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Data and code for "Freeness Reined in by a Single Qubit"

Alexander Altland, Francisco Divi, Tobias Micklitz, Maedeh Rezaei

Published Dec 24, 2025• 1 citation
Publisher

Abstract

Abstract Free probability provides a framework for describing correlations between non-commuting observables in complex quantum systems whose Hilbert-space states follow maximum-entropy distributions. We examine the robustness of this framework under a minimal deviation from freeness: the coupling of a single ancilla qubit to a Haar-distributed quantum circuit of dimension $D_0\gg 1$. We find that, even in this setting, the correlation functions predicted by free probability theory receive corrections of order $O(1)$. These modifications persist at long times, when the dynamics of the coupled system is already ergodic. We trace their origin to non-uniformly distributed stationary quantum states, which we characterize analytically and confirm numerically. Data and code for Freeness Reined in by a Single Qubit Here, we provide the numerical code and precomputed Monte-Carlo data used to reproduce the spectral form factor (SFF) and two-point correlation function (CBA) figures in the companion manuscript "Freeness Reined in by a Single Qubit." The repository includes Precomputed Monte-Carlo data Spectral form factor data: estimates of $\overline{|⟨U^t⟩|²}$ for discrete times $t = 1,\dots,300$. Two-point correlator ("CBA") data: values of $\overline{⟨A B(t)⟩}$ stored as a complex array (the plots use the real part). Results are provided for multiple ancilla-environment coupling strengths $g \in \{0.5,\,0.6,\,0.8,\,1.0\}$, internally mapped to the rate $\gamma = g^{2}/2$. Default model size: an environment register of $6$ qubits plus one ancilla qubit, yielding a total Hilbert-space dimension $D = 128$. The numerics are performed using Python code included in the repository. A Jupyter notebook loads the data and reproduces all plots in publication-ready form. Files 01\_sff\_cba.py: Monte-Carlo simulation script (generates Data/*.pkl) 02\_plots.ipynb: plotting notebook (loads Data/*.pkl and generates figures) Data/: precomputed dataset sff\_cba\_multi\_g\_env6\_samples1000000.pkl Results/: output figures produced by the notebook: SFF\_multig\_env6\_samples1000000.(pdf|png) CBA\_env6\_samples1000000.(pdf|png) The Jupyter notebook automatically loads the data stored in the folder Data/ and writes the resulting figures to Results/.

DOI
10.5281/zenodo.18046231

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Abstract Free probability provides a framework for describing correlations between non-commuting observables in complex quantum systems whose Hilbert-space states follow maximum-entropy distributions. We examine the robustness of this framework under a minimal deviation from freeness: the coupling of a single ancilla qubit to a Haar-distributed quantum circuit of dimension D0≫1D_0\gg 1D0​≫1. We find that, even in this setting, the correlation functions predicted by free probability theory receive corrections of order O(1)O(1)O(1). These modifications persist at long times, when the dynamics of the coupled system is already ergodic. We trace their origin to non-uniformly distributed stationary quantum states, which we characterize analytically and confirm numerically. Data and code for Freeness Reined in by a Single Qubit Here, we provide the numerical code and precomputed Monte-Carlo data used to reproduce the spectral form factor (SFF) and two-point correlation function (CBA) figures in the companion manuscript "Freeness Reined in by a Single Qubit." The repository includes Precomputed Monte-Carlo data Spectral form factor data: estimates of ∣⟨Ut⟩∣2‾\overline{|⟨U^t⟩|²}∣⟨Ut⟩∣2​ for discrete times t=1,…,300t = 1,\dots,300t=1,…,300. Two-point correlator ("CBA") data: values of ⟨AB(t)⟩‾\overline{⟨A B(t)⟩}⟨AB(t)⟩​ stored as a complex array (the plots use the real part). Results are provided for multiple ancilla-environment coupling strengths g∈{0.5, 0.6, 0.8, 1.0}g \in \{0.5,\,0.6,\,0.8,\,1.0\}g∈{0.5,0.6,0.8,1.0}, internally mapped to the rate γ=g2/2\gamma = g^{2}/2γ=g2/2. Default model size: an environment register of 666 qubits plus one ancilla qubit, yielding a total Hilbert-space dimension D=128D = 128D=128. The numerics are performed using Python code included in the repository. A Jupyter notebook loads the data and reproduces all plots in publication-ready form. Files 01_sff_cba.py: Monte-Carlo simulation script (generates Data/.pkl) 02_plots.ipynb: plotting notebook (loads Data/.pkl and generates figures) Data/: precomputed dataset sff_cba_multi_g_env6_samples1000000.pkl Results/: output figures produced by the notebook: SFF_multig_env6_samples1000000.(pdf|png) CBA_env6_samples1000000.(pdf|png) The Jupyter notebook automatically loads the data stored in the folder Data/ and writes the resulting figures to Results/.
This archive contains the numerical data and scripts used to generate the figures for the paper “Energy-selective quantum search with Ising Hamiltonian phase oracles”. The main numerical dataset is used for the scaling analysis of the correlation-induced resonance displacement φ₀ in the zero-field random Ising ensemble. The simulations use dimensionless units with σ_J = 1 and the Hamiltonian H(s) = 2 sum_{i<j} J_ij s_i s_j, where the couplings are independent Gaussian random variables, J_ij ~ N(0, 1), and h_i = 0. For each random Ising instance, the archive stores the truncated characteristic-function coefficients g_m = 2^{-n} Tr exp(-i m T H), m = 0, ..., 15. These coefficients are used to extract the local displacement φ₀ from the zero of the imaginary part of the response function C(φ) = 2 sum_{m=0}^{15} g_m (-1)^m exp(i m φ) - 1. The quantity plotted in the paper is the disorder-averaged second moment <φ₀²> and the scaled combination n⁶<φ₀²>. The Monte Carlo average is over independent disorder realizations only. For each realization, the trace over spin configurations is evaluated deterministically. The data cover even system sizes n = 18, 20, 22, ..., 36. For the largest systems, the computation splits the spins into two equal halves, performs GPU block contractions, and uses the zero-field spin-flip symmetry H(s)=H(-s) to reduce the sum by a factor of two. Archive contents - scripts/01_generate_phi0_coefficients.py Generates the characteristic-function coefficient files g_m for the random Ising ensemble. - scripts/02_analyze_phi0_scaling.py Computes φ₀, bootstrap confidence intervals, the finite-n covariance-sum comparison, and the scaling plots. - scripts/03_make_gaussian_amplification.py Generates the Gaussian amplification figure used as Fig. 1 in the paper. - data/phi0_coefficients/ Directory for the NumPy .npz files containing the characteristic-function coefficients. The production files have names of the form Nq_<n>_K_<K>.npz. The .npz files are saved without pickle objects. They can be read with standard NumPy using np.load. Typical usage To regenerate the Gaussian amplification figure: python scripts/03_make_gaussian_amplification.py --output-dir output/fig1_gaussian To analyze the φ₀ scaling data: python scripts/02_analyze_phi0_scaling.py \ --data-dir data/phi0_coefficients \ --output-dir output/phi0_scaling To run a short test calculation of the characteristic-function coefficients: python scripts/01_generate_phi0_coefficients.py \ --smoke-test \ --output-dir output/smoke_test_coefficients To regenerate the full production dataset on a CUDA-capable GPU: python scripts/01_generate_phi0_coefficients.py \ --production \ --output-dir data/phi0_coefficients The production calculations were performed using CuPy on an NVIDIA GPU. The analysis and plotting scripts use NumPy, pandas, and Matplotlib.
Python notebooks reproducing the numerical results and figures of the manuscript "Qubit encodings in the p-orbital-valley spectrum for enhanced coherence and tunable two-qubit interaction" by John H. Caporaletti and J. P. Kestner (University of Maryland Baltimore County). The manuscript proposes the pOv qubit, an encoding in the two-level subspace spanned by the lowest p-orbital state in the excited valley and the excited p-orbital in the ground valley of an anisotropic silicon quantum dot, coupled by disorder-induced valley-orbit coupling (VOC). This archive contains three self-contained notebooks: VOC_std_calc.ipynb — evaluates the VOC statistics of Appendix VI A using the discretized effective-mass theory of alloy disorder: the vertical envelope function is obtained from a finite-difference solve of the heterostructure confinement potential, and the standard deviations of the disorder-induced generator coefficients, together with the mean valley splitting and mean VOC magnitude, follow from the lattice sum over independent cells. The correlation matrix of the VOC generators is also reported. fig6_Qeff_hist.ipynb — reproduces Fig. 6. For each of 10410^4104 alloy-disorder realizations it finds the pair of dot orientations that switches off the two-qubit coupling Omega_zz while maximizing the effective two-qubit quality factor Q_eff, and histograms the result. fig7_inf_scatter_plot.ipynb — reproduces Fig. 7. It synthesizes the R_ZZ(π\piπ/2) entangling gate for each disorder realization and evaluates its infidelity against a Monte-Carlo ensemble of quasistatic two-level-fluctuator dipole configurations, comparing a single-segment analytic pulse with a three-segment composite pulse obtained from a two-stage optimization (L-BFGS-B on the deterministic infidelity with analytic gradients, then SLSQP minimizing total gate time subject to a deterministic-infidelity budget). All notebooks use the device parameters of Table I, run top to bottom with no input files, and require only NumPy, SciPy, and Matplotlib (developed under Python 3.12). All random draws are seeded, so the published figures reproduce exactly.
Free probability provides a framework for describing correlations between non-commuting observables in complex quantum systems whose Hilbert-space states follow maximum-entropy distributions. We examine the robustness of this framework under a minimal deviation from freeness: the coupling of a single ancilla qubit to a Haar-distributed quantum circuit of dimension D0≫1D0 \gg 1D0≫1. We find that, even in this setting, the correlation functions predicted by free probability theory receive corrections of order O(1)O(1)O(1). These modifications persist at long times, when the dynamics of the coupled system is already ergodic. We trace their origin to non-uniformly distributed stationary quantum states, which we characterize analytically and confirm numerically.
This repository contains the numerical data and computational codes supporting the results presented in the manuscript: "Engineered non-Gaussian Coherence as a Thermodynamic Resource for Quantum Batteries". The archive (data_QB.zip) includes all datasets, Jupyter notebooks, and figure files required to reproduce the results reported in both the main text and the Supplemental Material. Each figure is organised in a separate directory containing the corresponding data (.txt), code (.ipynb), and generated plots (.png/.pdf). The provided materials are sufficient to reproduce all figures and results without restriction. Detailed instructions and directory structure are described in the accompanying README.txt file. This dataset is made openly available to ensure transparency and reproducibility of the reported results.
The role of entanglement and quantum correlations in complex physical systems and quantum information processing devices has become a topic of intense study in the past two decades. In this work we present tools for learning about entanglement and quantum correlations in dynamical systems where the quantum states are mixed and the eigenvalue spectrum is highly degenerate. We apply these results to the deterministic quantum computing with one qubit (DQC1) computation model and show that the states generated in a DQC1 circuit have an eigenvalue structure that makes them difficult to entangle, even when they are relatively far from the completely mixed state. Our results strengthen the conjecture that it may be possible to find quantum algorithms that do not generate entanglement and yet still have an exponential advantage over their classical counterparts.
Shortcuts to adiabaticity with a quantum control field — Code & data========================================================== This repository contains a Jupyter notebook and supporting data files (csv) to reproduce the numerical results and figures in our manuscript (arXiv 2508.06304), which investigates adiabatic shortcuts mediated by a quantum control field.The code implements a Landau–Zener two-level system coupled to a spectator qubit, time evolution of the full system (LZ qubit + spectator) via QuTiP, parameter sweeps (parallelized with joblib), and plotting utilities for the manuscript figures. Related publication----------------------------This code is complementary to the preprint: https://arxiv.org/abs/2508.06304. Requirements---------------------The simulations were developed for and tested on Python 3.9+. Recommended packages and minimum versions: - Python >= 3.9- numpy >= 1.22- scipy >= 1.8- matplotlib >= 3.5- qutip >= 4.6 (install a recent stable qutip; some installations require `pip install qutip` or `conda` channels)- joblib >= 1.2- tqdm >= 4.60- pandas (optional, for CSV handling)- jupyterlab / notebook (for interactive use) Running the notebook (interactive) Open the notebook with JupyterLab or Jupyter Notebook. Execute cells sequentially in the order indicated by the notebook outline.The sections “Definitions” and “Figure imports and setup” must be executed before generating any data or figures, as they define all operators, and helper functions. The notebook is organized into separate sections for each figure of the manuscript, allowing you to re-run only the figure(s) of interest. Note: Figure 4 (Density plot) is configured to load precomputed data files to reduce computational cost. To reproduce Figure 4 without recomputing the full dataset, ensure the required .csv files are downloaded and located in the same directory as the notebook. To explore different parameter regimes: Modify the relevant parameter definitions (e.g., anneal_window, g, epsilon, omega_c, x0, x0_vals, q_vals). To include dissipation of the spectator (ancilla) qubit, set the parameter kappa> 0. For questions about reproducing the figures or about parameter choices used in the paper, contact the corresponding authors listed in the preprint. If you reuse this data/code, please cite the associated preprint and this code archive.