[DEMO] QML for Phase Detection

[SaverioMonaco]

Title:
Supervised and Unsupervised Quantum Machine Learning models for the phase detection of the ANNNI spin model

Summary:
Showcase of two foundational models in QML for Phase Detection for Spin Systems. One follows a supervised approach, the other follows an unsupervised approach.

Relevant references:

Possible Drawbacks:
QCNN models is considered now outdated.

Related GitHub Issues:

---

If you are writing a demonstration, please answer these questions to facilitate the marketing process.

• GOALS — Why are we working on this now?

  Eg. Promote a new PL feature or show a PL implementation of a recent paper.
  Show a working example of the founding models in QML for Phase Detection. Show how MPS and QML can be used together.

• AUDIENCE — Who is this for?

  Eg. Chemistry researchers, PL educators, beginners in quantum computing.
  Beginners in Quantum Computing, Chemistry researchers

• KEYWORDS — What words should be included in the marketing post?
  ANNNI, QCNN, QAD, Phase Detection

• Which of the following types of documentation is most similar to your file?
  (more details here)

• [ x] Tutorial
• Demo
• How-to

[sc-80710]

[SaverioMonaco]

• The image annni_sketch.png is a rough sketch for the thumbnail. It needs to be updated by replacing $J2$ with $\kappa$ and $J$ with $J_1$ to ensure consistency with the Hamiltonian described in Eq. (1).

• The other two images, annni_pd.png and annni_pd_analytical.png, were originally generated using matplotlib.pyplot. I imported them as images to reduce code volume. I will add the code to reproduce these images shortly.

[SaverioMonaco]

• annni_pd.png

import numpy as np
import matplotlib.pyplot as plt
from matplotlib.colors import ListedColormap, BoundaryNorm

# Kosterlitz-Thouless transition line
def kt_transition(k):
    return 1.05 * np.sqrt((k - 0.5) * (k - 0.1))

# Ising transition line
def ising_transition(k):
    return np.where(k == 0, 1, (1 - k) * (1 - np.sqrt((1 - 3 * k + 4 * k**2) / (1 - k))) / np.maximum(k, 1e-9))

# Floating Phase transition line
def bkt_transition(k):
    return 1.05 * (k - 0.5)
    
# Get the phase from the DMRG transition lines
def get_phase(k, h):
    if k < .5 and h < ising_transition(k):
        return 0
    elif k > .5 and h < kt_transition(k):
        return 1
    return 2

# Generate the phase diagram data from DMRG transition lines
ks100, hs100 = np.linspace(0, 1, 100), np.linspace(0, 2, 100)
K100, H100 = np.meshgrid(ks100, hs100)
img_dmrg = np.vectorize(get_phase)(K100, H100)

colors = ['#80bfff', '#fff2a8',  '#80f090', '#da8080',]
phase_labels = ["Ferromagnetic", "Antiphase", "Paramagnetic", "Trash Class",]
cmap = ListedColormap(colors)

bounds = [-0.5, 0.5, 1.5, 2.5, 3.5]
norm = BoundaryNorm(bounds, cmap.N)

# Plot the phase diagram
plt.figure(figsize=(4, 4))
plt.imshow(img_dmrg, cmap=cmap, aspect="auto", origin="lower", extent=[0, 1, 0, 2], norm=norm)

# Plot the transition lines.
k_vals1 = np.linspace(0.0, 0.5, 50)
k_vals2 = np.linspace(0.5, 1.0, 50)
plt.plot(k_vals1, ising_transition(k_vals1), 'k', lw=2)
plt.plot(k_vals2, kt_transition(k_vals2), 'k', lw=2)
plt.plot(k_vals2, bkt_transition(k_vals2), 'k', ls = '--', lw=2)

# Create legend entries
for color, phase in zip(colors, phase_labels[:-1]):
    plt.scatter([], [], color=color, label=phase, edgecolors='black')
plt.plot([], [], 'k', label='Transition Lines')
plt.text(0.15, 0.5, s=r'$h_I$', fontsize=12)
plt.text(0.70, 0.5, s=r'$h_C$', fontsize=12)
plt.text(0.8, 0.2, s=r'$h_{BKT}$', fontsize=12)

plt.legend(), plt.xlabel("k"), plt.ylabel("h"), plt.title("Phase diagram"), plt.show();

• annni_pd_analytical.png

import numpy as np
import matplotlib.pyplot as plt
from matplotlib.colors import ListedColormap, BoundaryNorm

# Kosterlitz-Thouless transition line
def kt_transition(k):
    return 1.05 * np.sqrt((k - 0.5) * (k - 0.1))

# Ising transition line
def ising_transition(k):
    return np.where(k == 0, 1, (1 - k) * (1 - np.sqrt((1 - 3 * k + 4 * k**2) / (1 - k))) / np.maximum(k, 1e-9))

# Floating Phase transition line
def bkt_transition(k):
    return 1.05 * (k - 0.5)
    
# Get the phase from the DMRG transition lines
def get_phase(k, h):
    if k < .5 and h < ising_transition(k):
        return 0
    elif k > .5 and h < kt_transition(k):
        return 1
    return 2

# Generate the phase diagram data from DMRG transition lines
ks100, hs100 = np.linspace(0, 1, 100), np.linspace(0, 2, 100)
K100, H100 = np.meshgrid(ks100, hs100)
img_dmrg = np.vectorize(get_phase)(K100, H100)

colors = ['#80bfff', '#fff2a8',  '#80f090', '#da8080',]
phase_labels = ["Ferromagnetic", "Antiphase", "Paramagnetic", "Trash Class",]
cmap = ListedColormap(colors)

bounds = [-0.5, 0.5, 1.5, 2.5, 3.5]
norm = BoundaryNorm(bounds, cmap.N)

# Plot the phase diagram
plt.figure(figsize=(4, 4))
plt.imshow(img_dmrg, cmap=cmap, aspect="auto", origin="lower", extent=[0, 1, 0, 2], norm=norm)

# Plot the transition lines.
k_vals1 = np.linspace(0.0, 0.5, 50)
k_vals2 = np.linspace(0.5, 1.0, 50)
plt.plot(k_vals1, ising_transition(k_vals1), 'k', lw=2)
plt.plot(k_vals2, kt_transition(k_vals2), 'k', lw=2)
plt.plot(k_vals2, bkt_transition(k_vals2), 'k', ls = '--', lw=2)

# Create legend entries
for color, phase in zip(colors, phase_labels[:-1]):
    plt.scatter([], [], color=color, label=phase, edgecolors='black')
plt.plot([], [], 'k', label='Transition Lines')

# Highlight analytical points (k = 0 and h = 0)
plt.scatter(np.zeros_like(hs100), hs100, color='red', s=20, label="Analytical Points")  # y-axis points
plt.scatter(ks100, np.zeros_like(ks100), color='red', s=20)  # x-axis point

plt.text(0.15, 0.5, s=r'$h_I$', fontsize=12)
plt.text(0.70, 0.5, s=r'$h_C$', fontsize=12)
plt.text(0.8, 0.2, s=r'$h_{BKT}$', fontsize=12)

plt.legend(), plt.xlabel("k"), plt.ylabel("h"), plt.title("Phase diagram"), plt.show();

[github-actions]

Thank you for opening this pull request.

You can find the built site at this link.

Deployment Info:

• Pull Request ID: 1335
• Deployment SHA: b9b821b74315284c59892abafce6aece3c892ebf
  (The Deployment SHA refers to the latest commit hash the docs were built from)

Note: It may take several minutes for updates to this pull request to be reflected on the deployed site.

[SaverioMonaco]

Final touches:

• Acknowledgement to DESY
• Training of the QCNN with the full training set instead of sampling each epoch for a smoother loss curve
• Adapted hyperparameters of the training function of the QCNN

[SaverioMonaco]

Fix of a comment in the QAD training function

[CatalinaAlbornoz]

Amazing job @SaverioMonaco !!!
I'm excited to see this demo go live.