Operator-Level Quantum Acceleration of Non-Logconcave Sampling

This repository provides three self-contained folders—fig4/, fig5/, and fig6/—for reproducing the figures from Operator-Level Quantum Acceleration of Non-Logconcave Sampling (PDF).

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fig4/ — MALA vs. Quantum-accelerated LD (with Gibbs) on Müller–Brown potential and panel visualization

Contents

• QSVT_filter_construction/ Helper routines used by the quantum-accelerated pipeline:

  • smoothStep.m
  • chebyshev_coeffs_fft.m
  • filter_total.m

• Q_accelerated_LD.m Runs the quantum-accelerated Langevin dynamics (QSVT-based pipeline). Saves to data/:

  • beta04QSVT.mat, beta06QSVT.mat, beta08QSVT.mat (variable: probability_final)
  • gibbs_04.mat, gibbs_06.mat, gibbs_08.mat (variable: true distribution gibbs_state)
  • All_Overlap_QSVT.mat (variable: All_Overlap, 1×3)

• MALA.m Runs MALA sampling for β ∈ {0.4, 0.6, 0.8}. Saves to data/:

  • beta04MALA.mat, beta06MALA.mat, beta08MALA.mat (variable: density_storage)
  • All_Overlap_MALA.mat (variable: All_Overlap, 1×3)

• draw_fig4.m Loads the above .mat files and draws the 3×3 panel (MALA / QSVT / Gibbs).

• data/ Created automatically after you run the MATLAB scripts.

How to run (order)

1. Open MATLAB and cd into fig4/.

2. Run Q_accelerated_LD.m (this creates the QSVT and Gibbs files in data/).

3. Run MALA.m (this creates the MALA files and All_Overlap_MALA.mat).

4. Run draw_fig4.m to generate the figure.

   • Tip: If parfor is unavailable, change it to for in the scripts.

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fig5/ — Inverse spectral gaps vs. β for different dynamics

Contents

• main_fig_5.m (MATLAB) Computes spectral gaps for a range of β and stores results only (no plotting). Saves to data/ (each file also contains the vector beta):

  • spectral_gap.mat (variable: spectral_gap)
  • spectral_gap_cla.mat (variable: spectral_gap_cla)
  • spectral_gap_witten_lap.mat (variable: spectral_gap_witten_lap)
  • spectral_gap_Lx_witten.mat (variable: spectral_gap_Lx_witten)

• draw_fig5.py (Python) Reads the .mat files under data/ and plots 1/Gap on a semilog-y axis, writing log-beta-gap.pdf.

  • Requires: numpy, matplotlib, scipy (and optionally LaTeX if usetex=True).

• log-beta-gap.pdf Produced by draw_fig5.py.

• data/ Output folder created by main_fig_5.m.

How to run (order)

1. Open MATLAB and cd into fig5/, then run:

   main_fig_5

   This will populate fig5/data/ with the four .mat files.

2. Run the Python plotter:

   # From the same folder (or anywhere):
   python draw_fig5.py

   The script saves log-beta-gap.pdf next to itself.

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fig6/ Lindbladian-based warm-start generation — (generate data + plot)

This folder contains scripts to (i) run the Lindblad dynamics for chosen inverse temperatures β and save the time–overlap data, and (ii) generate the final figure from the saved outputs.

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Contents

• main1_fig6.py (Python)
  Runs the simulation for a specified β, saves overlap data and (optionally) a quick diagnostic plot.
  You typically run this script multiple times, changing β each time.

• figure_beta_git.py (Python)
  Reads the saved results produced by main1_fig6.py (for multiple β values) and generates the final figure for the paper / GitHub.

• *.pdf (outputs)
  Final figure produced by figure_beta_git.py (filename depends on the script settings).

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Requirements

Install dependencies (recommended: use a virtual environment):

pip install numpy scipy matplotlib h5py