Quantum Generative Adversarial Network to Generate Fake MAV (Micro Aerial Vehicle) Navigation Data

Authors

Michel Barbeau

Joaquin Garcia-Alfaro

Resources

Version: August 30, 2019

Associated Paper available on https://arxiv.org/abs/1907.03038.

Sample videocaptures.

Reused code from:

Variational classifier - Iris classification

Quantum Generative Adversarial Network

Gaussian transformation

1. Preamble

PennyLane and other relevant package are imported.

import pennylane as qml
from pennylane import numpy as np
from pennylane.optimize import GradientDescentOptimizer
import math

A six-qubit device is create. With probability amplitude encoding, up to $2^6$ single scalar values can be reprented in probability amplitudes in the input circuit quantum state.

num_qubits = 6
dev = qml.device('default.qubit', wires=num_qubits)

2. Pre-processing of real data

Read the raw real data, assuming $m$ input files. Each file may contain up to $2^{6}$ single scalar values. Each line comprises a timestamp (not used), a $x$-velocity, $y$-velocity, and $z$-velocity.

m = 6 # number of real data files
real_values = np.zeros((m,2**num_qubits))
for i in range(m):
    # read real data from one file
    data = np.loadtxt('Parrot_Mambo_Data/realdata'+str(i)+'.txt')
    selected_data = data[:,1:4] # omit the timestamp
    # reshape three-column matrix into an array
    real_values[i][0:len(selected_data)*3] = \
        selected_data.reshape(len(selected_data)*3)
    print("Raw data: ",real_values[i,:])

Raw data:  [-0.08376908  0.33940566  0.00183541 -0.08376908  0.33940566  0.00183541
 -0.08376908  0.33940566  0.00183541  0.00109504 -0.01502317  0.01236341
  0.00109504 -0.01502317  0.01236341  0.00109504 -0.01502317  0.01236341
 -0.03160943  0.01453508  0.01444724 -0.03160943  0.01453508  0.01444724
 -0.03160943  0.01453508  0.01444724 -0.03160943  0.01453508  0.01444724
 -0.02090058  0.01354719  0.00997251 -0.02090058  0.01354719  0.00997251
 -0.02090058  0.01354719  0.00997251 -0.00556759 -0.00805624  0.01183656
 -0.00556759 -0.00805624  0.01183656 -0.00556759 -0.00805624  0.01183656
  0.01629235 -0.00671357  0.01200437  0.01629235 -0.00671357  0.01200437
  0.01629235 -0.00671357  0.01200437  0.01629235 -0.00671357  0.01200437
  0.01775298  0.00872391  0.01240711  0.        ]
Raw data:  [ 0.07393985 -0.08155741 -0.00174763 -0.01699297  0.0439186  -0.00723943
 -0.01699297  0.0439186  -0.00723943 -0.0467293   0.02906708  0.02376857
 -0.0467293   0.02906708  0.02376857  0.00972683  0.05553944  0.01813706
  0.00972683  0.05553944  0.01813706  0.00972683  0.05553944  0.01813706
  0.00267601 -0.00370482  0.01482126  0.00267601 -0.00370482  0.01482126
 -0.01471184 -0.0770198  -0.00371389 -0.01471184 -0.0770198  -0.00371389
 -0.01471184 -0.0770198  -0.00371389  0.01513966  0.00751151  0.0016375
  0.          0.          0.          0.          0.          0.
  0.          0.          0.          0.          0.          0.
  0.          0.          0.          0.          0.          0.
  0.          0.          0.          0.        ]
Raw data:  [ 9.180749e-02 -7.005025e-02  1.178688e-02  9.180749e-02 -7.005025e-02
  1.178688e-02  1.077333e-02 -1.144817e-02  1.415802e-02  1.077333e-02
 -1.144817e-02  1.415802e-02  1.077333e-02 -1.144817e-02  1.415802e-02
 -5.388897e-02  2.071080e-02  1.527728e-02 -5.388897e-02  2.071080e-02
  1.527728e-02 -3.625565e-02  2.723628e-02  1.610531e-02 -3.625565e-02
  2.723628e-02  1.610531e-02 -3.625565e-02  2.723628e-02  1.610531e-02
  9.066000e-05 -1.490826e-02 -2.906900e-04  9.066000e-05 -1.490826e-02
 -2.906900e-04  9.066000e-05 -1.490826e-02 -2.906900e-04  3.568355e-02
 -3.052060e-02  2.274893e-02  3.568355e-02 -3.052060e-02  2.274893e-02
  3.568355e-02 -3.052060e-02  2.274893e-02 -7.671050e-03  3.214242e-02
  9.254490e-03  0.000000e+00  0.000000e+00  0.000000e+00  0.000000e+00
  0.000000e+00  0.000000e+00  0.000000e+00  0.000000e+00  0.000000e+00
  0.000000e+00  0.000000e+00  0.000000e+00  0.000000e+00]
Raw data:  [-0.00509642 -0.00832376  0.01506707 -0.00509642 -0.00832376  0.01506707
 -0.00509642 -0.00832376  0.01506707 -0.02719803  0.03381     0.01706991
 -0.02719803  0.03381     0.01706991  0.00247172  0.00940448  0.01620432
  0.00247172  0.00940448  0.01620432  0.00247172  0.00940448  0.01620432
 -0.01173865 -0.00884636  0.01537707 -0.01173865 -0.00884636  0.01537707
 -0.01173865 -0.00884636  0.01537707  0.01738478 -0.01649329  0.01408078
  0.01738478 -0.01649329  0.01408078  0.01618808 -0.0008779   0.016907
  0.          0.          0.          0.          0.          0.
  0.          0.          0.          0.          0.          0.
  0.          0.          0.          0.          0.          0.
  0.          0.          0.          0.        ]
Raw data:  [-9.5720300e-03  2.7768114e-01  8.9101400e-03 -9.5720300e-03
  2.7768114e-01  8.9101400e-03 -9.5720300e-03  2.7768114e-01
  8.9101400e-03  2.2996000e-02 -1.3513980e-02  8.8767600e-03
  2.2996000e-02 -1.3513980e-02  8.8767600e-03 -1.3623320e-02
  6.5740600e-03  1.1207880e-02 -1.3623320e-02  6.5740600e-03
  1.1207880e-02 -1.3623320e-02  6.5740600e-03  1.1207880e-02
 -2.0980570e-02  2.6495010e-02  1.3232210e-02 -2.0980570e-02
  2.6495010e-02  1.3232210e-02 -2.0980570e-02  2.6495010e-02
  1.3232210e-02  5.8310000e-05  1.8326290e-02  1.0773430e-02
  5.8310000e-05  1.8326290e-02  1.0773430e-02  1.3626400e-03
  2.7880800e-03  1.2730800e-02  1.3626400e-03  2.7880800e-03
  1.2730800e-02  1.3626400e-03  2.7880800e-03  1.2730800e-02
  5.7880700e-03 -1.5020210e-02  1.2070470e-02  5.7880700e-03
 -1.5020210e-02  1.2070470e-02 -5.3494500e-03 -3.8512300e-03
  1.1064610e-02  0.0000000e+00  0.0000000e+00  0.0000000e+00
  0.0000000e+00  0.0000000e+00  0.0000000e+00  0.0000000e+00]
Raw data:  [-0.00068315 -0.03387602  0.01362513 -0.00068315 -0.03387602  0.01362513
 -0.00068315 -0.03387602  0.01362513 -0.03181319  0.02339145  0.01939554
 -0.03181319  0.02339145  0.01939554 -0.0176115   0.01941477  0.01413892
 -0.0176115   0.01941477  0.01413892 -0.0176115   0.01941477  0.01413892
 -0.00012692  0.01502633  0.00993828 -0.00012692  0.01502633  0.00993828
 -0.00012692  0.01502633  0.00993828  0.00801174 -0.00790758  0.01288757
  0.00801174 -0.00790758  0.01288757  0.00737117  0.00776267  0.01545549
  0.          0.          0.          0.          0.          0.
  0.          0.          0.          0.          0.          0.
  0.          0.          0.          0.          0.          0.
  0.          0.          0.          0.        ]

This example uses probability amplitude encoding, which requires normalized data. Let $x_0,\ldots,x_{n-1}$ be the real data values, their normal form is

$x_0/m,\ldots,x_{n-1}/m$

where

$m = \sqrt{x_0^2 + \ldots +x_{n-1}^2}$.

# Normalization function
def normalize(values):
    m = np.sqrt(np.sum(values ** 2))
    return values / m
for i in range(len(real_values)):
   real_values[i,:] = normalize(real_values[i,:])
   # check consistency of probability amplitudes
   assert np.abs(1-np.sum(real_values[i,:] ** 2))<=0.0000001
print("Normalized data: ",real_values)

Normalized data:  [[-1.36124893e-01  5.51534757e-01  2.98254428e-03 -1.36124893e-01
   5.51534757e-01  2.98254428e-03 -1.36124893e-01  5.51534757e-01
   2.98254428e-03  1.77944181e-03 -2.44126760e-02  2.00905616e-02
   1.77944181e-03 -2.44126760e-02  2.00905616e-02  1.77944181e-03
  -2.44126760e-02  2.00905616e-02 -5.13653759e-02  2.36195290e-02
   2.34767888e-02 -5.13653759e-02  2.36195290e-02  2.34767888e-02
  -5.13653759e-02  2.36195290e-02  2.34767888e-02 -5.13653759e-02
   2.36195290e-02  2.34767888e-02 -3.39634770e-02  2.20142061e-02
   1.62053452e-02 -3.39634770e-02  2.20142061e-02  1.62053452e-02
  -3.39634770e-02  2.20142061e-02  1.62053452e-02 -9.04734293e-03
  -1.30914033e-02  1.92344295e-02 -9.04734293e-03 -1.30914033e-02
   1.92344295e-02 -9.04734293e-03 -1.30914033e-02  1.92344295e-02
   2.64750956e-02 -1.09095623e-02  1.95071211e-02  2.64750956e-02
  -1.09095623e-02  1.95071211e-02  2.64750956e-02 -1.09095623e-02
   1.95071211e-02  2.64750956e-02 -1.09095623e-02  1.95071211e-02
   2.88486218e-02  1.41763681e-02  2.01615742e-02  0.00000000e+00]
 [ 3.19283423e-01 -3.52177196e-01 -7.54652991e-03 -7.33782072e-02
   1.89647138e-01 -3.12609505e-02 -7.33782072e-02  1.89647138e-01
  -3.12609505e-02 -2.01784165e-01  1.25516035e-01  1.02636270e-01
  -2.01784165e-01  1.25516035e-01  1.02636270e-01  4.20019189e-02
   2.39827678e-01  7.83185604e-02  4.20019189e-02  2.39827678e-01
   7.83185604e-02  4.20019189e-02  2.39827678e-01  7.83185604e-02
   1.15554148e-02 -1.59979715e-02  6.40004359e-02  1.15554148e-02
  -1.59979715e-02  6.40004359e-02 -6.35279439e-02 -3.32583112e-01
  -1.60371371e-02 -6.35279439e-02 -3.32583112e-01 -1.60371371e-02
  -6.35279439e-02 -3.32583112e-01 -1.60371371e-02  6.53753352e-02
   3.24358330e-02  7.07097196e-03  0.00000000e+00  0.00000000e+00
   0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00
   0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00
   0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00
   0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00
   0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00]
 [ 4.01126866e-01 -3.06064758e-01  5.14994390e-02  4.01126866e-01
  -3.06064758e-01  5.14994390e-02  4.70710189e-02 -5.00195414e-02
   6.18594647e-02  4.70710189e-02 -5.00195414e-02  6.18594647e-02
   4.70710189e-02 -5.00195414e-02  6.18594647e-02 -2.35452615e-01
   9.04899839e-02  6.67497548e-02 -2.35452615e-01  9.04899839e-02
   6.67497548e-02 -1.58408810e-01  1.19001223e-01  7.03675977e-02
  -1.58408810e-01  1.19001223e-01  7.03675977e-02 -1.58408810e-01
   1.19001223e-01  7.03675977e-02  3.96113233e-04 -6.51374262e-02
  -1.27008775e-03  3.96113233e-04 -6.51374262e-02 -1.27008775e-03
   3.96113233e-04 -6.51374262e-02 -1.27008775e-03  1.55909181e-01
  -1.33351131e-01  9.93950166e-02  1.55909181e-01 -1.33351131e-01
   9.93950166e-02  1.55909181e-01 -1.33351131e-01  9.93950166e-02
  -3.35164837e-02  1.40437215e-01  4.04348770e-02  0.00000000e+00
   0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00
   0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00
   0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00]
 [-5.13822540e-02 -8.39203893e-02  1.51906636e-01 -5.13822540e-02
  -8.39203893e-02  1.51906636e-01 -5.13822540e-02 -8.39203893e-02
   1.51906636e-01 -2.74211326e-01  3.40873399e-01  1.72099327e-01
  -2.74211326e-01  3.40873399e-01  1.72099327e-01  2.49199526e-02
   9.48162397e-02  1.63372423e-01  2.49199526e-02  9.48162397e-02
   1.63372423e-01  2.49199526e-02  9.48162397e-02  1.63372423e-01
  -1.18349409e-01 -8.91892577e-02  1.55032065e-01 -1.18349409e-01
  -8.91892577e-02  1.55032065e-01 -1.18349409e-01 -8.91892577e-02
   1.55032065e-01  1.75273855e-01 -1.66285827e-01  1.41962832e-01
   1.75273855e-01 -1.66285827e-01  1.41962832e-01  1.63208691e-01
  -8.85101322e-03  1.70456864e-01  0.00000000e+00  0.00000000e+00
   0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00
   0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00
   0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00
   0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00
   0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00]
 [-1.95053898e-02  5.65844327e-01  1.81566244e-02 -1.95053898e-02
   5.65844327e-01  1.81566244e-02 -1.95053898e-02  5.65844327e-01
   1.81566244e-02  4.68600646e-02 -2.75380925e-02  1.80886044e-02
   4.68600646e-02 -2.75380925e-02  1.80886044e-02 -2.77608999e-02
   1.33962809e-02  2.28388407e-02 -2.77608999e-02  1.33962809e-02
   2.28388407e-02 -2.77608999e-02  1.33962809e-02  2.28388407e-02
  -4.27531251e-02  5.39901670e-02  2.69639161e-02 -4.27531251e-02
   5.39901670e-02  2.69639161e-02 -4.27531251e-02  5.39901670e-02
   2.69639161e-02  1.18821115e-04  3.73443700e-02  2.19535409e-02
   1.18821115e-04  3.73443700e-02  2.19535409e-02  2.77671762e-03
   5.68140585e-03  2.59421686e-02  2.77671762e-03  5.68140585e-03
   2.59421686e-02  2.77671762e-03  5.68140585e-03  2.59421686e-02
   1.17946310e-02 -3.06074104e-02  2.45965822e-02  1.17946310e-02
  -3.06074104e-02  2.45965822e-02 -1.09008337e-02 -7.84783817e-03
   2.25468925e-02  0.00000000e+00  0.00000000e+00  0.00000000e+00
   0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00]
 [-6.12357736e-03 -3.03655755e-01  1.22132090e-01 -6.12357736e-03
  -3.03655755e-01  1.22132090e-01 -6.12357736e-03 -3.03655755e-01
   1.22132090e-01 -2.85165088e-01  2.09674820e-01  1.73856532e-01
  -2.85165088e-01  2.09674820e-01  1.73856532e-01 -1.57864865e-01
   1.74028904e-01  1.26737569e-01 -1.57864865e-01  1.74028904e-01
   1.26737569e-01 -1.57864865e-01  1.74028904e-01  1.26737569e-01
  -1.13767758e-03  1.34692080e-01  8.90841344e-02 -1.13767758e-03
   1.34692080e-01  8.90841344e-02 -1.13767758e-03  1.34692080e-01
   8.90841344e-02  7.18151353e-02 -7.08814724e-02  1.15520796e-01
   7.18151353e-02 -7.08814724e-02  1.15520796e-01  6.60732339e-02
   6.95825371e-02  1.38538957e-01  0.00000000e+00  0.00000000e+00
   0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00
   0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00
   0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00
   0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00
   0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00]]

3. Discriminator

3.1 Elementary Circuit Layer

The parameter $W$ is a matrix containg the rotation angles applied to the circuit. It has "num_qubits" rows and three colums.

def layer(num_wires,W):
    for i in range(num_wires):
       qml.Rot(W[i, 0], W[i, 1], W[i, 2], wires=i)
    for i in range(num_wires):
        qml.CNOT(wires=[i, (i+1)%num_wires])

3.2 Quantum node for real data

The first parameter, "real_values", is an array that contains the $n$ input real data points. The normalized input data is encoded in the amplitudes of "num_qubits". The second parameter "weights" is a matrix containing the rotation angles applied to the rotation gates. The matrix has one row per layer and "num_qubits" columns. The wire 0 is the output of the circuit. The output ranges in the continuous interval +1 down to -1, respectively corresponding to qubits $|0>$ and $|1>$ (i.e., $ket 0$ and $ket 1$). Intermediate values represent superpositions of qubits $|0>$ and $|1>$. The optimizer aims at finding rotation angles such that the output of the circuit is approaching $+1$, which corresponds to qubit $|0>$.

@qml.qnode(dev)
def real_disc_circuit(real_values,weights):
    # initialize the input state real data
    qml.QubitStateVector(real_values,wires=range(num_qubits))
    # assign weights to layers
    for W in weights:
        layer(num_qubits,W)
    # measure
    return qml.expval.PauliZ(0)

@qml.qnode(dev)
def gen_disc_circuit(fake_values,disc_weights):
    # initialize with generator data
    qml.QubitStateVector(fake_values, wires=range(num_qubits))
    # discriminator layers
    for W in disc_weights:
        layer(W)
    # measure
    return qml.expval.PauliZ(0)

3.3 Initial circuit values.

The discriminator circuit is initialized with random rotation angles contained in variable "disc_weights".

np.random.seed(0)
num_layers = 2
# discriminator weights
disc_weights = 0.01 * np.random.randn(num_layers, num_qubits, 3)
# test the discriminator circuit
r = real_disc_circuit(real_values[0,:],disc_weights)
assert(r>=-1 and r<=1)
print("Discriminator expectation (test mode): ", r)

Discriminator expectation (test mode):  -0.3186664047343165

3.4 Discriminator Optimization

Probability of correctly classifying real data

The output of the discriminator $r$ is a value in the continuous interval $+1$ down to $-1$, coreesponding to $|0>$ and $|1>$. The output is interpreted as follows. When the output is $+1$, the data is accepted as rrue. When it is -1, the data is rejected and considered fake. The output $r$ is converted to a probability value, in the interval $[0,1]$, using the following conversion:

$p_R=\frac{r+1}{2}.$

This is called the probability of real true. Parameter "values" is an arry of $n$ normalized data points. Parameter "disc_weights" is a matrix of angles used in the discriinator circuit.

def prob_real_true(real_values,disc_weights):
    # get measurement
    r = real_disc_circuit(real_values,disc_weights)
    assert(r>=-1 and r<=1)
    # convert "r" to a probability
    p = (r + 1) / 2
    assert(p>=0 and r<=1)
    return p

Discriminator cost function

The discriminator aims to maximize the probability $p_R$ of accepting true data while minimizing the probability $p_F$ of accepting fake data. During the optimization of the discriminator, the optimizer, being gradient descent, tries to minimize the cost represented by the term $p_F - p_R$.

def disc_cost(real_values,disc_weights):
    cost = - prob_real_true(real_values,disc_weights)
    return cost

Generate initial fake data

# random fake values
fake_values =  0.01 * np.random.randn(2**num_qubits)
# normalized fake values
norm_fake_values = normalize(fake_values)
# check consistency of probability amplitudes
assert np.abs(1-np.sum(norm_fake_values ** 2))<=0.0000001
print("Normalized fake data: ",norm_fake_values)
# test the generator circuit
#r = gen_disc_circuit(norm_fake_values,disc_weights)
#assert(r>=-1 and r<=1)
#print("Generator expectation (test mode): ", r)

Normalized fake data:  [ 0.16451982  0.16078746 -0.05179503 -0.04042524 -0.14021706 -0.18989097
 -0.22816986  0.26086616 -0.0681529  -0.0585812  -0.16752924  0.10396939
 -0.21581743 -0.02844855 -0.11974568  0.05173828 -0.06830708 -0.15787927
 -0.00376865  0.0572784   0.00889497  0.04044786 -0.08482431 -0.04850733
 -0.08992433 -0.04808101 -0.10873745 -0.23084601  0.02372619 -0.05372789
 -0.21799721  0.06188525 -0.12132788  0.00694636  0.09749716  0.01724816
  0.15236561 -0.16512627  0.05380287 -0.09157578 -0.11644678 -0.07740629
 -0.04166216  0.00751067 -0.1558089   0.12046243  0.06227041 -0.20543318
  0.19901555  0.25352654  0.15763153 -0.02406033 -0.14318569  0.14100587
 -0.05391457  0.16347067  0.02785144  0.13060042  0.04765486  0.09448603
  0.00140411  0.23881436  0.01697124  0.05375576]

Discriminator optimization

The outcome of the optimization of the discriminator is a matrix of rotation angles (weights) actualizing the discriminator circuit such that the probability that real data is reconized as tru is maximized.

# create the optimizer
opt = GradientDescentOptimizer(0.1)
for i in range(50):
    j = 0
    disc_weights = opt.step(lambda v: disc_cost(real_values[j,:],v),disc_weights)
    cost = disc_cost(real_values[j,:],disc_weights)
    if i % 5 == 0:
        print("Step {}: cost = {}".format(i+1, cost))
p_R = prob_real_true(real_values[0,:],disc_weights)
assert(p_R>=0 and p_R<=1)
print("Probability of real true: ", p_R)
#p_F = prob_fake_true(norm_fake_values, disc_weights)
#assert(p_F>=0 and p_F<=1)
#print("Probability of fake true: ",p_F)

Step 1: cost = -0.36000213013053106
Step 6: cost = -0.4801209071665075
Step 11: cost = -0.6267257784058955
Step 16: cost = -0.7646215871267537
Step 21: cost = -0.8631753136938661
Step 26: cost = -0.9200803809217414
Step 31: cost = -0.9492141305426495
Step 36: cost = -0.9635297581772699
Step 41: cost = -0.9706756627495225
Step 46: cost = -0.9744598940355289
Probability of real true:  0.9762990269792955

4. Generator

The generator produces fake data with entropy, i.e., with uncertainty.

The fake data can be generated by a qubit-quantum circuit, but it is not usable for perpetrating attacks in the classical data world. Qubit-uantum circuits cannot generate continuous-domain classical data.

The generator is built using photonic quantum computing and a circuit containing only Gaussian operations that can generate data in a continuous domain.

A photonic quantum computing Gaussian device is created:

dev_gaussian = qml.device('default.gaussian', wires=2**num_qubits)

4.1 Construction of Photonic Quantum Node

The input on each wire is the vacuum state $|0>$, i.e., no photon on the wire. The first gate is a displacement gate, with parameter $\alpha$, that phase shift the qumod. The parameter $\alpha$ is a specified in the polar form, as a magnitude (mag_alpha) and as an angle (phase_alpha). This is an active transformation that modifies the photonic energy of the system.

The second gate rotate the qumode by an angle $\phi$. The measured mean number of photons is $< \hat{a}{\dagger} \hat{a}>$, i.e., the average number of photons in the final state.

#@qml.qnode(dev_gaussian)
#def mean_photon_gaussian(params):
#    qml.Displacement(params[0],params[1], wires=0)
#    qml.Rotation(params[2], wires=0)
#    return qml.expval.MeanPhoton(0)

@qml.qnode(dev_gaussian)
def mean_photon_gaussian(params):
    for i in range(2**num_qubits):
        qml.Displacement(params[i][0],params[i][1], wires=i)
        qml.Rotation(params[i][2], wires=i)
    #qml.Displacement(params[0][0],params[0][1], wires=1)
    #qml.Rotation(params[0][2], wires=1)
    return [qml.expval.MeanPhoton(i) for i in range(2**num_qubits)]

Using arbitrary displacement and rotatio verify the generator circuit:

init_params = 0.1*np.ones([2**num_qubits,3],dtype=float)
mean_photon_gaussian(init_params)

array([0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01,
       0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01,
       0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01,
       0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01,
       0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01,
       0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01])

4.2 Generator Cost Function

Using a Gaussian device, displacement and rotation angle pairs (in "params") are applied succesively to the photonic quantum node. The expected numbers of output photons are measured for each case. Measurement results are stored into variable "fake_values". There is one fake value per probability amplitude, i.e., $2^{6}$.

The fake values are normalized and applied to the discriminator circuit, using the rotation angles determined during its optimization.

def gen_cost(params,disc_weights):
    # calculate expected number of photons
    fake_values = mean_photon_gaussian(params)
    # normalize fake values
    norm_fake_values = normalize(fake_values)
    assert np.abs(1-np.sum(norm_fake_values ** 2))<=0.0000001
    # determine the probability of recognizing them as true values
    cost = - prob_real_true(fake_values,disc_weights)
    return cost

Verify cost function:

# generator weights
gen_weights = 0.1*np.ones([2**num_qubits,3],dtype=float)
gen_cost(gen_weights,disc_weights)

-0.4999946815954298

4.3 Generator optimization

Perform gradient descent optimization iterations:

# initialize the optimizer
opt = qml.GradientDescentOptimizer(stepsize=0.1)
# set the number of steps
steps = 20
print("...")
for i in range(steps):
    # update the circuit parameters
    gen_weights = opt.step(lambda v: gen_cost(v,disc_weights),gen_weights)
    cost = gen_cost(gen_weights,disc_weights)
    if i % 5 == 0:
        print("Step {}: cost = {}".format(i+1, cost))

5. Experimental Results

The performance of the photonic-circuit design described in Section III has been validated through simulation on a classical computing platform. Simulations were conducted using an Intel Xeon 32-core 2.70 GHz server, with 256 GB of memory. We generated genuine navigation for a Parrot Mambo MAV. In the scenario, the MAV takes off one meter, does two circles on the horizontal plane, then lands. The navigation data consists of x, y and z velocity triples. The whole scenario generates less than 64 real number values.

The above figure plots the discriminator and generator learning time (ms) versus the number of qubits available. The x axis represents the number of qubits. The left y axis refers to the learning time (ms). The right y axis shows the corresponding probability of real true, for the discriminator, and probability of fake true, for the generator. Hundred optimization iterations were done for each case. Negligible error margins are included, but not visible since they are very tiny. The discriminator is trained with six different genuine navigation data sets. A navigation data set is picked at random at every optimization iteration. The discriminator optimization time grows exponentially. Due to the exponential complexity of the generator circuit (in $\mathcal{O}(2^{n})$), the optimization time also grows exponentially. On our simulation platform, it becomes unpractical from six qubits. The learning time becomes in the order of days. Amplitude encoding has also $\mathcal{O}(2^{n})$ time complexity, but it is only executed once at the start of the optimization process.

6. Conclusion

We have investigated the use of QGAN designs to generate fake MAV (Micro Aerial Vehicle) navigation data. We assume the same approach to discriminate between genuine and fake MAV navigation data. The goal pursued by the adversary is to generate fake data that is accepted as true by a trained discriminator. The discriminator must accept, with high probability, true navigation data. The discriminator must reject, with high probability, fake navigation data. The elaborated quantum circuits have been evaluated running on a classical computing platform. As demonstrated in the experimental results section, the exponentially growing time complexity in the number of qubits is an obstacle to scalability. We identified hurdles that must be overcome by the upcoming evolution of quantum machine learning. The main hurdle for the adversary is the generation of navigation data in classical continuous domains, i.e., real numbers; and the cost of the transformation into the quantum format at every optimization iteration. Further research is needed to improve and find alternatives to the photonic quantum design used in this initial study.

References

If using this code for research purposes, please cite:

M. Barbeau and J. Garcia-Alfaro. Faking and Discriminating the Navigation Data of a Micro Aerial Vehicle Using Quantum Generative Adversarial Networks. IEEE GLOBECOM 2019 Workshop on Quantum Communications and Information Technology 2019 (5th QCIT workshop of the Emerging Technical Committee on Quantum Communications and Information Technology), December 2019.

@inproceedings{barbeau-alfaro2019faking,
  title={Faking and Discriminating the Navigation Data of a Micro Aerial Vehicle Using Quantum Generative Adversarial Networks},
  author={Barbeau, Michel and Garcia-Alfaro, Joaquin},
  booktitle={IEEE GLOBECOM 2019 Workshop on Quantum Communications and Information Technology 2019 (5th QCIT workshop of the Emerging Technical Committee on Quantum Communications and Information Technology)},
  pages={1--6},
  year={2019},
  organization={IEEE}
}