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 can be achieved 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 previously mentioned, a ComplexWaveFunction object requires a dictionary that contains the unitary 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 under 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
)


By default, QuCumber will attempt to run on a GPU if one is available (if one is not available, QuCumber will fall back to CPU). If one wishes to guarantee that QuCumber runs on the CPU, add the flag gpu=False in the ComplexWaveFunction object instantiation. Set gpu=True in the line above to run this tutorial on a GPU.

Now the hyperparameters of the training process can be specified.

1. epochs: the total number of training cycles that will be performed (default = 100)

2. pos_batch_size: the number of data points used in the positive phase of the gradient (default = 100)

3. neg_batch_size: the number of data points used in the negative phase of the gradient (default = pos_batch_size)

4. k: the number of contrastive divergence steps (default = 1)

5. 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.

1. period: the frequency of the training evaluators being calculated (e.g. period=200 means that the MetricEvaluator will compute the desired metrics every 200 epochs)

2. 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 entire 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 period epochs must be given to the MetricEvaluator. Note that some of the coefficients aren’t being evaluated as they are commented out. This is simply to avoid cluttering the output, and may be uncommented by the user.

[7]:

period = 2
space = nn_state.generate_hilbert_space(nv)

callbacks = [
MetricEvaluator(
period,
{
"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 function 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 = 0.623747     KL = 0.226386   normα = 0.272518
Epoch: 4        Fidelity = 0.744691     KL = 0.142639   normα = 0.248872
Epoch: 6        Fidelity = 0.818254     KL = 0.094584   normα = 0.263589
Epoch: 8        Fidelity = 0.867098     KL = 0.067506   normα = 0.278453
Epoch: 10       Fidelity = 0.900217     KL = 0.051592   normα = 0.281094
Epoch: 12       Fidelity = 0.922993     KL = 0.041311   normα = 0.276052
Epoch: 14       Fidelity = 0.937807     KL = 0.034972   normα = 0.274676
Epoch: 16       Fidelity = 0.947232     KL = 0.030543   normα = 0.283873
Epoch: 18       Fidelity = 0.955277     KL = 0.027313   normα = 0.278906
Epoch: 20       Fidelity = 0.959930     KL = 0.025034   normα = 0.290271
Epoch: 22       Fidelity = 0.963333     KL = 0.023719   normα = 0.296183
Epoch: 24       Fidelity = 0.969419     KL = 0.021086   normα = 0.276108
Epoch: 26       Fidelity = 0.972300     KL = 0.020200   normα = 0.290305
Epoch: 28       Fidelity = 0.974777     KL = 0.018635   normα = 0.284231
Epoch: 30       Fidelity = 0.976208     KL = 0.017865   normα = 0.282036
Epoch: 32       Fidelity = 0.978382     KL = 0.016862   normα = 0.282498
Epoch: 34       Fidelity = 0.980578     KL = 0.015977   normα = 0.279435
Epoch: 36       Fidelity = 0.980983     KL = 0.015545   normα = 0.277835
Epoch: 38       Fidelity = 0.982651     KL = 0.014751   normα = 0.280070
Epoch: 40       Fidelity = 0.983155     KL = 0.014353   normα = 0.276912
Epoch: 42       Fidelity = 0.983996     KL = 0.013827   normα = 0.278844
Epoch: 44       Fidelity = 0.982731     KL = 0.015100   normα = 0.305219
Epoch: 46       Fidelity = 0.984791     KL = 0.013417   normα = 0.293674
Epoch: 48       Fidelity = 0.985395     KL = 0.012845   normα = 0.280658
Epoch: 50       Fidelity = 0.986767     KL = 0.012093   normα = 0.277599
Epoch: 52       Fidelity = 0.987795     KL = 0.011650   normα = 0.278886
Epoch: 54       Fidelity = 0.987057     KL = 0.011843   normα = 0.271735
Epoch: 56       Fidelity = 0.987125     KL = 0.011552   normα = 0.280304
Epoch: 58       Fidelity = 0.987295     KL = 0.011382   normα = 0.288229
Epoch: 60       Fidelity = 0.988201     KL = 0.011201   normα = 0.266736
Epoch: 62       Fidelity = 0.989181     KL = 0.010504   normα = 0.288520
Epoch: 64       Fidelity = 0.989308     KL = 0.010293   normα = 0.292218
Epoch: 66       Fidelity = 0.989321     KL = 0.009901   normα = 0.282069
Epoch: 68       Fidelity = 0.989347     KL = 0.009836   normα = 0.275723
Epoch: 70       Fidelity = 0.989494     KL = 0.009838   normα = 0.293840
Epoch: 72       Fidelity = 0.990115     KL = 0.009225   normα = 0.282556
Epoch: 74       Fidelity = 0.990199     KL = 0.009095   normα = 0.278911
Epoch: 76       Fidelity = 0.989979     KL = 0.009214   normα = 0.273241
Epoch: 78       Fidelity = 0.989633     KL = 0.009275   normα = 0.274384
Epoch: 80       Fidelity = 0.989972     KL = 0.008976   normα = 0.275430
Epoch: 82       Fidelity = 0.989920     KL = 0.008871   normα = 0.285605
Epoch: 84       Fidelity = 0.991177     KL = 0.008183   normα = 0.282607
Epoch: 86       Fidelity = 0.991249     KL = 0.008095   normα = 0.276934
Epoch: 88       Fidelity = 0.990857     KL = 0.008273   normα = 0.272151
Epoch: 90       Fidelity = 0.990802     KL = 0.008071   normα = 0.280823
Epoch: 92       Fidelity = 0.991090     KL = 0.007838   normα = 0.279963
Epoch: 94       Fidelity = 0.990995     KL = 0.007861   normα = 0.275772
Epoch: 96       Fidelity = 0.990326     KL = 0.008202   normα = 0.289882
Epoch: 98       Fidelity = 0.991012     KL = 0.007690   normα = 0.277037
Epoch: 100      Fidelity = 0.991736     KL = 0.007292   normα = 0.275516


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]:

# Note that the key given to the *MetricEvaluator* must be
# what comes after callbacks[0].
fidelities = callbacks[0].Fidelity

# Alternatively, we may use the usual dictionary/list subscripting
# syntax. This is useful in cases where the name of the metric
# may contain special characters or spaces.
KLs = callbacks[0]["KL"]
coeffs = callbacks[0]["normα"]
epoch = np.arange(period, epochs + 1, period)

[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”.