Note

Click here to download the full example code

# 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 with TensorFlow.

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.:

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

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:

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.

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:

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: 0.1847 Fidelity: 0.8335
Rep: 22 Cost: 0.0875 Fidelity: 0.8396
Rep: 23 Cost: 0.0989 Fidelity: 0.8629
Rep: 24 Cost: 0.1784 Fidelity: 0.9070
Rep: 25 Cost: 0.0620 Fidelity: 0.9116
Rep: 26 Cost: 0.2740 Fidelity: 0.8740
Rep: 27 Cost: 0.2471 Fidelity: 0.8896
Rep: 28 Cost: 0.0817 Fidelity: 0.8494
Rep: 29 Cost: 0.1848 Fidelity: 0.8073
Rep: 30 Cost: 0.1303 Fidelity: 0.8201
Rep: 31 Cost: 0.1411 Fidelity: 0.8797
Rep: 32 Cost: 0.1534 Fidelity: 0.8854
Rep: 33 Cost: 0.0715 Fidelity: 0.8683
Rep: 34 Cost: 0.1118 Fidelity: 0.8842
Rep: 35 Cost: 0.0392 Fidelity: 0.9238
Rep: 36 Cost: 0.0819 Fidelity: 0.9486
Rep: 37 Cost: 0.0544 Fidelity: 0.9613
Rep: 38 Cost: 0.0351 Fidelity: 0.9603
Rep: 39 Cost: 0.0774 Fidelity: 0.9587
Rep: 40 Cost: 0.0470 Fidelity: 0.9463
Rep: 41 Cost: 0.0431 Fidelity: 0.9470
Rep: 42 Cost: 0.0677 Fidelity: 0.9427
Rep: 43 Cost: 0.0457 Fidelity: 0.9546
Rep: 44 Cost: 0.0496 Fidelity: 0.9588
Rep: 45 Cost: 0.0614 Fidelity: 0.9704
Rep: 46 Cost: 0.0402 Fidelity: 0.9745
Rep: 47 Cost: 0.0528 Fidelity: 0.9796
Rep: 48 Cost: 0.0496 Fidelity: 0.9800
Rep: 49 Cost: 0.0327 Fidelity: 0.9795
Rep: 50 Cost: 0.0602 Fidelity: 0.9776
Rep: 51 Cost: 0.0217 Fidelity: 0.9736
Rep: 52 Cost: 0.0438 Fidelity: 0.9743
Rep: 53 Cost: 0.0534 Fidelity: 0.9692
Rep: 54 Cost: 0.0204 Fidelity: 0.9733
Rep: 55 Cost: 0.0386 Fidelity: 0.9735
Rep: 56 Cost: 0.0623 Fidelity: 0.9805
Rep: 57 Cost: 0.0103 Fidelity: 0.9814
Rep: 58 Cost: 0.1102 Fidelity: 0.9766
Rep: 59 Cost: 0.0534 Fidelity: 0.9843
Rep: 60 Cost: 0.1465 Fidelity: 0.9795
Rep: 61 Cost: 0.1809 Fidelity: 0.9706
Rep: 62 Cost: 0.0830 Fidelity: 0.9746
Rep: 63 Cost: 0.1393 Fidelity: 0.9650
Rep: 64 Cost: 0.1944 Fidelity: 0.9472
Rep: 65 Cost: 0.1117 Fidelity: 0.9304
Rep: 66 Cost: 0.0847 Fidelity: 0.8987
Rep: 67 Cost: 0.1435 Fidelity: 0.8823
Rep: 68 Cost: 0.1213 Fidelity: 0.8799
Rep: 69 Cost: 0.0597 Fidelity: 0.8848
Rep: 70 Cost: 0.0998 Fidelity: 0.9123
Rep: 71 Cost: 0.0864 Fidelity: 0.9475
Rep: 72 Cost: 0.0409 Fidelity: 0.9743
Rep: 73 Cost: 0.0403 Fidelity: 0.9758
Rep: 74 Cost: 0.0861 Fidelity: 0.9672
Rep: 75 Cost: 0.0686 Fidelity: 0.9614
Rep: 76 Cost: 0.0842 Fidelity: 0.9496
Rep: 77 Cost: 0.0896 Fidelity: 0.9457
Rep: 78 Cost: 0.0403 Fidelity: 0.9574
Rep: 79 Cost: 0.0358 Fidelity: 0.9667
Rep: 80 Cost: 0.0818 Fidelity: 0.9668
Rep: 81 Cost: 0.0613 Fidelity: 0.9766
Rep: 82 Cost: 0.0713 Fidelity: 0.9885
Rep: 83 Cost: 0.0723 Fidelity: 0.9858
Rep: 84 Cost: 0.0355 Fidelity: 0.9737
Rep: 85 Cost: 0.0340 Fidelity: 0.9656
Rep: 86 Cost: 0.0604 Fidelity: 0.9607
Rep: 87 Cost: 0.0492 Fidelity: 0.9572
Rep: 88 Cost: 0.0551 Fidelity: 0.9607
Rep: 89 Cost: 0.0542 Fidelity: 0.9647
Rep: 90 Cost: 0.0412 Fidelity: 0.9657
Rep: 91 Cost: 0.0335 Fidelity: 0.9725
Rep: 92 Cost: 0.0661 Fidelity: 0.9821
Rep: 93 Cost: 0.0557 Fidelity: 0.9849
Rep: 94 Cost: 0.0536 Fidelity: 0.9834
Rep: 95 Cost: 0.0516 Fidelity: 0.9841
Rep: 96 Cost: 0.0498 Fidelity: 0.9841
Rep: 97 Cost: 0.0430 Fidelity: 0.9824
Rep: 98 Cost: 0.0591 Fidelity: 0.9819
Rep: 99 Cost: 0.0536 Fidelity: 0.9803
Rep: 100 Cost: 0.0459 Fidelity: 0.9739
Rep: 101 Cost: 0.0448 Fidelity: 0.9713
Rep: 102 Cost: 0.0447 Fidelity: 0.9742
Rep: 103 Cost: 0.0374 Fidelity: 0.9734
Rep: 104 Cost: 0.0545 Fidelity: 0.9693
Rep: 105 Cost: 0.0503 Fidelity: 0.9708
Rep: 106 Cost: 0.0397 Fidelity: 0.9755
Rep: 107 Cost: 0.0354 Fidelity: 0.9768
Rep: 108 Cost: 0.0535 Fidelity: 0.9767
Rep: 109 Cost: 0.0462 Fidelity: 0.9790
Rep: 110 Cost: 0.0477 Fidelity: 0.9808
Rep: 111 Cost: 0.0438 Fidelity: 0.9812
Rep: 112 Cost: 0.0479 Fidelity: 0.9813
Rep: 113 Cost: 0.0431 Fidelity: 0.9815
Rep: 114 Cost: 0.0482 Fidelity: 0.9811
Rep: 115 Cost: 0.0426 Fidelity: 0.9809
Rep: 116 Cost: 0.0491 Fidelity: 0.9801
Rep: 117 Cost: 0.0459 Fidelity: 0.9793
Rep: 118 Cost: 0.0416 Fidelity: 0.9794
Rep: 119 Cost: 0.0362 Fidelity: 0.9790
Rep: 120 Cost: 0.0516 Fidelity: 0.9773
Rep: 121 Cost: 0.0473 Fidelity: 0.9770
Rep: 122 Cost: 0.0395 Fidelity: 0.9785
Rep: 123 Cost: 0.0356 Fidelity: 0.9787
Rep: 124 Cost: 0.0494 Fidelity: 0.9772
Rep: 125 Cost: 0.0436 Fidelity: 0.9779
Rep: 126 Cost: 0.0437 Fidelity: 0.9802
Rep: 127 Cost: 0.0397 Fidelity: 0.9806
Rep: 128 Cost: 0.0464 Fidelity: 0.9793
Rep: 129 Cost: 0.0418 Fidelity: 0.9797
Rep: 130 Cost: 0.0441 Fidelity: 0.9810
Rep: 131 Cost: 0.0390 Fidelity: 0.9810
Rep: 132 Cost: 0.0472 Fidelity: 0.9801
Rep: 133 Cost: 0.0434 Fidelity: 0.9800
Rep: 134 Cost: 0.0411 Fidelity: 0.9801
Rep: 135 Cost: 0.0365 Fidelity: 0.9800
Rep: 136 Cost: 0.0478 Fidelity: 0.9796
Rep: 137 Cost: 0.0432 Fidelity: 0.9796
Rep: 138 Cost: 0.0411 Fidelity: 0.9796
Rep: 139 Cost: 0.0371 Fidelity: 0.9797
Rep: 140 Cost: 0.0459 Fidelity: 0.9798
Rep: 141 Cost: 0.0409 Fidelity: 0.9801
Rep: 142 Cost: 0.0432 Fidelity: 0.9804
Rep: 143 Cost: 0.0389 Fidelity: 0.9807
Rep: 144 Cost: 0.0445 Fidelity: 0.9806
Rep: 145 Cost: 0.0402 Fidelity: 0.9808
Rep: 146 Cost: 0.0428 Fidelity: 0.9811
Rep: 147 Cost: 0.0381 Fidelity: 0.9812
Rep: 148 Cost: 0.0452 Fidelity: 0.9807
Rep: 149 Cost: 0.0413 Fidelity: 0.9806
Rep: 150 Cost: 0.0408 Fidelity: 0.9809
Rep: 151 Cost: 0.0363 Fidelity: 0.9809
Rep: 152 Cost: 0.0459 Fidelity: 0.9805
Rep: 153 Cost: 0.0415 Fidelity: 0.9805
Rep: 154 Cost: 0.0406 Fidelity: 0.9808
Rep: 155 Cost: 0.0365 Fidelity: 0.9809
Rep: 156 Cost: 0.0447 Fidelity: 0.9808
Rep: 157 Cost: 0.0400 Fidelity: 0.9810
Rep: 158 Cost: 0.0418 Fidelity: 0.9812
Rep: 159 Cost: 0.0377 Fidelity: 0.9814
Rep: 160 Cost: 0.0434 Fidelity: 0.9813
Rep: 161 Cost: 0.0391 Fidelity: 0.9814
Rep: 162 Cost: 0.0420 Fidelity: 0.9814
Rep: 163 Cost: 0.0376 Fidelity: 0.9815
Rep: 164 Cost: 0.0435 Fidelity: 0.9813
Rep: 165 Cost: 0.0394 Fidelity: 0.9814
Rep: 166 Cost: 0.0410 Fidelity: 0.9814
Rep: 167 Cost: 0.0365 Fidelity: 0.9815
Rep: 168 Cost: 0.0440 Fidelity: 0.9814
Rep: 169 Cost: 0.0398 Fidelity: 0.9815
Rep: 170 Cost: 0.0403 Fidelity: 0.9817
Rep: 171 Cost: 0.0361 Fidelity: 0.9818
Rep: 172 Cost: 0.0439 Fidelity: 0.9816
Rep: 173 Cost: 0.0396 Fidelity: 0.9817
Rep: 174 Cost: 0.0403 Fidelity: 0.9821
Rep: 175 Cost: 0.0362 Fidelity: 0.9822
Rep: 176 Cost: 0.0435 Fidelity: 0.9817
Rep: 177 Cost: 0.0393 Fidelity: 0.9818
Rep: 178 Cost: 0.0402 Fidelity: 0.9823
Rep: 179 Cost: 0.0360 Fidelity: 0.9824
Rep: 180 Cost: 0.0434 Fidelity: 0.9817
Rep: 181 Cost: 0.0393 Fidelity: 0.9818
Rep: 182 Cost: 0.0398 Fidelity: 0.9824
Rep: 183 Cost: 0.0356 Fidelity: 0.9824
Rep: 184 Cost: 0.0434 Fidelity: 0.9818
Rep: 185 Cost: 0.0393 Fidelity: 0.9819
Rep: 186 Cost: 0.0395 Fidelity: 0.9826
Rep: 187 Cost: 0.0352 Fidelity: 0.9827
Rep: 188 Cost: 0.0434 Fidelity: 0.9820
Rep: 189 Cost: 0.0393 Fidelity: 0.9821
Rep: 190 Cost: 0.0392 Fidelity: 0.9829
Rep: 191 Cost: 0.0351 Fidelity: 0.9829
Rep: 192 Cost: 0.0433 Fidelity: 0.9821
Rep: 193 Cost: 0.0392 Fidelity: 0.9822
Rep: 194 Cost: 0.0390 Fidelity: 0.9830
Rep: 195 Cost: 0.0348 Fidelity: 0.9831
Rep: 196 Cost: 0.0432 Fidelity: 0.9821
Rep: 197 Cost: 0.0391 Fidelity: 0.9822
Rep: 198 Cost: 0.0387 Fidelity: 0.9832
Rep: 199 Cost: 0.0345 Fidelity: 0.9832
```

## 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")
```

Out:

```
Text(0.5, 0, 'Step')
```

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()
```

Out:

```
/home/circleci/project/venv/lib/python3.7/site-packages/numpy/core/_asarray.py:136: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify 'dtype=object' when creating the ndarray
```

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()
```

## References¶

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).

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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