Quantum Generative Adversarial Networks with Cirq + TensorFlow

This demo constructs a Quantum Generative Adversarial Network (QGAN) (Lloyd and Weedbrook (2018), Dallaire-Demers and Killoran (2018)) using two subcircuits, a generator and a discriminator. The generator attempts to generate synthetic quantum data to match a pattern of “real” data, while the discriminator tries to discern real data from fake data (see image below). The gradient of the discriminator’s output provides a training signal for the generator to improve its fake generated data.


../_images/qgan.png

Using Cirq + TensorFlow

PennyLane allows us to mix and match quantum devices and classical machine learning software. For this demo, we will link together Google’s Cirq and TensorFlow libraries.

We begin by importing PennyLane, NumPy, and TensorFlow.

import pennylane as qml
import numpy as np
import tensorflow as tf

We also declare a 3-qubit simulator device running in Cirq.

dev  = qml.device('cirq.simulator', wires=3)

Generator and Discriminator

In classical GANs, the starting point is to draw samples either from some “real data” distribution, or from the generator, and feed them to the discriminator. In this QGAN example, we will use a quantum circuit to generate the real data.

For this simple example, our real data will be a qubit that has been rotated (from the starting state \(\left|0\right\rangle\)) to some arbitrary, but fixed, state.

def real(phi, theta, omega):
    qml.Rot(phi, theta, omega, wires=0)

For the generator and discriminator, we will choose the same basic circuit structure, but acting on different wires.

Both the real data circuit and the generator will output on wire 0, which will be connected as an input to the discriminator. Wire 1 is provided as a workspace for the generator, while the discriminator’s output will be on wire 2.

def generator(w):
    qml.RX(w[0], wires=0)
    qml.RX(w[1], wires=1)
    qml.RY(w[2], wires=0)
    qml.RY(w[3], wires=1)
    qml.RZ(w[4], wires=0)
    qml.RZ(w[5], wires=1)
    qml.CNOT(wires=[0, 1])
    qml.RX(w[6], wires=0)
    qml.RY(w[7], wires=0)
    qml.RZ(w[8], wires=0)


def discriminator(w):
    qml.RX(w[0], wires=0)
    qml.RX(w[1], wires=2)
    qml.RY(w[2], wires=0)
    qml.RY(w[3], wires=2)
    qml.RZ(w[4], wires=0)
    qml.RZ(w[5], wires=2)
    qml.CNOT(wires=[1, 2])
    qml.RX(w[6], wires=2)
    qml.RY(w[7], wires=2)
    qml.RZ(w[8], wires=2)

We create two QNodes. One where the real data source is wired up to the discriminator, and one where the generator is connected to the discriminator. In order to pass TensorFlow Variables into the quantum circuits, we specify the "tf" interface.

@qml.qnode(dev, interface="tf")
def real_disc_circuit(phi, theta, omega, disc_weights):
    real(phi, theta, omega)
    discriminator(disc_weights)
    return qml.expval(qml.PauliZ(2))


@qml.qnode(dev, interface="tf")
def gen_disc_circuit(gen_weights, disc_weights):
    generator(gen_weights)
    discriminator(disc_weights)
    return qml.expval(qml.PauliZ(2))

QGAN cost functions

There are two cost functions of interest, corresponding to the two stages of QGAN training. These cost functions are built from two pieces: the first piece is the probability that the discriminator correctly classifies real data as real. The second piece is the probability that the discriminator classifies fake data (i.e., a state prepared by the generator) as real.

The discriminator is trained to maximize the probability of correctly classifying real data, while minimizing the probability of mistakenly classifying fake data.

The generator is trained to maximize the probability that the discriminator accepts fake data as real.

def prob_real_true(disc_weights):
    true_disc_output = real_disc_circuit(phi, theta, omega, disc_weights)
    # convert to probability
    prob_real_true = (true_disc_output + 1) / 2
    return prob_real_true


def prob_fake_true(gen_weights, disc_weights):
    fake_disc_output = gen_disc_circuit(gen_weights, disc_weights)
    # convert to probability
    prob_fake_true = (fake_disc_output + 1) / 2
    return prob_fake_true


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


def gen_cost(gen_weights):
    return -prob_fake_true(gen_weights, disc_weights)

Training the QGAN

We initialize the fixed angles of the “real data” circuit, as well as the initial parameters for both generator and discriminator. These are chosen so that the generator initially prepares a state on wire 0 that is very close to the \(\left| 1 \right\rangle\) state.

phi = np.pi / 6
theta = np.pi / 2
omega = np.pi / 7
np.random.seed(0)
eps = 1e-2
init_gen_weights = np.array([np.pi] + [0] * 8) + \
                   np.random.normal(scale=eps, size=(9,))
init_disc_weights = np.random.normal(size=(9,))

gen_weights = tf.Variable(init_gen_weights)
disc_weights = tf.Variable(init_disc_weights)

We begin by creating the optimizer:

opt = tf.keras.optimizers.SGD(0.1)

In the first stage of training, we optimize the discriminator while keeping the generator parameters fixed.

cost = lambda: disc_cost(disc_weights)

for step in range(50):
    opt.minimize(cost, disc_weights)
    if step % 5 == 0:
        cost_val = cost().numpy()
        print("Step {}: cost = {}".format(step, cost_val))

Out:

Step 0: cost = -0.10942014679312706
Step 5: cost = -0.3899883683770895
Step 10: cost = -0.6660191221162677
Step 15: cost = -0.8550836374051869
Step 20: cost = -0.9454460255801678
Step 25: cost = -0.9805878275074065
Step 30: cost = -0.9931367967510596
Step 35: cost = -0.9974893236067146
Step 40: cost = -0.9989861474605277
Step 45: cost = -0.9994997430330841

At the discriminator’s optimum, the probability for the discriminator to correctly classify the real data should be close to one.

print("Prob(real classified as real): ", prob_real_true(disc_weights).numpy())

Out:

Prob(real classified as real):  0.9998971883615013

For comparison, we check how the discriminator classifies the generator’s (still unoptimized) fake data:

print("Prob(fake classified as real): ", prob_fake_true(gen_weights, disc_weights).numpy())

Out:

Prob(fake classified as real):  0.00024289608700200915

In the adversarial game we now have to train the generator to better fool the discriminator. For this demo, we only perform one stage of the game. For more complex models, we would continue training the models in an alternating fashion until we reach the optimum point of the two-player adversarial game.

cost = lambda: gen_cost(gen_weights)

for step in range(200):
    opt.minimize(cost, gen_weights)
    if step % 5 == 0:
        cost_val = cost().numpy()
        print("Step {}: cost = {}".format(step, cost_val))

Out:

Step 0: cost = -0.00026636153052095324
Step 5: cost = -0.00042660595499910414
Step 10: cost = -0.0006873353268019855
Step 15: cost = -0.0011113660875707865
Step 20: cost = -0.001800207479391247
Step 25: cost = -0.0029180023120716214
Step 30: cost = -0.004727608757093549
Step 35: cost = -0.007646640995517373
Step 40: cost = -0.012325902469456196
Step 45: cost = -0.019754532724618912
Step 50: cost = -0.03136847913265228
Step 55: cost = -0.04909771494567394
Step 60: cost = -0.07520400732755661
Step 65: cost = -0.11169053986668587
Step 70: cost = -0.15917328000068665
Step 75: cost = -0.21566066145896912
Step 80: cost = -0.27637405693531036
Step 85: cost = -0.3354172706604004
Step 90: cost = -0.38835008442401886
Step 95: cost = -0.4337175488471985
Step 100: cost = -0.4728485941886902
Step 105: cost = -0.5087772309780121
Step 110: cost = -0.5451968759298325
Step 115: cost = -0.585662230849266
Step 120: cost = -0.6327883154153824
Step 125: cost = -0.6872449368238449
Step 130: cost = -0.7468433082103729
Step 135: cost = -0.8066394627094269
Step 140: cost = -0.8607337176799774
Step 145: cost = -0.9048400558531284
Step 150: cost = -0.937667777761817
Step 155: cost = -0.9604097697883844
Step 160: cost = -0.9753705067560077
Step 165: cost = -0.984874417539686
Step 170: cost = -0.9907763195224106
Step 175: cost = -0.9943897356279194
Step 180: cost = -0.9965827631531283
Step 185: cost = -0.9979065986117348
Step 190: cost = -0.9987028733594343
Step 195: cost = -0.9991812075313646

At the optimum of the generator, the probability for the discriminator to be fooled should be close to 1.

print("Prob(fake classified as real): ", prob_fake_true(gen_weights, disc_weights).numpy())

Out:

Prob(fake classified as real):  0.9994220492662862

At the joint optimum the discriminator cost will be close to zero, indicating that the discriminator assigns equal probability to both real and generated data.

print("Discriminator cost: ", disc_cost(disc_weights).numpy())

# The generator has successfully learned how to simulate the real data
# enough to fool the discriminator.

Out:

Discriminator cost:  -0.00047513909521512687

Total running time of the script: ( 0 minutes 31.196 seconds)

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