Double-Bracket Algorithmic Cooling (dbac) package

Code accompanying the paper Double-Bracket Algorithmic Cooling (M. Alghadeer, K. U. Giang, S. Cao, S. D. Fasciati, M. Piscitelli, N. Ng, P. J. Leek, M. Gluza, M. Bakr).

Background

Algorithmic cooling lowers the entropy of a qubit using only local operations and a "colder" ancilla, and is the basis of established techniques like heat-bath algorithmic cooling (HBAC). This paper introduces double-bracket algorithmic cooling (DBAC), which instead removes coherence from an already-pure qubit state, extending algorithmic cooling from a thermodynamic (entropy) notion of cooling to a genuinely quantum one.

DBAC simulates quantum imaginary-time evolution (QITE) by recursively synthesizing unitaries that implement Riemannian steepest-descent ("double-bracket") flows towards the ground state, one small step at a time. Each step needs an extra copy of the current state to act as a quantum instruction: that copy is consumed via density matrix exponentiation (DME), a protocol that uses repeated partial SWAPs to approximate the unitary $e^{-it\rho}$ generated by a state $\rho$, without ever synthesizing $\rho$ as a gate. Because the resulting circuit does not depend on the input state and needs no measurements or feedback, cooling performance is governed entirely by how many quantum instructions (copies of the state) are available -- a trade-off the paper relates to the Nernst principle ("cooling to absolute zero costs infinite resources").

Repository layout

src/dbac/          Physics and numerics: DME, DB-QITE/GCI recursion, HBAC, state utilities.
src/dbac/plotting/ All plotting code, factored out of the physics and shared across scripts/notebooks.
scripts/           The three DBAC scripts (computation + plotting), runnable standalone.
notebooks/ Narrated walkthroughs that call scripts/ and dbac.plotting and explain the physics.
figures/           All generated figures (PDF).
docs/              Sphinx documentation (`pip install -e ".[docs]" && make -C docs html`).

src/dbac/ -- physics and numerics (no plotting)

src/dbac/plotting/ -- all plotting

scripts/ -- one script per result, runnable from the command line

The three DBAC scripts do the computation (calling into dbac), calls dbac.plotting to plot it, and saves the resulting figures to figures/:

pip install -e .
python scripts/sup_figure_S2.py      # DME channel implementation benchmark
python scripts/sup_figure_S4.py       # DBAC with ideal (M -> infinity) DME, sweep over k
python scripts/figure_4.py           # Main result: cooling vs. number of quantum instructions (Fig. 4)
python scripts/sup_figure_S5.py       # Bloch-sphere illustration of the DBAC recursion

notebooks/ -- narrated walkthroughs

notebooks/not_in_use/ holds earlier exploratory notebooks kept for reference; they are not part of the maintained code path.

Setup

python3 -m venv .venv
source .venv/bin/activate
pip install -e ".[notebooks]"
jupyter notebook notebooks/

Figures that use LaTeX-rendered labels require a local LaTeX installation (matplotlib's text.usetex); without one, either install texlive or disable it via dbac.plotting.use_paper_style(use_tex=False).

Citation

If you use this code, please cite:

@article{alghadeer2025dbac,
  title   = {Double-Bracket Algorithmic Cooling},
  author  = {Alghadeer, Mohammed and Giang, Khanh Uyen and Cao, Shuxiang and
             Fasciati, Simone D. and Piscitelli, Michele and Ng, Nelly and
             Leek, Peter J. and Gluza, Marek and Bakr, Mustafa},
  journal = {arXiv preprint arXiv:2510.00302},
  year    = {2025}
}

Notebooks only save their plots when SAVE_PLOTS = True in notebooks/config.py (off by default); PDFs then go to figures/.