Neural Quantum States for 1D Spin Systems

Research code and exploratory notebooks for neural quantum states, matrix-product states, conditional mutual information (CMI), and one-dimensional quantum spin models.

The main focus is the rotated 1D cluster-state family, with the transverse-field Ising model (TFIM) used as a reference problem. This is the code behind my M.Sc. research at the University of Waterloo and Perimeter Institute; a manuscript is in preparation.

Result: measurement-induced correlations in the cluster state

Measure every qubit of the 1D cluster state in a basis rotated by an angle $\theta$ about the $y$ axis, and ask how much the two ends of a block still share once the $m$ sites between them are known: $I_m = I(X_0 : X_{m+1} \mid X_1, \dots, X_m)$.

The CMI is exactly zero in the $Z$ basis ($\theta = 0$) and the $X$ basis ($\theta = \pi/2$), where the outcomes are independent uniform bits. In between it decays exponentially in $m$, and the decay slows by about two orders of magnitude as $\theta$ approaches $\pi/2$, where odd and even $m$ also separate. The figure comes from cluster_mps_cmi_length_scan.py, which computes the probabilities exactly from the bond-dimension-2 MPS, with no sampling:

python scripts/cluster_state/cluster_mps_cmi_length_scan.py --max-m 12 --n-theta 31 --fit-max 10

Repository structure

nqs_project/
├── README.md
├── LICENSE
├── requirements.txt
├── CITATION.cff
├── docs/                 # figures used in this README
├── tests/                # pytest checks for the CMI script
├── notebooks/
│   ├── cluster_state/
│   │   ├── mbqc.ipynb
│   │   ├── rcs_mps_dmrg.ipynb
│   │   └── netket_rbm_exploratory.ipynb
│   └── tfim/
│       └── tfim_rnn_vmc.ipynb
└── scripts/
    └── cluster_state/
        ├── cluster_mps_cmi_length_scan.py
        └── hes_mps_cmi_length.py

Contents

Cluster-state notebooks

• mbqc.ipynb — small PennyLane/NetworkX exploration of cluster states and rotated-basis measurement probabilities.
• rcs_mps_dmrg.ipynb — MPS/DMRG calculations for the rotated cluster-state Hamiltonian using TeNPy.
• netket_rbm_exploratory.ipynb — exploratory NetKet RBM/VMC calculations together with exact-diagonalization and CMI diagnostics.

TFIM notebook

• tfim_rnn_vmc.ipynb — exact diagonalization, measurement distributions, CMI, autoregressive RNN learning, and an RNN-VMC prototype for the 1D TFIM.

Reproducible scripts

• cluster_mps_cmi_length_scan.py — exact bond-dimension-2 cluster-state MPS calculation of CMI decay versus measurement angle.
• hes_mps_cmi_length.py — exact MPS calculation of endpoint/local CMI for the entanglement-swapping stabilizer Hamiltonian.

Setup

The two scripts need only NumPy and Matplotlib. The notebooks need the full stack in requirements.txt (NetKet, JAX, PyTorch, TeNPy, PennyLane).

python -m venv .venv
source .venv/bin/activate  # Windows: .venv\Scripts\activate
pip install -r requirements.txt   # or: pip install numpy matplotlib pytest

Then start JupyterLab:

jupyter lab

Example script usage

python scripts/cluster_state/cluster_mps_cmi_length_scan.py --max-m 12 --n-theta 21
python scripts/cluster_state/hes_mps_cmi_length.py --mode endpoint --max-m 18 --n-theta 16

Tests

pytest tests

The tests compare the transfer-matrix probabilities and CMI with a brute-force state-vector calculation on an explicit chain (agreement to 1e-12), and check normalization and the vanishing CMI in the $Z$ and $X$ bases. CI runs them and a short version of both scripts on every push.

Generated CSV/PNG outputs are intentionally excluded from Git by .gitignore so that the repository stays code-focused.

Notes

The notebooks are exploratory research records rather than a stable Python package. The standalone scripts are the cleaner entry points for reproducible CMI calculations.

Citation

If this code is useful to you, please cite it using CITATION.cff.