Detection of Quantum Phase Transitions in the ANNNI (Frustrated TFIM) Model

A reproduction and extension, in Qiskit, of the VQE methodology from:

K. Lively, T. Bode, J. Szangolies, J.-X. Zhu, B. Fauseweh, "Noise robust detection of quantum phase transitions," Phys. Rev. Research 6, 043254 (2024). https://doi.org/10.1103/PhysRevResearch.6.043254

This is a reproduction/extension study, not novel research. The physical model, the phase-transition detection method (Hellmann-Feynman derivative of the energy), and the overall experimental logic (classically-optimized parameters, executed once on noisy hardware) all belong to the paper above. Attempt to implement on real hardware failed due to failing IBM account verification (their end) and no funds available to buy access plan.

What's here

File	What it does	Depends on
ANNNI_Ideal.py	Main entry point. Sweeps J2/J1 = 0.40-0.60, optimizing a VQE ansatz against a noiseless statevector simulator for each point. Produces the "ideal simulation" reference data.	Nothing (run first)
ANNNI_Noisy.py	Takes the already-optimized parameters from the file above, binds them as fixed circuits, and measures them under a realistic IBM device noise model (no re-optimization see "Why no optimization on hardware" below).	Output of ANNNI_Ideal.py

Scripts must be run in that order each one after the first reads a .npz file the previous one writes. This isn't automatic just because the files are in one repository; see "Running on your own machine" below.

Data files included

• annni_N12_h0.1_periodic_reps2_ideal.npz ideal (noiseless) simulation results: energy, dE/dJ2, and the full optimized parameter set for all 21 points, against both VQE and exact diagonalization.
• annni_N12_h0.1_periodic_reps2_noisy.npz the same 21 points executed under simulated IBM device noise (FakeSherbrooke).
• annni_three_way_comparison.png
• annni_scan_ideal.png

These are included so the results can be inspected or replotted without re-running.

Findings

1. Warm-starting across a phase transition can silently converge to the wrong state. Continuing a J2-sweep's optimizer from the previous point's parameters works well within a phase, but can get trapped describing the old phase's local minimum after crossing a transition.

2. Optimizer tolerances can cause silent zero-iteration "convergence." With loose tolerances, L-BFGS-B can accept a warm-started point whose gradient already looks small for the new Hamiltonian and terminate in nit=0 real steps silently reusing the previous point's answer rather than re-optimizing. Confirmed by instrumenting the optimizer directly; fixed by tightening convergence tolerances.

Why no optimization on real/noisy hardware

ANNNI_Noisy.py and annni_hardware_execution.py(N/A) never re-optimize circuit parameters they only measure fixed, already-optimal ones. reason:

• The exact-gradient method used for ideal-simulation optimization (ReverseEstimatorGradient) works by inspecting the classical simulator's internal statevector a real, noisy-simulated backend never exposes this, so it cannot be used outside noiseless simulation.

This mirrors the parent paper's own methodology: all parameter optimization is classical and offline; hardware is used only to execute and measure.

Setup

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

Tested against the exact versions pinned in requirements.txt; Qiskit's API has changed across versions before (e.g. EfficientSU2 is deprecated as of Qiskit 2.1 in favor of the efficient_su2 function), so an unpinned pip install qiskit some months from now may not run this code unmodified.

Running on your own machine

python ANNNI_Ideal.py      # ~20-25 min; produces the _exactgrad.npz file
python ANNNI_Noisy.py       # ~25-30 min; noisy simulation is slow per point.
                                    #   Checkpointed  if interrupted, run again to resume.
python plot_three_way.py           # seconds; produces annni_three_way_comparison.png (this is the last cell of ANNNI_Noisy.py)

Run all three from the same directory — they use relative filenames, no path configuration needed.

Status: real-hardware execution is implemented and calibration-tested in simulation, but not yet run against real hardware pending IBM Open Plan account verification.

Repository does not include

Note: This reproduction covers ground-state energy and its derivative (dE/dJ2) only.

annni_hardware_execution.py (failed IBM account verififcation ) Earlier, superseded versions of the sweep script (single-blind-reset and targeted-reset-point variants, both replaced by the multi-start strategy in ANNNI_Ideal.py for the reasons in "Findings" above) and an earlier plotting script (plot_annni_scan.py v1, superseded by plot_three_way.py) were intentionally left out to keep the repository to its final, working state rather than its debugging history. Also It does not include TFIM 1D VQE nor the similar study of frustrated TFIM for N=4, 8, 12 ,16 using QuSpin.

These frustrated Ising-type models have been more successfully studied on trapped-ion systems, given their all-to-all connectivity better suits the model's interactions, one such example (Kirmani et al., 2025). Anyone looking to work in this direction would benefit from exploring those modalities. Open to discussion and comments.