Note

Click here to download the full example code

# Quantum gate synthesis¶

This demonstration works through the process used to produce the gate synthesis 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 action of the variational quantum circuit is close to some specified target unitary.

Note

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

For arbitrary gate synthesis 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
described below:

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 = 10
# gate cutoff
gate_cutoff = 4
# Number of layers
depth = 15
# Number of steps in optimization routine performing gradient descent
reps = 200
# Learning rate
lr = 0.025
# Standard deviation of initial parameters
passive_sd = 0.1
active_sd = 0.001
```

Note that, unlike in state learning, we must also specify a *gate
cutoff* \(d\). This restricts the target unitary to its action on a
\(d\)-dimensional subspace of the truncated Fock space, where
\(d\leq D\), where \(D\) is the overall simulation Fock basis
cutoff. As a result, we restrict the gate synthesis optimization to only
\(d\) input-output relations.

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

We use TensorFlow to create the variables corresponding to the gate
parameters. Note that 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)
np.random.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
```

We must also specify the input states to the variational quantum circuit - these are the Fock state \(\ket{i}\), \(i=0,\dots,d\), allowing us to optimize the circuit parameters to learn the target unitary acting on all input Fock states within the \(d\)-dimensional subspace.

```
in_ket = np.zeros([gate_cutoff, cutoff])
np.fill_diagonal(in_ket, 1)
# Apply circuit of layers with corresponding depth
with prog.context as q:
Ket(in_ket) | 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 gate synthesis, we minimize the overlaps in the Fock basis between the target and learnt unitaries via the following cost function:

where \(V\) is the target unitary, \(U(\vec{\theta})\) is the learnt unitary, and \(d\) is the gate cutoff. Note that this is a generalization of state preparation to more than one input-output relation.

For our target unitary, let’s use Strawberry Fields to generate a 4x4 random unitary:

```
from strawberryfields.utils import random_interferometer
# define unitary up to gate_cutoff
random_unitary = random_interferometer(gate_cutoff)
print(random_unitary)
# extend unitary up to cutoff
target_unitary = np.identity(cutoff, dtype=np.complex128)
target_unitary[:gate_cutoff, :gate_cutoff] = random_unitary
```

Out:

```
[[ 0.23648826-0.48221431j 0.06829648+0.04447898j 0.51150074-0.09529866j
0.55205719-0.35974699j]
[-0.11148167+0.69780321j -0.24943828+0.08410701j 0.46705929-0.43192981j
0.16220654-0.01817602j]
[-0.22351926-0.25918352j 0.24364996-0.05375623j -0.09259829-0.53810588j
0.27267708+0.66941977j]
[ 0.11519953-0.28596729j -0.90164923-0.22099186j -0.09627758-0.13105595j
-0.0200152 +0.12766128j]]
```

This matches the gate cutoff of \(d=4\) that we chose above when defining our hyperparameters.

Now, we instantiate the Strawberry Fields TensorFlow backend:

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

Here, we use the `batch_size`

argument to perform the optimization in
parallel - each batch calculates the variational quantum circuit acting
on a different input Fock state: \(U(\vec{\theta}) | n\rangle\).

```
in_state = np.arange(gate_cutoff)
# extract action of the target unitary acting on
# the allowed input fock states.
target_kets = np.array([target_unitary[:, i] for i in in_state])
target_kets = tf.constant(target_kets, dtype=tf.complex64)
```

Using this target unitary, we define the cost function we would like to
minimize. We must use TensorFlow functions to manipulate this data, as well
as a `GradientTape`

to keep track of the corresponding gradients!

```
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()
# overlaps
overlaps = tf.math.real(tf.einsum('bi,bi->b', tf.math.conj(target_kets), ket))
mean_overlap = tf.reduce_mean(overlaps)
# Objective function to minimize
cost = tf.abs(tf.reduce_sum(overlaps - 1))
return cost, overlaps, 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.

```
# Using Adam algorithm for optimization
opt = tf.keras.optimizers.Adam(learning_rate=lr)
```

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

```
overlap_progress = []
cost_progress = []
# Run optimization
for i in range(reps):
# reset the engine if it has already been executed
if eng.run_progs:
eng.reset()
# one repetition of the optimization
with tf.GradientTape() as tape:
loss, overlaps_val, ket_val = cost(weights)
# calculate the mean overlap
# This gives us an idea of how the optimization is progressing
mean_overlap_val = np.mean(overlaps_val)
# store cost at each step
cost_progress.append(loss)
overlap_progress.append(overlaps_val)
# 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 progress
print("Rep: {} Cost: {:.4f} Mean overlap: {:.4f}".format(i, loss, mean_overlap_val))
```

Out:

```
Rep: 0 Cost: 4.2947 Mean overlap: -0.0737
Rep: 1 Cost: 3.2125 Mean overlap: 0.1969
Rep: 2 Cost: 4.3774 Mean overlap: -0.0944
Rep: 3 Cost: 3.6844 Mean overlap: 0.0789
Rep: 4 Cost: 3.7196 Mean overlap: 0.0701
Rep: 5 Cost: 3.2360 Mean overlap: 0.1910
Rep: 6 Cost: 3.1782 Mean overlap: 0.2055
Rep: 7 Cost: 3.1090 Mean overlap: 0.2228
Rep: 8 Cost: 3.0612 Mean overlap: 0.2347
Rep: 9 Cost: 3.0602 Mean overlap: 0.2349
Rep: 10 Cost: 2.7062 Mean overlap: 0.3234
Rep: 11 Cost: 2.7678 Mean overlap: 0.3081
Rep: 12 Cost: 2.2194 Mean overlap: 0.4451
Rep: 13 Cost: 2.0853 Mean overlap: 0.4787
Rep: 14 Cost: 2.2453 Mean overlap: 0.4387
Rep: 15 Cost: 1.6812 Mean overlap: 0.5797
Rep: 16 Cost: 1.9051 Mean overlap: 0.5237
Rep: 17 Cost: 1.4323 Mean overlap: 0.6419
Rep: 18 Cost: 1.4057 Mean overlap: 0.6486
Rep: 19 Cost: 1.2089 Mean overlap: 0.6978
Rep: 20 Cost: 1.1891 Mean overlap: 0.7027
Rep: 21 Cost: 1.2000 Mean overlap: 0.7000
Rep: 22 Cost: 1.1956 Mean overlap: 0.7011
Rep: 23 Cost: 1.1900 Mean overlap: 0.7025
Rep: 24 Cost: 1.1602 Mean overlap: 0.7100
Rep: 25 Cost: 1.0940 Mean overlap: 0.7265
Rep: 26 Cost: 1.0667 Mean overlap: 0.7333
Rep: 27 Cost: 1.0020 Mean overlap: 0.7495
Rep: 28 Cost: 0.9791 Mean overlap: 0.7552
Rep: 29 Cost: 0.9648 Mean overlap: 0.7588
Rep: 30 Cost: 0.9356 Mean overlap: 0.7661
Rep: 31 Cost: 0.9526 Mean overlap: 0.7618
Rep: 32 Cost: 0.8805 Mean overlap: 0.7799
Rep: 33 Cost: 0.9031 Mean overlap: 0.7742
Rep: 34 Cost: 0.8137 Mean overlap: 0.7966
Rep: 35 Cost: 0.8456 Mean overlap: 0.7886
Rep: 36 Cost: 0.8192 Mean overlap: 0.7952
Rep: 37 Cost: 0.8201 Mean overlap: 0.7950
Rep: 38 Cost: 0.8197 Mean overlap: 0.7951
Rep: 39 Cost: 0.7660 Mean overlap: 0.8085
Rep: 40 Cost: 0.7770 Mean overlap: 0.8058
Rep: 41 Cost: 0.7360 Mean overlap: 0.8160
Rep: 42 Cost: 0.7346 Mean overlap: 0.8163
Rep: 43 Cost: 0.7328 Mean overlap: 0.8168
Rep: 44 Cost: 0.7027 Mean overlap: 0.8243
Rep: 45 Cost: 0.7108 Mean overlap: 0.8223
Rep: 46 Cost: 0.6900 Mean overlap: 0.8275
Rep: 47 Cost: 0.6831 Mean overlap: 0.8292
Rep: 48 Cost: 0.6847 Mean overlap: 0.8288
Rep: 49 Cost: 0.6627 Mean overlap: 0.8343
Rep: 50 Cost: 0.6624 Mean overlap: 0.8344
Rep: 51 Cost: 0.6518 Mean overlap: 0.8370
Rep: 52 Cost: 0.6354 Mean overlap: 0.8412
Rep: 53 Cost: 0.6388 Mean overlap: 0.8403
Rep: 54 Cost: 0.6310 Mean overlap: 0.8422
Rep: 55 Cost: 0.6186 Mean overlap: 0.8453
Rep: 56 Cost: 0.6168 Mean overlap: 0.8458
Rep: 57 Cost: 0.6052 Mean overlap: 0.8487
Rep: 58 Cost: 0.5878 Mean overlap: 0.8531
Rep: 59 Cost: 0.5823 Mean overlap: 0.8544
Rep: 60 Cost: 0.5790 Mean overlap: 0.8552
Rep: 61 Cost: 0.5666 Mean overlap: 0.8583
Rep: 62 Cost: 0.5546 Mean overlap: 0.8614
Rep: 63 Cost: 0.5487 Mean overlap: 0.8628
Rep: 64 Cost: 0.5416 Mean overlap: 0.8646
Rep: 65 Cost: 0.5304 Mean overlap: 0.8674
Rep: 66 Cost: 0.5214 Mean overlap: 0.8696
Rep: 67 Cost: 0.5165 Mean overlap: 0.8709
Rep: 68 Cost: 0.5098 Mean overlap: 0.8726
Rep: 69 Cost: 0.4999 Mean overlap: 0.8750
Rep: 70 Cost: 0.4911 Mean overlap: 0.8772
Rep: 71 Cost: 0.4850 Mean overlap: 0.8788
Rep: 72 Cost: 0.4789 Mean overlap: 0.8803
Rep: 73 Cost: 0.4711 Mean overlap: 0.8822
Rep: 74 Cost: 0.4627 Mean overlap: 0.8843
Rep: 75 Cost: 0.4551 Mean overlap: 0.8862
Rep: 76 Cost: 0.4484 Mean overlap: 0.8879
Rep: 77 Cost: 0.4420 Mean overlap: 0.8895
Rep: 78 Cost: 0.4360 Mean overlap: 0.8910
Rep: 79 Cost: 0.4307 Mean overlap: 0.8923
Rep: 80 Cost: 0.4261 Mean overlap: 0.8935
Rep: 81 Cost: 0.4217 Mean overlap: 0.8946
Rep: 82 Cost: 0.4175 Mean overlap: 0.8956
Rep: 83 Cost: 0.4135 Mean overlap: 0.8966
Rep: 84 Cost: 0.4106 Mean overlap: 0.8974
Rep: 85 Cost: 0.4089 Mean overlap: 0.8978
Rep: 86 Cost: 0.4093 Mean overlap: 0.8977
Rep: 87 Cost: 0.4116 Mean overlap: 0.8971
Rep: 88 Cost: 0.4185 Mean overlap: 0.8954
Rep: 89 Cost: 0.4284 Mean overlap: 0.8929
Rep: 90 Cost: 0.4437 Mean overlap: 0.8891
Rep: 91 Cost: 0.4490 Mean overlap: 0.8877
Rep: 92 Cost: 0.4441 Mean overlap: 0.8890
Rep: 93 Cost: 0.4154 Mean overlap: 0.8962
Rep: 94 Cost: 0.3849 Mean overlap: 0.9038
Rep: 95 Cost: 0.3642 Mean overlap: 0.9089
Rep: 96 Cost: 0.3601 Mean overlap: 0.9100
Rep: 97 Cost: 0.3686 Mean overlap: 0.9079
Rep: 98 Cost: 0.3814 Mean overlap: 0.9047
Rep: 99 Cost: 0.3918 Mean overlap: 0.9020
Rep: 100 Cost: 0.3890 Mean overlap: 0.9027
Rep: 101 Cost: 0.3765 Mean overlap: 0.9059
Rep: 102 Cost: 0.3562 Mean overlap: 0.9110
Rep: 103 Cost: 0.3395 Mean overlap: 0.9151
Rep: 104 Cost: 0.3304 Mean overlap: 0.9174
Rep: 105 Cost: 0.3291 Mean overlap: 0.9177
Rep: 106 Cost: 0.3333 Mean overlap: 0.9167
Rep: 107 Cost: 0.3396 Mean overlap: 0.9151
Rep: 108 Cost: 0.3465 Mean overlap: 0.9134
Rep: 109 Cost: 0.3496 Mean overlap: 0.9126
Rep: 110 Cost: 0.3499 Mean overlap: 0.9125
Rep: 111 Cost: 0.3426 Mean overlap: 0.9144
Rep: 112 Cost: 0.3324 Mean overlap: 0.9169
Rep: 113 Cost: 0.3190 Mean overlap: 0.9203
Rep: 114 Cost: 0.3071 Mean overlap: 0.9232
Rep: 115 Cost: 0.2975 Mean overlap: 0.9256
Rep: 116 Cost: 0.2909 Mean overlap: 0.9273
Rep: 117 Cost: 0.2868 Mean overlap: 0.9283
Rep: 118 Cost: 0.2845 Mean overlap: 0.9289
Rep: 119 Cost: 0.2838 Mean overlap: 0.9290
Rep: 120 Cost: 0.2848 Mean overlap: 0.9288
Rep: 121 Cost: 0.2888 Mean overlap: 0.9278
Rep: 122 Cost: 0.2966 Mean overlap: 0.9258
Rep: 123 Cost: 0.3116 Mean overlap: 0.9221
Rep: 124 Cost: 0.3308 Mean overlap: 0.9173
Rep: 125 Cost: 0.3530 Mean overlap: 0.9118
Rep: 126 Cost: 0.3528 Mean overlap: 0.9118
Rep: 127 Cost: 0.3306 Mean overlap: 0.9173
Rep: 128 Cost: 0.2866 Mean overlap: 0.9283
Rep: 129 Cost: 0.2533 Mean overlap: 0.9367
Rep: 130 Cost: 0.2439 Mean overlap: 0.9390
Rep: 131 Cost: 0.2555 Mean overlap: 0.9361
Rep: 132 Cost: 0.2737 Mean overlap: 0.9316
Rep: 133 Cost: 0.2785 Mean overlap: 0.9304
Rep: 134 Cost: 0.2655 Mean overlap: 0.9336
Rep: 135 Cost: 0.2411 Mean overlap: 0.9397
Rep: 136 Cost: 0.2240 Mean overlap: 0.9440
Rep: 137 Cost: 0.2216 Mean overlap: 0.9446
Rep: 138 Cost: 0.2291 Mean overlap: 0.9427
Rep: 139 Cost: 0.2360 Mean overlap: 0.9410
Rep: 140 Cost: 0.2332 Mean overlap: 0.9417
Rep: 141 Cost: 0.2215 Mean overlap: 0.9446
Rep: 142 Cost: 0.2068 Mean overlap: 0.9483
Rep: 143 Cost: 0.1970 Mean overlap: 0.9507
Rep: 144 Cost: 0.1943 Mean overlap: 0.9514
Rep: 145 Cost: 0.1964 Mean overlap: 0.9509
Rep: 146 Cost: 0.1992 Mean overlap: 0.9502
Rep: 147 Cost: 0.1992 Mean overlap: 0.9502
Rep: 148 Cost: 0.1956 Mean overlap: 0.9511
Rep: 149 Cost: 0.1886 Mean overlap: 0.9529
Rep: 150 Cost: 0.1805 Mean overlap: 0.9549
Rep: 151 Cost: 0.1729 Mean overlap: 0.9568
Rep: 152 Cost: 0.1670 Mean overlap: 0.9582
Rep: 153 Cost: 0.1629 Mean overlap: 0.9593
Rep: 154 Cost: 0.1602 Mean overlap: 0.9599
Rep: 155 Cost: 0.1586 Mean overlap: 0.9603
Rep: 156 Cost: 0.1581 Mean overlap: 0.9605
Rep: 157 Cost: 0.1588 Mean overlap: 0.9603
Rep: 158 Cost: 0.1613 Mean overlap: 0.9597
Rep: 159 Cost: 0.1671 Mean overlap: 0.9582
Rep: 160 Cost: 0.1782 Mean overlap: 0.9554
Rep: 161 Cost: 0.1975 Mean overlap: 0.9506
Rep: 162 Cost: 0.2244 Mean overlap: 0.9439
Rep: 163 Cost: 0.2522 Mean overlap: 0.9369
Rep: 164 Cost: 0.2572 Mean overlap: 0.9357
Rep: 165 Cost: 0.2291 Mean overlap: 0.9427
Rep: 166 Cost: 0.1776 Mean overlap: 0.9556
Rep: 167 Cost: 0.1401 Mean overlap: 0.9650
Rep: 168 Cost: 0.1342 Mean overlap: 0.9664
Rep: 169 Cost: 0.1540 Mean overlap: 0.9615
Rep: 170 Cost: 0.1779 Mean overlap: 0.9555
Rep: 171 Cost: 0.1819 Mean overlap: 0.9545
Rep: 172 Cost: 0.1622 Mean overlap: 0.9595
Rep: 173 Cost: 0.1354 Mean overlap: 0.9662
Rep: 174 Cost: 0.1232 Mean overlap: 0.9692
Rep: 175 Cost: 0.1301 Mean overlap: 0.9675
Rep: 176 Cost: 0.1448 Mean overlap: 0.9638
Rep: 177 Cost: 0.1523 Mean overlap: 0.9619
Rep: 178 Cost: 0.1451 Mean overlap: 0.9637
Rep: 179 Cost: 0.1301 Mean overlap: 0.9675
Rep: 180 Cost: 0.1182 Mean overlap: 0.9704
Rep: 181 Cost: 0.1162 Mean overlap: 0.9709
Rep: 182 Cost: 0.1223 Mean overlap: 0.9694
Rep: 183 Cost: 0.1296 Mean overlap: 0.9676
Rep: 184 Cost: 0.1324 Mean overlap: 0.9669
Rep: 185 Cost: 0.1285 Mean overlap: 0.9679
Rep: 186 Cost: 0.1208 Mean overlap: 0.9698
Rep: 187 Cost: 0.1133 Mean overlap: 0.9717
Rep: 188 Cost: 0.1091 Mean overlap: 0.9727
Rep: 189 Cost: 0.1089 Mean overlap: 0.9728
Rep: 190 Cost: 0.1114 Mean overlap: 0.9721
Rep: 191 Cost: 0.1149 Mean overlap: 0.9713
Rep: 192 Cost: 0.1178 Mean overlap: 0.9706
Rep: 193 Cost: 0.1194 Mean overlap: 0.9702
Rep: 194 Cost: 0.1191 Mean overlap: 0.9702
Rep: 195 Cost: 0.1176 Mean overlap: 0.9706
Rep: 196 Cost: 0.1148 Mean overlap: 0.9713
Rep: 197 Cost: 0.1119 Mean overlap: 0.9720
Rep: 198 Cost: 0.1090 Mean overlap: 0.9728
Rep: 199 Cost: 0.1064 Mean overlap: 0.9734
```

## Results and visualisation¶

Plotting the cost vs. optimization step:

```
from matplotlib import pyplot as plt
# %matplotlib inline
plt.rcParams['font.family'] = 'serif'
plt.rcParams['font.sans-serif'] = ['Computer Modern Roman']
plt.style.use('default')
plt.plot(cost_progress)
plt.ylabel('Cost')
plt.xlabel('Step')
plt.show()
```

We can use matrix plots to plot the real and imaginary components of the target unitary \(V\) and learnt unitary \(U\).

```
learnt_unitary = ket_val.numpy().T[:gate_cutoff, :gate_cutoff]
target_unitary = target_unitary[:gate_cutoff, :gate_cutoff]
fig, ax = plt.subplots(1, 4, figsize=(7, 4))
ax[0].matshow(target_unitary.real, cmap=plt.get_cmap('Reds'))
ax[1].matshow(target_unitary.imag, cmap=plt.get_cmap('Greens'))
ax[2].matshow(learnt_unitary.real, cmap=plt.get_cmap('Reds'))
ax[3].matshow(learnt_unitary.imag, cmap=plt.get_cmap('Greens'))
ax[0].set_xlabel(r'$\mathrm{Re}(V)$')
ax[1].set_xlabel(r'$\mathrm{Im}(V)$')
ax[2].set_xlabel(r'$\mathrm{Re}(U)$')
ax[3].set_xlabel(r'$\mathrm{Im}(U)$')
fig.show()
```

## Process fidelity¶

The process fidelity between the two unitaries is defined by

where:

\(\left|\Psi(V)\right\rangle\) is the action of \(V\) on one half of a maximally entangled state \(\left|\phi\right\rangle\):

\(V\) is the target unitary,

\(U\) the learnt unitary.

```
I = np.identity(gate_cutoff)
phi = I.flatten()/np.sqrt(gate_cutoff)
psiV = np.kron(I, target_unitary) @ phi
psiU = np.kron(I, learnt_unitary) @ phi
```

Therefore, after 200 repetitions, the learnt unitary synthesized via a variational quantum circuit has the following process fidelity to the target unitary:

```
print(np.abs(np.vdot(psiV, psiU))**2)
```

Out:

```
0.9481729779485119
```

## Circuit parameters¶

We can also query the optimal variational circuit parameters \(\vec{\theta}\) that resulted in the learnt unitary. For example, to determine the maximum squeezing magnitude in the variational quantum circuit:

```
print(np.max(np.abs(weights[:, 0])))
```

Out:

```
0.39022577
```

## Further results¶

After downloading the tutorial, even more refined results can be obtained by
increasing the number of repetitions (`reps`

), changing the depth of the
circuit or altering the gate cutoff!

## 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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