This is a static, non-editable tutorial.

We recommend you install QuCumber if you want to run the examples locally. You can then get an archive file containing the examples from the relevant release here. Alternatively, you can launch an interactive online version, though it may be a bit slow:

# Reconstruction of a complex wavefunction¶

In this tutorial, a walkthrough of how to reconstruct a **complex** wavefunction via training a *Restricted Boltzmann Machine* (RBM), the neural network behind qucumber, will be presented.

## The wavefunction to be reconstructed¶

The simple wavefunction below describing two qubits (coefficients stored in *qubits_psi.txt*) will be reconstructed.

where the exact values of and used for this tutorial are

The example dataset, *qubits_train.txt*, comprises of 500 measurements made in various bases (X, Y and Z). A corresponding file containing the bases for each data point in *qubits_train.txt*, *qubits_train_bases.txt*, is also required. As per convention, spins are represented in binary notation with zero and one denoting spin-down and spin-up, respectively.

## Using qucumber to reconstruct the wavefunction¶

### Imports¶

To begin the tutorial, first import the required Python packages.

```
[1]:
```

```
import numpy as np
import torch
import matplotlib.pyplot as plt
from qucumber.nn_states import ComplexWaveFunction
from qucumber.callbacks import MetricEvaluator
import qucumber.utils.unitaries as unitaries
import qucumber.utils.cplx as cplx
import qucumber.utils.training_statistics as ts
import qucumber.utils.data as data
```

The Python class *ComplexWaveFunction* contains generic properties of a RBM meant to reconstruct a complex wavefunction, the most notable one being the gradient function required for stochastic gradient descent.

To instantiate a *ComplexWaveFunction* object, one needs to specify the number of visible and hidden units in the RBM. The number of visible units, *num_visible*, is given by the size of the physical system, i.e. the number of spins or qubits (2 in this case), while the number of hidden units, *num_hidden*, can be varied to change the expressiveness of the neural network.

**Note:** The optimal *num_hidden* : *num_visible* ratio will depend on the system. For the two-qubit wavefunction described above, good results are yielded when this ratio is 1.

On top of needing the number of visible and hidden units, a *ComplexWaveFunction* object requires the user to input a dictionary containing the unitary operators (2x2) that will be used to rotate the qubits in and out of the computational basis, Z, during the training process. The *unitaries* utility will take care of creating this dictionary.

The *MetricEvaluator* class and *training_statistics* utility are built-in amenities that will allow the user to evaluate the training in real time.

Lastly, the *cplx* utility allows qucumber to be able to handle complex numbers. Currently, Pytorch does not support complex numbers.

### Training¶

To evaluate the training in real time, the fidelity between the true wavefunction of the system and the wavefunction that qucumber reconstructs, , will be calculated along with the Kullback-Leibler (KL) divergence (the RBM’s cost function). First, the training data and the true wavefunction of this system need to be loaded using the *data* utility.

```
[2]:
```

```
train_path = "qubits_train.txt"
train_bases_path = "qubits_train_bases.txt"
psi_path = "qubits_psi.txt"
bases_path = "qubits_bases.txt"
train_samples, true_psi, train_bases, bases = data.load_data(
train_path, psi_path, train_bases_path, bases_path
)
```

The file *qubits_bases.txt* contains every unique basis in the *qubits_train_bases.txt* file. Calculation of the full KL divergence in every basis requires the user to specify each unique basis.

As previouosly mentioned, a *ComplexWaveFunction* object requires a dictionary that contains the unitariy operators that will be used to rotate the qubits in and out of the computational basis, Z, during the training process. In the case of the provided dataset, the unitaries required are the well-known , and gates. The dictionary needed can be created with the following command.

```
[3]:
```

```
unitary_dict = unitaries.create_dict()
# unitary_dict = unitaries.create_dict(unitary_name=torch.tensor([[real part],
# [imaginary part]],
# dtype=torch.double)
```

If the user wishes to add their own unitary operators from their experiment to *unitary_dict*, uncomment the block above. When *unitaries.create_dict()* is called, it will contain the identity and the and gates by default with the keys “Z”, “X” and “Y”, respectively.

The number of visible units in the RBM is equal to the number of qubits. The number of hidden units will also be taken to be the number of visible units.

```
[4]:
```

```
nv = train_samples.shape[-1]
nh = nv
nn_state = ComplexWaveFunction(
num_visible=nv, num_hidden=nh, unitary_dict=unitary_dict, gpu=False
)
# nn_state = ComplexWaveFunction(num_visible=nv, num_hidden=nh, unitary_dict=unitary_dict)
```

By default, qucumber will attempt to run on a GPU if one is available (if one is not available, qucumber will default to CPU). If one wishes to run qucumber on a CPU, add the flag “gpu = False” in the *ComplexWaveFunction* object instantiation. Uncomment the line above to run this tutorial on a GPU.

Now the hyperparameters of the training process can be specified.

**epochs**: the total number of training cycles that will be performed (default = 100)**pos_batch_size**: the number of data points used in the positive phase of the gradient (default = 100)**neg_batch_size**: the number of data points used in the negative phase of the gradient (default =*pos_batch_size*)**k**: the number of contrastive divergence steps (default = 1)**lr**: the learning rate (default = 0.001)**Note:**For more information on the hyperparameters above, it is strongly encouraged that the user to read through the brief, but thorough theory document on RBMs. One does not have to specify these hyperparameters, as their default values will be used without the user overwriting them. It is recommended to keep with the default values until the user has a stronger grasp on what these hyperparameters mean. The quality and the computational efficiency of the training will highly depend on the choice of hyperparameters. As such, playing around with the hyperparameters is almost always necessary.

The two-qubit example in this tutorial should be extremely easy to train, regardless of the choice of hyperparameters. However, the hyperparameters below will be used.

```
[5]:
```

```
epochs = 100
pbs = 50 # pos_batch_size
nbs = 50 # neg_batch_size
lr = 0.1
k = 5
```

For evaluating the training in real time, the *MetricEvaluator* will be called to calculate the training evaluators every 10 epochs. The *MetricEvaluator* requires the following arguments.

**log_every**: the frequency of the training evaluators being calculated is controlled by the*log_every*argument (e.g.*log_every*= 200 means that the*MetricEvaluator*will update the user every 200 epochs)- A dictionary of functions you would like to reference to evaluate the training (arguments required for these functions are keyword arguments placed after the dictionary)

The following additional arguments are needed to calculate the fidelity and KL divergence in the *training_statistics* utility.

**target_psi**(the true wavefunction of the system)**space**(the hilbert space of the system)

The training evaluators can be printed out via the *verbose=True* statement.

Although the fidelity and KL divergence are excellent training evaluators, they are not practical to calculate in most cases; the user may not have access to the target wavefunction of the system, nor may generating the hilbert space of the system be computationally feasible. However, evaluating the training in real time is extremely convenient.

Any custom function that the user would like to use to evaluate the training can be given to the *MetricEvaluator*, thus avoiding having to calculate fidelity and/or KL divergence. As an example, functions that calculate the the norm of each of the reconstructed wavefunction’s coefficients are presented. Any custom function given to *MetricEvaluator* must take the neural-network state (in this case, the *ComplexWaveFunction* object) and keyword arguments. Although the given example requires the
hilbert space to be computed, the scope of the *MetricEvaluator*’s ability to be able to handle any function should still be evident.

```
[6]:
```

```
def alpha(nn_state, space, **kwargs):
rbm_psi = nn_state.psi(space)
normalization = nn_state.compute_normalization(space).sqrt_()
alpha_ = cplx.norm(
torch.tensor([rbm_psi[0][0], rbm_psi[1][0]], device=nn_state.device)
/ normalization
)
return alpha_
def beta(nn_state, space, **kwargs):
rbm_psi = nn_state.psi(space)
normalization = nn_state.compute_normalization(space).sqrt_()
beta_ = cplx.norm(
torch.tensor([rbm_psi[0][1], rbm_psi[1][1]], device=nn_state.device)
/ normalization
)
return beta_
def gamma(nn_state, space, **kwargs):
rbm_psi = nn_state.psi(space)
normalization = nn_state.compute_normalization(space).sqrt_()
gamma_ = cplx.norm(
torch.tensor([rbm_psi[0][2], rbm_psi[1][2]], device=nn_state.device)
/ normalization
)
return gamma_
def delta(nn_state, space, **kwargs):
rbm_psi = nn_state.psi(space)
normalization = nn_state.compute_normalization(space).sqrt_()
delta_ = cplx.norm(
torch.tensor([rbm_psi[0][3], rbm_psi[1][3]], device=nn_state.device)
/ normalization
)
return delta_
```

Now the hilbert space of the system must be generated for the fidelity and KL divergence and the dictionary of functions the user would like to compute every “*log_every*” epochs must be given to the *MetricEvaluator*.

```
[7]:
```

```
log_every = 2
space = nn_state.generate_hilbert_space(nv)
callbacks = [
MetricEvaluator(
log_every,
{
"Fidelity": ts.fidelity,
"KL": ts.KL,
"normα": alpha,
# "normβ": beta,
# "normγ": gamma,
# "normδ": delta,
},
target_psi=true_psi,
bases=bases,
verbose=True,
space=space,
)
]
```

Now the training can begin. The *ComplexWaveFunction* object has a property called *fit* which takes care of this.

```
[8]:
```

```
nn_state.fit(
train_samples,
epochs=epochs,
pos_batch_size=pbs,
neg_batch_size=nbs,
lr=lr,
k=k,
input_bases=train_bases,
callbacks=callbacks,
)
```

```
Epoch: 2 Fidelity = 0.609290 KL = 0.251365 normα = 0.328561
Epoch: 4 Fidelity = 0.761520 KL = 0.130756 normα = 0.268571
Epoch: 6 Fidelity = 0.838254 KL = 0.082158 normα = 0.253614
Epoch: 8 Fidelity = 0.882321 KL = 0.059876 normα = 0.251210
Epoch: 10 Fidelity = 0.909942 KL = 0.046754 normα = 0.259010
Epoch: 12 Fidelity = 0.929755 KL = 0.036943 normα = 0.258238
Epoch: 14 Fidelity = 0.942398 KL = 0.030965 normα = 0.262524
Epoch: 16 Fidelity = 0.951771 KL = 0.027162 normα = 0.250808
Epoch: 18 Fidelity = 0.958536 KL = 0.023913 normα = 0.257332
Epoch: 20 Fidelity = 0.963766 KL = 0.021570 normα = 0.259561
Epoch: 22 Fidelity = 0.968224 KL = 0.019610 normα = 0.263528
Epoch: 24 Fidelity = 0.970991 KL = 0.018523 normα = 0.256601
Epoch: 26 Fidelity = 0.974414 KL = 0.017127 normα = 0.256168
Epoch: 28 Fidelity = 0.976543 KL = 0.016030 normα = 0.261707
Epoch: 30 Fidelity = 0.977891 KL = 0.015551 normα = 0.254371
Epoch: 32 Fidelity = 0.978768 KL = 0.015847 normα = 0.242762
Epoch: 34 Fidelity = 0.981126 KL = 0.013974 normα = 0.259100
Epoch: 36 Fidelity = 0.981330 KL = 0.013704 normα = 0.262152
Epoch: 38 Fidelity = 0.981989 KL = 0.013495 normα = 0.256111
Epoch: 40 Fidelity = 0.983230 KL = 0.012711 normα = 0.269000
Epoch: 42 Fidelity = 0.984230 KL = 0.012301 normα = 0.273091
Epoch: 44 Fidelity = 0.984190 KL = 0.012178 normα = 0.260481
Epoch: 46 Fidelity = 0.985494 KL = 0.011501 normα = 0.269168
Epoch: 48 Fidelity = 0.985192 KL = 0.011518 normα = 0.269217
Epoch: 50 Fidelity = 0.986754 KL = 0.010896 normα = 0.264035
Epoch: 52 Fidelity = 0.987266 KL = 0.011288 normα = 0.263792
Epoch: 54 Fidelity = 0.987998 KL = 0.010323 normα = 0.268859
Epoch: 56 Fidelity = 0.987993 KL = 0.009957 normα = 0.267215
Epoch: 58 Fidelity = 0.988191 KL = 0.009873 normα = 0.273412
Epoch: 60 Fidelity = 0.988129 KL = 0.009665 normα = 0.264751
Epoch: 62 Fidelity = 0.986891 KL = 0.010557 normα = 0.254349
Epoch: 64 Fidelity = 0.988134 KL = 0.009504 normα = 0.264387
Epoch: 66 Fidelity = 0.987419 KL = 0.009870 normα = 0.263105
Epoch: 68 Fidelity = 0.988299 KL = 0.009280 normα = 0.264269
Epoch: 70 Fidelity = 0.989128 KL = 0.008897 normα = 0.267263
Epoch: 72 Fidelity = 0.989572 KL = 0.008808 normα = 0.268239
Epoch: 74 Fidelity = 0.988996 KL = 0.008901 normα = 0.256966
Epoch: 76 Fidelity = 0.988131 KL = 0.009278 normα = 0.261540
Epoch: 78 Fidelity = 0.989458 KL = 0.008405 normα = 0.263894
Epoch: 80 Fidelity = 0.988967 KL = 0.008628 normα = 0.261319
Epoch: 82 Fidelity = 0.989337 KL = 0.008332 normα = 0.271008
Epoch: 84 Fidelity = 0.989170 KL = 0.008351 normα = 0.270172
Epoch: 86 Fidelity = 0.989256 KL = 0.008291 normα = 0.273329
Epoch: 88 Fidelity = 0.989592 KL = 0.008019 normα = 0.269137
Epoch: 90 Fidelity = 0.989677 KL = 0.007917 normα = 0.266120
Epoch: 92 Fidelity = 0.989260 KL = 0.008174 normα = 0.261290
Epoch: 94 Fidelity = 0.990006 KL = 0.007627 normα = 0.266389
Epoch: 96 Fidelity = 0.989957 KL = 0.007610 normα = 0.270035
Epoch: 98 Fidelity = 0.990623 KL = 0.007506 normα = 0.278505
Epoch: 100 Fidelity = 0.990754 KL = 0.007096 normα = 0.266719
```

All of these training evaluators can be accessed after the training has completed, as well. The code below shows this, along with plots of each training evaluator versus the training cycle number (epoch).

```
[9]:
```

```
fidelities = callbacks[0].Fidelity
KLs = callbacks[0].KL
coeffs = callbacks[0].normα
# Please note that the key given to the *MetricEvaluator* must be what comes after callbacks[0].
epoch = np.arange(log_every, epochs + 1, log_every)
```

```
[10]:
```

```
# Some parameters to make the plots look nice
params = {'text.usetex': True,
'font.family': 'serif',
'legend.fontsize': 14,
'figure.figsize': (10, 3),
'axes.labelsize': 16,
'xtick.labelsize':14,
'ytick.labelsize':14,
'lines.linewidth':2,
'lines.markeredgewidth': 0.8,
'lines.markersize': 5,
'lines.marker': "o",
"patch.edgecolor": "black"
}
plt.rcParams.update(params)
plt.style.use('seaborn-deep')
```

```
[11]:
```

```
fig, axs = plt.subplots(nrows=1, ncols=3, figsize=(14, 3))
ax = axs[0]
ax.plot(epoch, fidelities, "o", color = "C0", markeredgecolor="black")
ax.set_ylabel(r'Fidelity')
ax.set_xlabel(r'Epoch')
ax = axs[1]
ax.plot(epoch, KLs, "o", color = "C1", markeredgecolor="black")
ax.set_ylabel(r'KL Divergence')
ax.set_xlabel(r'Epoch')
ax = axs[2]
ax.plot(epoch, coeffs, "o", color = "C2", markeredgecolor="black")
ax.set_ylabel(r'$\vert\alpha\vert$')
ax.set_xlabel(r'Epoch')
plt.tight_layout()
plt.savefig("complex_fid_KL.pdf")
plt.show()
```

It should be noted that one could have just ran *nn_state.fit(train_samples)* and just used the default hyperparameters and no training evaluators.

At the end of the training process, the network parameters (the weights, visible biases and hidden biases) are stored in the *ComplexWaveFunction* object. One can save them to a pickle file, which will be called *saved_params.pt*, with the following command.

```
[12]:
```

```
nn_state.save("saved_params.pt")
```

This saves the weights, visible biases and hidden biases as torch tensors with the following keys: “weights”, “visible_bias”, “hidden_bias”.