np.pi, unit='rad')
}
)
carr_2 = pulse.Carrier(
name="carrier",
desc="Carrier on drive 2",
params={
'freq': Qty(value=lo_freq_q2, min_val=0.9 * lo_freq_q2, max_val=1.1 * lo_freq_q2, unit='Hz 2pi'),
'framechange': Qty(value=0.0, min_val=-np.pi, max_val=3 * np.pi, unit='rad')
}
)
Instructions¶
The instruction to be optimised is a CNOT gates controlled by qubit 1.
# CNOT comtrolled by qubit 1
cnot12 = gates.Instruction(
name="cnot12", targets=[0, 1], t_start=0.0, t_end=t_final, channels=["d1", "d2"],
ideal=np.array([
[1,0,0,0],
[0,1,0,0],
[0,0,0,1],
[0,0,1,0]
])
)
cnot12.add_component(gauss_env_single_1, "d1")
cnot12.add_component(carr_1, "d1")
cnot12.add_component(gauss_env_single_2, "d2")
cnot12.add_component(carr_2, "d2")
cnot12.comps["d1"]["carrier"].params["framechange"].set_value(
(-sideband * t_final) * 2 * np.pi % (2 * np.pi)
)
The experiment¶
All components are collected in the parameter map and the experiment is set up.
parameter_map = PMap(instructions=[cnot12], model=model, generator=generator)
exp = Exp(pmap=parameter_map)
Calculate and print the propagator before the optimisation.
unitaries = exp.compute_propagators()
print(unitaries[cnot12.get_key()])
tf.Tensor(
[[ 5.38699071e-01-7.17750563e-02j -8.34752005e-01+8.73275022e-02j
-6.95346256e-03-2.15875540e-03j -4.35619589e-03+3.35449682e-03j
-1.06942994e-02+4.11831376e-03j -6.46672021e-05-3.73989900e-05j
-1.67838080e-04-2.08026492e-04j -6.43312053e-05-7.70584828e-07j
-3.76227149e-07-6.49845314e-07j]
[-8.22954017e-01+1.64865789e-01j -5.35373070e-01+9.17248769e-02j
-7.01716357e-03+7.68563193e-03j -1.04194796e-02+4.75452421e-03j
-1.61239175e-02-5.34774092e-03j -2.42060738e-04-1.19946128e-05j
3.81855912e-05+8.66289943e-06j -1.30621879e-04-2.10380577e-04j
-8.82654253e-07-1.33276919e-06j]
[-7.61570279e-03+7.68089055e-04j -4.61417534e-03+9.02462832e-03j
3.59132066e-01-9.32828470e-01j -9.10153028e-05-6.83262609e-05j
-2.24711912e-04+8.79671466e-05j 2.62921224e-02-1.48696337e-03j
-4.75883791e-04-4.20508543e-05j 3.46114778e-05+1.64470496e-04j
2.10121296e-04+1.48066297e-04j]
[ 4.65531318e-03-6.63491197e-05j 8.62792565e-03+8.22022317e-03j
-5.58701973e-05+1.08666061e-04j 6.94902895e-02-7.11528641e-01j
-6.81737268e-01-1.53183314e-01j -2.09824678e-03-1.43761730e-03j
1.48197730e-02-1.51149441e-02j -6.85074400e-03+1.43594091e-03j
4.07440635e-05-6.43168354e-05j]
[ 9.49155432e-03+6.86731461e-03j 4.92068252e-03+1.60041286e-02j
1.71300460e-04+1.83910737e-04j -6.94165643e-01-7.98008223e-02j
1.68675369e-01-6.94722446e-01j 2.75768137e-03-5.72343874e-03j
-6.67593164e-03+1.87532770e-03j 1.07707017e-02+7.28665794e-03j
1.40030301e-04-6.25646793e-05j]
[ 3.43460967e-05+8.01438338e-05j 1.86345824e-04+1.52916372e-04j
-1.74936595e-02-1.96833938e-02j -2.61695107e-03-5.33671505e-04j
1.02116861e-03-6.21800378e-03j -4.07849502e-01+9.12571012e-01j
7.51460471e-05-1.15167196e-04j 2.32056836e-04-2.97650209e-04j
2.03278960e-04+1.15047574e-02j]
[ 2.54853797e-04-1.25904275e-04j 6.64845849e-05-1.08876861e-05j
2.38628329e-04-2.95318799e-04j -2.10696691e-02+5.90348860e-05j
4.21445291e-03+6.01993253e-03j -1.32690530e-04-2.44975772e-05j
5.90859776e-01+4.84056180e-01j -6.08336007e-01-2.14442516e-01j
3.13146026e-03+2.83895304e-03j]
[ 2.96366741e-05-8.10052801e-05j 2.39607442e-04-8.47647458e-05j
-2.60360838e-04+2.04175607e-04j 4.95127881e-03+5.19423708e-03j
-5.00047077e-03-1.18242204e-02j -3.71631612e-04-5.78977628e-05j
-6.29480118e-01-1.40758384e-01j -7.57820104e-01+9.68476237e-02j
1.32060361e-03+7.25998662e-03j]
[ 8.28054635e-07-3.59336781e-07j 1.64602058e-06-1.47364829e-06j
-2.13361477e-04+2.05358711e-04j -5.70978380e-05+4.73283539e-05j
-1.48466829e-04-3.89352221e-06j 1.00811226e-02-5.54615336e-03j
4.21887172e-03+1.38103179e-03j 3.74182763e-03+6.21303072e-03j
-5.89257172e-01+8.07818774e-01j]], shape=(9, 9), dtype=complex128)
Dynamics¶
The system is initialised in the state \(|0,1\rangle\) so that a transition to \(|1,1\rangle\) should be visible.
psi_init = [[0] * 9]
psi_init[0][0] = 1
init_state = tf.transpose(tf.constant(psi_init, tf.complex128))
print(init_state)
tf.Tensor(
[[1.+0.j]
[0.+0.j]
[0.+0.j]
[0.+0.j]
[0.+0.j]
[0.+0.j]
[0.+0.j]
[0.+0.j]
[0.+0.j]], shape=(9, 1), dtype=complex128)
def plot_dynamics(exp, psi_init, seq):
"""
Plotting code for time-resolved populations.
Parameters
----------
psi_init: tf.Tensor
Initial state or density matrix.
seq: list
List of operations to apply to the initial state.
"""
model = exp.pmap.model
dUs = exp.partial_propagators
psi_t = psi_init.numpy()
pop_t = exp.populations(psi_t, model.lindbladian)
for gate in seq:
for du in dUs[gate]:
psi_t = np.matmul(du.numpy(), psi_t)
pops = exp.populations(psi_t, model.lindbladian)
pop_t = np.append(pop_t, pops, axis=1)
fig, axs = plt.subplots(1, 1)
ts = exp.ts
dt = ts[1] - ts[0]
ts = np.linspace(0.0, dt*pop_t.shape[1], pop_t.shape[1])
axs.plot(ts / 1e-9, pop_t.T)
axs.grid(linestyle="--")
axs.tick_params(
direction="in", left=True, right=True, top=True, bottom=True
)
axs.set_xlabel('Time [ns]')
axs.set_ylabel('Population')
plt.legend(model.state_labels)
pass
def getQubitsPopulation(population: np.array, dims: List[int]) -> np.array:
"""
Splits the population of all levels of a system into the populations of levels per subsystem.
Parameters
----------
population: np.array
The time dependent population of each energy level. First dimension: level index, second dimension: time.
dims: List[int]
The number of levels for each subsystem.
Returns
-------
np.array
The time-dependent population of energy levels for each subsystem. First dimension: subsystem index, second
dimension: level index, third dimension: time.
"""
numQubits = len(dims)
# create a list of all levels
qubit_levels = []
for dim in dims:
qubit_levels.append(list(range(dim)))
combined_levels = list(itertools.product(*qubit_levels))
# calculate populations
qubitsPopulations = np.zeros((numQubits, dims[0], population.shape[1]))
for idx, levels in enumerate(combined_levels):
for i in range(numQubits):
qubitsPopulations[i, levels[i]] += population[idx]
return qubitsPopulations
def plotSplittedPopulation(
exp: Exp,
psi_init: tf.Tensor,
sequence: List[str]
) -> None:
"""
Plots time dependent populations for multiple qubits in separate plots.
Parameters
----------
exp: Experiment
The experiment containing the model and propagators
psi_init: np.array
Initial state vector
sequence: List[str]
List of gate names that will be applied to the state
-------
"""
# calculate the time dependent level population
model = exp.pmap.model
dUs = exp.partial_propagators
psi_t = psi_init.numpy()
pop_t = exp.populations(psi_t, model.lindbladian)
for gate in sequence:
for du in dUs[gate]:
psi_t = np.matmul(du, psi_t)
pops = exp.populations(psi_t, model.lindbladian)
pop_t = np.append(pop_t, pops, axis=1)
dims = [s.hilbert_dim for s in model.subsystems.values()]
splitted = getQubitsPopulation(pop_t, dims)
# timestamps
dt = exp.ts[1] - exp.ts[0]
ts = np.linspace(0.0, dt * pop_t.shape[1], pop_t.shape[1])
# create both subplots
titles = list(exp.pmap.model.subsystems.keys())
fig, axs = plt.subplots(1, len(splitted), sharey="all")
for idx, ax in enumerate(axs):
ax.plot(ts / 1e-9, splitted[idx].T)
ax.tick_params(direction="in", left=True, right=True, top=False, bottom=True)
ax.set_xlabel("Time [ns]")
ax.set_ylabel("Population")
ax.set_title(titles[idx])
ax.legend([str(x) for x in np.arange(dims[idx])])
ax.grid()
plt.tight_layout()
plt.show()
sequence = [cnot12.get_key()]
plot_dynamics(exp, init_state, sequence)
plotSplittedPopulation(exp, init_state, sequence)
Open-loop optimal control¶
Now, open-loop optimisation with DRAG enabled is set up.
generator.devices['AWG'].enable_drag_2()
opt_gates = [cnot12.get_key()]
exp.set_opt_gates(opt_gates)
gateset_opt_map=[
[(cnot12.get_key(), "d1", "gauss1", "amp")],
[(cnot12.get_key(), "d1", "gauss1", "freq_offset")],
[(cnot12.get_key(), "d1", "gauss1", "xy_angle")],
[(cnot12.get_key(), "d1", "gauss1", "delta")],
[(cnot12.get_key(), "d1", "carrier", "framechange")],
[(cnot12.get_key(), "d2", "gauss2", "amp")],
[(cnot12.get_key(), "d2", "gauss2", "freq_offset")],
[(cnot12.get_key(), "d2", "gauss2", "xy_angle")],
[(cnot12.get_key(), "d2", "gauss2", "delta")],
[(cnot12.get_key(), "d2", "carrier", "framechange")]
]
parameter_map.set_opt_map(gateset_opt_map)
parameter_map.print_parameters()
cnot12[0, 1]-d1-gauss1-amp : 800.000 mV
cnot12[0, 1]-d1-gauss1-freq_offset : -53.000 MHz 2pi
cnot12[0, 1]-d1-gauss1-xy_angle : -444.089 arad
cnot12[0, 1]-d1-gauss1-delta : -1.000
cnot12[0, 1]-d1-carrier-framechange : 4.712 rad
cnot12[0, 1]-d2-gauss2-amp : 30.000 mV
cnot12[0, 1]-d2-gauss2-freq_offset : -53.000 MHz 2pi
cnot12[0, 1]-d2-gauss2-xy_angle : -444.089 arad
cnot12[0, 1]-d2-gauss2-delta : -1.000
cnot12[0, 1]-d2-carrier-framechange : 0.000 rad
As a fidelity function we choose unitary fidelity as well as LBFG-S (a wrapper of the scipy implementation) from our library.
import os
import tempfile
from c3.optimizers.optimalcontrol import OptimalControl
log_dir = os.path.join(tempfile.TemporaryDirectory().name, "c3logs")
opt = OptimalControl(
dir_path=log_dir,
fid_func=fidelities.unitary_infid_set,
fid_subspace=["Q1", "Q2"],
pmap=parameter_map,
algorithm=algorithms.lbfgs,
options={
"maxfun": 25
},
run_name="cnot12"
)
Start the optimisation
exp.set_opt_gates(opt_gates)
opt.set_exp(exp)
opt.optimize_controls()
C3:STATUS:Saving as: /tmp/tmpjx66lyg2/c3logs/cnot12/2021_12_08_T_12_27_05/open_loop.log
1 0.8790556354859858
2 0.9673489008768812
3 0.758622722337525
4 0.7679637459613755
5 0.6962301452070802
6 0.541321232138175
7 0.5682335581707882
8 0.382921410272719
9 0.43114251105289114
10 0.30099424375388173
11 0.32449492775751976
12 0.26537726105532744
13 0.2653362073570743
14 0.25121669688810866
15 0.23925168937407626
16 0.18551042816386099
17 0.1305543307431979
18 0.07413739981051659
19 0.031551815290153495
20 0.017447484467834062
21 0.007924221221055072
22 0.006483318391815374
23 0.005732979353259449
24 0.005594385264244273
25 0.0055582927728303755
26 0.005521343169743842
The final parameters and the fidelity are
parameter_map.print_parameters()
print(opt.current_best_goal)
cnot12[0, 1]-d1-gauss1-amp : 2.359 V
cnot12[0, 1]-d1-gauss1-freq_offset : -53.252 MHz 2pi
cnot12[0, 1]-d1-gauss1-xy_angle : 587.818 mrad
cnot12[0, 1]-d1-gauss1-delta : -743.473 m
cnot12[0, 1]-d1-carrier-framechange : -815.216 mrad
cnot12[0, 1]-d2-gauss2-amp : 56.719 mV
cnot12[0, 1]-d2-gauss2-freq_offset : -53.176 MHz 2pi
cnot12[0, 1]-d2-gauss2-xy_angle : -135.515 mrad
cnot12[0, 1]-d2-gauss2-delta : -519.864 m
cnot12[0, 1]-d2-carrier-framechange : 598.919 mrad
0.005521343169743842
Results of the optimisation¶
Plotting the dynamics with the same initial state:
plot_dynamics(exp, init_state, sequence)
plotSplittedPopulation(exp, init_state, sequence)
Now we plot the dynamics for the control in the excited state.
psi_init = [[0] * 9]
psi_init[0][4] = 1
init_state = tf.transpose(tf.constant(psi_init, tf.complex128))
print(init_state)
plot_dynamics(exp, init_state, sequence)
plotSplittedPopulation(exp, init_state, sequence)
tf.Tensor(
[[0.+0.j]
[0.+0.j]
[0.+0.j]
[0.+0.j]
[1.+0.j]
[0.+0.j]
[0.+0.j]
[0.+0.j]
[0.+0.j]], shape=(9, 1), dtype=complex128)
As intended, the dynamics of the target is dependent on the control qubit performing a flip if the control is excited and an identity otherwise.