Quantum state learning

This demonstration works through the process used to produce the state preparation results presented in “Machine learning method for state preparation and gate synthesis on photonic quantum computers”.

This tutorial uses the TensorFlow backend of Strawberry Fields, giving us access to a number of additional functionalities including: GPU integration, automatic gradient computation, built-in optimization algorithms, and other machine learning tools.

Variational quantum circuits

A key element of machine learning is optimization. We can use TensorFlow’s automatic differentiation tools to optimize the parameters of variational quantum circuits constructed using Strawberry Fields. In this approach, we fix a circuit architecture where the states, gates, and/or measurements may have learnable parameters \(\vec{\theta}\) associated with them. We then define a loss function based on the output state of this circuit. In this case, we define a loss function such that the fidelity of the output state of the variational circuit is maximized with respect to some target state.

Note

For more details on the TensorFlow backend in Strawberry Fields, please see Optimization & machine learning.

For arbitrary state preparation using optimization, we need to make use of a quantum circuit with a layer structure that is universal - that is, by ‘stacking’ the layers, we can guarantee that we can produce any CV state with at-most polynomial overhead. Therefore, the architecture we choose must consist of layers with each layer containing parameterized Gaussian and non-Gaussian gates. The non-Gaussian gates provide both the nonlinearity and the universality of the model. To this end, we employ the CV quantum neural network architecture as described in Killoran et al.:

layer

Here,

  • \(\mathcal{U}_i(\theta_i,\phi_i)\) is an N-mode linear optical interferometer composed of two-mode beamsplitters \(BS(\theta,\phi)\) and single-mode rotation gates \(R(\phi)=e^{i\phi\hat{n}}\),

  • \(\mathcal{D}(\alpha_i)\) are single mode displacements in the phase space by complex value \(\alpha_i\),

  • \(\mathcal{S}(r_i, \phi_i)\) are single mode squeezing operations of magnitude \(r_i\) and phase \(\phi_i\), and

  • \(\Phi(\lambda_i)\) is a single mode non-Gaussian operation, in this case chosen to be the Kerr interaction \(\mathcal{K}(\kappa_i)=e^{i\kappa_i\hat{n}^2}\) of strength \(\kappa_i\).

Hyperparameters

First, we must define the hyperparameters of our layer structure:

  • cutoff: the simulation Fock space truncation we will use in the optimization. The TensorFlow backend will perform numerical operations in this truncated Fock space when performing the optimization.

  • depth: The number of layers in our variational quantum circuit. As a general rule, increasing the number of layers (and thus, the number of parameters we are optimizing over) increases the optimizer’s chance of finding a reasonable local minimum in the optimization landscape.

  • reps: the number of steps in the optimization routine performing gradient descent

Some other optional hyperparameters include:

  • The standard deviation of initial parameters. Note that we make a distinction between the standard deviation of passive parameters (those that preserve photon number when changed, such as phase parameters), and active parameters (those that introduce or remove energy from the system when changed).

import numpy as np

import strawberryfields as sf
from strawberryfields.ops import *
from strawberryfields.utils import operation

# Cutoff dimension
cutoff = 9

# Number of layers
depth = 15

# Number of steps in optimization routine performing gradient descent
reps = 200

# Learning rate
lr = 0.05

# Standard deviation of initial parameters
passive_sd = 0.1
active_sd = 0.001

The layer parameters \(\vec{\theta}\)

We use TensorFlow to create the variables corresponding to the gate parameters. Note that we focus on a single mode circuit where each variable has shape (depth,), with each individual element representing the gate parameter in layer \(i\).

import tensorflow as tf

# set the random seed
tf.random.set_seed(42)

# squeeze gate
sq_r = tf.random.normal(shape=[depth], stddev=active_sd)
sq_phi = tf.random.normal(shape=[depth], stddev=passive_sd)

# displacement gate
d_r = tf.random.normal(shape=[depth], stddev=active_sd)
d_phi = tf.random.normal(shape=[depth], stddev=passive_sd)

# rotation gates
r1 = tf.random.normal(shape=[depth], stddev=passive_sd)
r2 = tf.random.normal(shape=[depth], stddev=passive_sd)

# kerr gate
kappa = tf.random.normal(shape=[depth], stddev=active_sd)

For convenience, we store the TensorFlow variables representing the weights as a tensor:

weights = tf.convert_to_tensor([r1, sq_r, sq_phi, r2, d_r, d_phi, kappa])
weights = tf.Variable(tf.transpose(weights))

Since we have a depth of 15 (so 15 layers), and each layer takes 7 different types of parameters, the final shape of our weights array should be \(\text{depth}\times 7\) or (15, 7):

print(weights.shape)

Out:

(15, 7)

Constructing the circuit

We can now construct the corresponding single-mode Strawberry Fields program:

# Single-mode Strawberry Fields program
prog = sf.Program(1)

# Create the 7 Strawberry Fields free parameters for each layer
sf_params = []
names = ["r1", "sq_r", "sq_phi", "r2", "d_r", "d_phi", "kappa"]

for i in range(depth):
    # For the ith layer, generate parameter names "r1_i", "sq_r_i", etc.
    sf_params_names = ["{}_{}".format(n, i) for n in names]
    # Create the parameters, and append them to our list ``sf_params``.
    sf_params.append(prog.params(*sf_params_names))

sf_params is now a nested list of shape (depth, 7), matching the shape of weights.

sf_params = np.array(sf_params)
print(sf_params.shape)

Out:

(15, 7)

Now, we can create a function to define the \(i\)th layer, acting on qumode q. We add the operation decorator so that the layer can be used as a single operation when constructing our circuit within the usual Strawberry Fields Program context

# layer architecture
@operation(1)
def layer(i, q):
    Rgate(sf_params[i][0]) | q
    Sgate(sf_params[i][1], sf_params[i][2]) | q
    Rgate(sf_params[i][3]) | q
    Dgate(sf_params[i][4], sf_params[i][5]) | q
    Kgate(sf_params[i][6]) | q
    return q

Now that we have defined our gate parameters and our layer structure, we can construct our variational quantum circuit.

# Apply circuit of layers with corresponding depth
with prog.context as q:
    for k in range(depth):
        layer(k) | q[0]

Performing the optimization

\(\newcommand{ket}[1]{\left|#1\right\rangle}\) With the Strawberry Fields TensorFlow backend calculating the resulting state of the circuit symbolically, we can use TensorFlow to optimize the gate parameters to minimize the cost function we specify. With state learning, the measure of distance between two quantum states is given by the fidelity of the output state \(\ket{\psi}\) with some target state \(\ket{\psi_t}\). This is defined as the overlap between the two states:

\[F = \left|\left\langle{\psi}\mid{\psi_t}\right\rangle\right|^2\]

where the output state can be written \(\ket{\psi}=U(\vec{\theta})\ket{\psi_0}\), with \(U(\vec{\theta})\) the unitary operation applied by the variational quantum circuit, and \(\ket{\psi_0}=\ket{0}\) the initial state.

Let’s first instantiate the TensorFlow backend, making sure to pass the Fock basis truncation cutoff.

eng = sf.Engine("tf", backend_options={"cutoff_dim": cutoff})

Now let’s define the target state as the single photon state \(\ket{\psi_t}=\ket{1}\):

import numpy as np

target_state = np.zeros([cutoff])
target_state[1] = 1
print(target_state)

Out:

[0. 1. 0. 0. 0. 0. 0. 0. 0.]

Using this target state, we calculate the fidelity with the state exiting the variational circuit. We must use TensorFlow functions to manipulate this data, as well as a GradientTape to keep track of the corresponding gradients!

We choose the following cost function:

\[C(\vec{\theta}) = \left| \langle \psi_t \mid U(\vec{\theta})\mid 0\rangle - 1\right|\]

By minimizing this cost function, the variational quantum circuit will prepare a state with high fidelity to the target state.

def cost(weights):
    # Create a dictionary mapping from the names of the Strawberry Fields
    # free parameters to the TensorFlow weight values.
    mapping = {p.name: w for p, w in zip(sf_params.flatten(), tf.reshape(weights, [-1]))}

    # Run engine
    state = eng.run(prog, args=mapping).state

    # Extract the statevector
    ket = state.ket()

    # Compute the fidelity between the output statevector
    # and the target state.
    fidelity = tf.abs(tf.reduce_sum(tf.math.conj(ket) * target_state)) ** 2

    # Objective function to minimize
    cost = tf.abs(tf.reduce_sum(tf.math.conj(ket) * target_state) - 1)
    return cost, fidelity, ket

Now that the cost function is defined, we can define and run the optimization. Below, we choose the Adam optimizer that is built into TensorFlow:

opt = tf.keras.optimizers.Adam(learning_rate=lr)

We then loop over all repetitions, storing the best predicted fidelity value.

fid_progress = []
best_fid = 0

for i in range(reps):
    # reset the engine if it has already been executed
    if eng.run_progs:
        eng.reset()

    with tf.GradientTape() as tape:
        loss, fid, ket = cost(weights)

    # Stores fidelity at each step
    fid_progress.append(fid.numpy())

    if fid > best_fid:
        # store the new best fidelity and best state
        best_fid = fid.numpy()
        learnt_state = ket.numpy()

    # one repetition of the optimization
    gradients = tape.gradient(loss, weights)
    opt.apply_gradients(zip([gradients], [weights]))

    # Prints progress at every rep
    if i % 1 == 0:
        print("Rep: {} Cost: {:.4f} Fidelity: {:.4f}".format(i, loss, fid))

Out:

Rep: 0 Cost: 0.9973 Fidelity: 0.0000
Rep: 1 Cost: 0.3459 Fidelity: 0.4297
Rep: 2 Cost: 0.5866 Fidelity: 0.2695
Rep: 3 Cost: 0.4118 Fidelity: 0.4013
Rep: 4 Cost: 0.5630 Fidelity: 0.1953
Rep: 5 Cost: 0.4099 Fidelity: 0.4548
Rep: 6 Cost: 0.2258 Fidelity: 0.6989
Rep: 7 Cost: 0.3994 Fidelity: 0.5251
Rep: 8 Cost: 0.1787 Fidelity: 0.7421
Rep: 9 Cost: 0.3777 Fidelity: 0.5672
Rep: 10 Cost: 0.2201 Fidelity: 0.6140
Rep: 11 Cost: 0.3580 Fidelity: 0.6169
Rep: 12 Cost: 0.3944 Fidelity: 0.5549
Rep: 13 Cost: 0.3197 Fidelity: 0.5456
Rep: 14 Cost: 0.1766 Fidelity: 0.6878
Rep: 15 Cost: 0.1305 Fidelity: 0.7586
Rep: 16 Cost: 0.1304 Fidelity: 0.7598
Rep: 17 Cost: 0.1256 Fidelity: 0.7899
Rep: 18 Cost: 0.2366 Fidelity: 0.8744
Rep: 19 Cost: 0.1744 Fidelity: 0.7789
Rep: 20 Cost: 0.1093 Fidelity: 0.7965
Rep: 21 Cost: 0.1846 Fidelity: 0.8335
Rep: 22 Cost: 0.0876 Fidelity: 0.8396
Rep: 23 Cost: 0.0985 Fidelity: 0.8630
Rep: 24 Cost: 0.1787 Fidelity: 0.9070
Rep: 25 Cost: 0.0620 Fidelity: 0.9116
Rep: 26 Cost: 0.2743 Fidelity: 0.8738
Rep: 27 Cost: 0.2477 Fidelity: 0.8895
Rep: 28 Cost: 0.0815 Fidelity: 0.8494
Rep: 29 Cost: 0.1855 Fidelity: 0.8072
Rep: 30 Cost: 0.1315 Fidelity: 0.8200
Rep: 31 Cost: 0.1403 Fidelity: 0.8799
Rep: 32 Cost: 0.1530 Fidelity: 0.8853
Rep: 33 Cost: 0.0718 Fidelity: 0.8679
Rep: 34 Cost: 0.1112 Fidelity: 0.8838
Rep: 35 Cost: 0.0394 Fidelity: 0.9237
Rep: 36 Cost: 0.0781 Fidelity: 0.9487
Rep: 37 Cost: 0.0619 Fidelity: 0.9613
Rep: 38 Cost: 0.0291 Fidelity: 0.9607
Rep: 39 Cost: 0.0669 Fidelity: 0.9595
Rep: 40 Cost: 0.0685 Fidelity: 0.9458
Rep: 41 Cost: 0.0317 Fidelity: 0.9466
Rep: 42 Cost: 0.0308 Fidelity: 0.9484
Rep: 43 Cost: 0.0729 Fidelity: 0.9612
Rep: 44 Cost: 0.0581 Fidelity: 0.9658
Rep: 45 Cost: 0.0272 Fidelity: 0.9766
Rep: 46 Cost: 0.0818 Fidelity: 0.9760
Rep: 47 Cost: 0.0123 Fidelity: 0.9828
Rep: 48 Cost: 0.0431 Fidelity: 0.9826
Rep: 49 Cost: 0.0866 Fidelity: 0.9775
Rep: 50 Cost: 0.0245 Fidelity: 0.9779
Rep: 51 Cost: 0.1784 Fidelity: 0.9657
Rep: 52 Cost: 0.2022 Fidelity: 0.9552
Rep: 53 Cost: 0.0907 Fidelity: 0.9511
Rep: 54 Cost: 0.1477 Fidelity: 0.9100
Rep: 55 Cost: 0.2128 Fidelity: 0.8746
Rep: 56 Cost: 0.1493 Fidelity: 0.8677
Rep: 57 Cost: 0.0704 Fidelity: 0.8736
Rep: 58 Cost: 0.1368 Fidelity: 0.8962
Rep: 59 Cost: 0.1268 Fidelity: 0.9239
Rep: 60 Cost: 0.0222 Fidelity: 0.9566
Rep: 61 Cost: 0.1432 Fidelity: 0.9641
Rep: 62 Cost: 0.1233 Fidelity: 0.9619
Rep: 63 Cost: 0.0487 Fidelity: 0.9633
Rep: 64 Cost: 0.0689 Fidelity: 0.9604
Rep: 65 Cost: 0.0488 Fidelity: 0.9584
Rep: 66 Cost: 0.0248 Fidelity: 0.9618
Rep: 67 Cost: 0.0967 Fidelity: 0.9660
Rep: 68 Cost: 0.0678 Fidelity: 0.9731
Rep: 69 Cost: 0.0859 Fidelity: 0.9768
Rep: 70 Cost: 0.0904 Fidelity: 0.9787
Rep: 71 Cost: 0.0312 Fidelity: 0.9789
Rep: 72 Cost: 0.0258 Fidelity: 0.9757
Rep: 73 Cost: 0.0826 Fidelity: 0.9704
Rep: 74 Cost: 0.0661 Fidelity: 0.9667
Rep: 75 Cost: 0.0554 Fidelity: 0.9651
Rep: 76 Cost: 0.0626 Fidelity: 0.9602
Rep: 77 Cost: 0.0358 Fidelity: 0.9513
Rep: 78 Cost: 0.0366 Fidelity: 0.9570
Rep: 79 Cost: 0.0524 Fidelity: 0.9734
Rep: 80 Cost: 0.0279 Fidelity: 0.9798
Rep: 81 Cost: 0.0962 Fidelity: 0.9768
Rep: 82 Cost: 0.0980 Fidelity: 0.9802
Rep: 83 Cost: 0.0127 Fidelity: 0.9884
Rep: 84 Cost: 0.0134 Fidelity: 0.9893
Rep: 85 Cost: 0.0874 Fidelity: 0.9864
Rep: 86 Cost: 0.0666 Fidelity: 0.9883
Rep: 87 Cost: 0.0601 Fidelity: 0.9885
Rep: 88 Cost: 0.0661 Fidelity: 0.9859
Rep: 89 Cost: 0.0317 Fidelity: 0.9830
Rep: 90 Cost: 0.0222 Fidelity: 0.9796
Rep: 91 Cost: 0.0763 Fidelity: 0.9769
Rep: 92 Cost: 0.0665 Fidelity: 0.9742
Rep: 93 Cost: 0.0377 Fidelity: 0.9702
Rep: 94 Cost: 0.0428 Fidelity: 0.9685
Rep: 95 Cost: 0.0415 Fidelity: 0.9703
Rep: 96 Cost: 0.0291 Fidelity: 0.9729
Rep: 97 Cost: 0.0673 Fidelity: 0.9749
Rep: 98 Cost: 0.0606 Fidelity: 0.9775
Rep: 99 Cost: 0.0385 Fidelity: 0.9815
Rep: 100 Cost: 0.0360 Fidelity: 0.9827
Rep: 101 Cost: 0.0580 Fidelity: 0.9801
Rep: 102 Cost: 0.0494 Fidelity: 0.9804
Rep: 103 Cost: 0.0504 Fidelity: 0.9832
Rep: 104 Cost: 0.0482 Fidelity: 0.9822
Rep: 105 Cost: 0.0444 Fidelity: 0.9772
Rep: 106 Cost: 0.0391 Fidelity: 0.9761
Rep: 107 Cost: 0.0526 Fidelity: 0.9784
Rep: 108 Cost: 0.0471 Fidelity: 0.9771
Rep: 109 Cost: 0.0444 Fidelity: 0.9726
Rep: 110 Cost: 0.0421 Fidelity: 0.9725
Rep: 111 Cost: 0.0441 Fidelity: 0.9755
Rep: 112 Cost: 0.0373 Fidelity: 0.9763
Rep: 113 Cost: 0.0525 Fidelity: 0.9757
Rep: 114 Cost: 0.0477 Fidelity: 0.9771
Rep: 115 Cost: 0.0422 Fidelity: 0.9794
Rep: 116 Cost: 0.0381 Fidelity: 0.9802
Rep: 117 Cost: 0.0503 Fidelity: 0.9797
Rep: 118 Cost: 0.0440 Fidelity: 0.9801
Rep: 119 Cost: 0.0470 Fidelity: 0.9811
Rep: 120 Cost: 0.0438 Fidelity: 0.9809
Rep: 121 Cost: 0.0436 Fidelity: 0.9789
Rep: 122 Cost: 0.0386 Fidelity: 0.9785
Rep: 123 Cost: 0.0489 Fidelity: 0.9797
Rep: 124 Cost: 0.0441 Fidelity: 0.9793
Rep: 125 Cost: 0.0430 Fidelity: 0.9768
Rep: 126 Cost: 0.0396 Fidelity: 0.9767
Rep: 127 Cost: 0.0449 Fidelity: 0.9789
Rep: 128 Cost: 0.0391 Fidelity: 0.9793
Rep: 129 Cost: 0.0474 Fidelity: 0.9774
Rep: 130 Cost: 0.0434 Fidelity: 0.9778
Rep: 131 Cost: 0.0418 Fidelity: 0.9802
Rep: 132 Cost: 0.0374 Fidelity: 0.9804
Rep: 133 Cost: 0.0475 Fidelity: 0.9785
Rep: 134 Cost: 0.0423 Fidelity: 0.9789
Rep: 135 Cost: 0.0435 Fidelity: 0.9808
Rep: 136 Cost: 0.0399 Fidelity: 0.9806
Rep: 137 Cost: 0.0438 Fidelity: 0.9784
Rep: 138 Cost: 0.0390 Fidelity: 0.9784
Rep: 139 Cost: 0.0452 Fidelity: 0.9802
Rep: 140 Cost: 0.0408 Fidelity: 0.9800
Rep: 141 Cost: 0.0428 Fidelity: 0.9780
Rep: 142 Cost: 0.0389 Fidelity: 0.9781
Rep: 143 Cost: 0.0436 Fidelity: 0.9800
Rep: 144 Cost: 0.0386 Fidelity: 0.9802
Rep: 145 Cost: 0.0448 Fidelity: 0.9785
Rep: 146 Cost: 0.0408 Fidelity: 0.9788
Rep: 147 Cost: 0.0417 Fidelity: 0.9807
Rep: 148 Cost: 0.0373 Fidelity: 0.9808
Rep: 149 Cost: 0.0452 Fidelity: 0.9791
Rep: 150 Cost: 0.0406 Fidelity: 0.9793
Rep: 151 Cost: 0.0421 Fidelity: 0.9810
Rep: 152 Cost: 0.0381 Fidelity: 0.9810
Rep: 153 Cost: 0.0436 Fidelity: 0.9791
Rep: 154 Cost: 0.0391 Fidelity: 0.9793
Rep: 155 Cost: 0.0429 Fidelity: 0.9810
Rep: 156 Cost: 0.0386 Fidelity: 0.9809
Rep: 157 Cost: 0.0429 Fidelity: 0.9792
Rep: 158 Cost: 0.0387 Fidelity: 0.9794
Rep: 159 Cost: 0.0423 Fidelity: 0.9810
Rep: 160 Cost: 0.0378 Fidelity: 0.9811
Rep: 161 Cost: 0.0435 Fidelity: 0.9795
Rep: 162 Cost: 0.0394 Fidelity: 0.9797
Rep: 163 Cost: 0.0413 Fidelity: 0.9813
Rep: 164 Cost: 0.0370 Fidelity: 0.9814
Rep: 165 Cost: 0.0438 Fidelity: 0.9798
Rep: 166 Cost: 0.0394 Fidelity: 0.9800
Rep: 167 Cost: 0.0412 Fidelity: 0.9815
Rep: 168 Cost: 0.0371 Fidelity: 0.9814
Rep: 169 Cost: 0.0430 Fidelity: 0.9799
Rep: 170 Cost: 0.0386 Fidelity: 0.9801
Rep: 171 Cost: 0.0417 Fidelity: 0.9815
Rep: 172 Cost: 0.0376 Fidelity: 0.9815
Rep: 173 Cost: 0.0422 Fidelity: 0.9801
Rep: 174 Cost: 0.0380 Fidelity: 0.9803
Rep: 175 Cost: 0.0417 Fidelity: 0.9816
Rep: 176 Cost: 0.0375 Fidelity: 0.9816
Rep: 177 Cost: 0.0421 Fidelity: 0.9804
Rep: 178 Cost: 0.0380 Fidelity: 0.9806
Rep: 179 Cost: 0.0414 Fidelity: 0.9817
Rep: 180 Cost: 0.0371 Fidelity: 0.9818
Rep: 181 Cost: 0.0421 Fidelity: 0.9807
Rep: 182 Cost: 0.0379 Fidelity: 0.9809
Rep: 183 Cost: 0.0412 Fidelity: 0.9818
Rep: 184 Cost: 0.0371 Fidelity: 0.9818
Rep: 185 Cost: 0.0417 Fidelity: 0.9808
Rep: 186 Cost: 0.0375 Fidelity: 0.9810
Rep: 187 Cost: 0.0413 Fidelity: 0.9819
Rep: 188 Cost: 0.0372 Fidelity: 0.9819
Rep: 189 Cost: 0.0413 Fidelity: 0.9810
Rep: 190 Cost: 0.0371 Fidelity: 0.9812
Rep: 191 Cost: 0.0414 Fidelity: 0.9820
Rep: 192 Cost: 0.0373 Fidelity: 0.9820
Rep: 193 Cost: 0.0410 Fidelity: 0.9813
Rep: 194 Cost: 0.0368 Fidelity: 0.9815
Rep: 195 Cost: 0.0413 Fidelity: 0.9821
Rep: 196 Cost: 0.0372 Fidelity: 0.9821
Rep: 197 Cost: 0.0408 Fidelity: 0.9815
Rep: 198 Cost: 0.0367 Fidelity: 0.9817
Rep: 199 Cost: 0.0412 Fidelity: 0.9821

Results and visualisation

Plotting the fidelity vs. optimization step:

from matplotlib import pyplot as plt

plt.rcParams["font.family"] = "serif"
plt.rcParams["font.sans-serif"] = ["Computer Modern Roman"]
plt.style.use("default")

plt.plot(fid_progress)
plt.ylabel("Fidelity")
plt.xlabel("Step")
../_images/sphx_glr_run_state_learner_001.png

We can use the following function to plot the Wigner function of our target and learnt state:

import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D


def wigner(rho):
    """This code is a modified version of the 'iterative' method
    of the wigner function provided in QuTiP, which is released
    under the BSD license, with the following copyright notice:

    Copyright (C) 2011 and later, P.D. Nation, J.R. Johansson,
    A.J.G. Pitchford, C. Granade, and A.L. Grimsmo.

    All rights reserved."""
    import copy

    # Domain parameter for Wigner function plots
    l = 5.0
    cutoff = rho.shape[0]

    # Creates 2D grid for Wigner function plots
    x = np.linspace(-l, l, 100)
    p = np.linspace(-l, l, 100)

    Q, P = np.meshgrid(x, p)
    A = (Q + P * 1.0j) / (2 * np.sqrt(2 / 2))

    Wlist = np.array([np.zeros(np.shape(A), dtype=complex) for k in range(cutoff)])

    # Wigner function for |0><0|
    Wlist[0] = np.exp(-2.0 * np.abs(A) ** 2) / np.pi

    # W = rho(0,0)W(|0><0|)
    W = np.real(rho[0, 0]) * np.real(Wlist[0])

    for n in range(1, cutoff):
        Wlist[n] = (2.0 * A * Wlist[n - 1]) / np.sqrt(n)
        W += 2 * np.real(rho[0, n] * Wlist[n])

    for m in range(1, cutoff):
        temp = copy.copy(Wlist[m])
        # Wlist[m] = Wigner function for |m><m|
        Wlist[m] = (2 * np.conj(A) * temp - np.sqrt(m) * Wlist[m - 1]) / np.sqrt(m)

        # W += rho(m,m)W(|m><m|)
        W += np.real(rho[m, m] * Wlist[m])

        for n in range(m + 1, cutoff):
            temp2 = (2 * A * Wlist[n - 1] - np.sqrt(m) * temp) / np.sqrt(n)
            temp = copy.copy(Wlist[n])
            # Wlist[n] = Wigner function for |m><n|
            Wlist[n] = temp2

            # W += rho(m,n)W(|m><n|) + rho(n,m)W(|n><m|)
            W += 2 * np.real(rho[m, n] * Wlist[n])

    return Q, P, W / 2

Computing the density matrices \(\rho = \left|\psi\right\rangle \left\langle\psi\right|\) of the target and learnt state,

rho_target = np.outer(target_state, target_state.conj())
rho_learnt = np.outer(learnt_state, learnt_state.conj())

Plotting the Wigner function of the target state:

fig = plt.figure()
ax = fig.add_subplot(111, projection="3d")
X, P, W = wigner(rho_target)
ax.plot_surface(X, P, W, cmap="RdYlGn", lw=0.5, rstride=1, cstride=1)
ax.contour(X, P, W, 10, cmap="RdYlGn", linestyles="solid", offset=-0.17)
ax.set_axis_off()
fig.show()
../_images/sphx_glr_run_state_learner_002.png

Plotting the Wigner function of the learnt state:

fig = plt.figure()
ax = fig.add_subplot(111, projection="3d")
X, P, W = wigner(rho_learnt)
ax.plot_surface(X, P, W, cmap="RdYlGn", lw=0.5, rstride=1, cstride=1)
ax.contour(X, P, W, 10, cmap="RdYlGn", linestyles="solid", offset=-0.17)
ax.set_axis_off()
fig.show()
../_images/sphx_glr_run_state_learner_003.png

References

  1. Juan Miguel Arrazola, Thomas R. Bromley, Josh Izaac, Casey R. Myers, Kamil Brádler, and Nathan Killoran. Machine learning method for state preparation and gate synthesis on photonic quantum computers. Quantum Science and Technology, 4 024004, (2019).

  2. Nathan Killoran, Thomas R. Bromley, Juan Miguel Arrazola, Maria Schuld, Nicolas Quesada, and Seth Lloyd. Continuous-variable quantum neural networks. Physical Review Research, 1(3), 033063., (2019).

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