Static Bias Controllers for XX Spin-1/2 Rings

SPDX-FileCopyrightText: Copyright (C) 2020-2023 SM Shermer lw1660@gmail.com
SPDX-FileCopyrightText: Copyright (C) 2020-2023 Frank C Langbein frank@langbein.org, Cardiff University
SPDX-FileCopyrightText: Copyright (C) 2022-2023 Sean Patrick O'Neil seanonei@usc.edu
SPDX-License-Identifier: CC-BY-SA-4.0

Propagation of information encoded in spin degrees of freedom through networks of coupled spins enables important applications in spintronics and quantum information processing. This data set contains the results of applying optimal control of information propagation in networks of spin-1/2 particles with uniform nearest neighbor XX-couplings forming a ring with a single excitation in the network as simple prototype of a router for spin-based information. The control is implemented via spatially distributed potentials, which remain constant during information transfer. The limited degrees of freedom makes finding a control that maximizes the transfer probability in a short time difficult.

It had been originally computed for

[1] FC Langbein, SG Schirmer, EA Jonckheere. Time optimal information transfer in spintronics networks. IEEE Conf Decision and Control, pp. 6454-6459, 2015. [DOI:10.1109/CDC.2015.7403236] [arXiv:1508.00928] [PDF:paper] [Details]

For further details see

[2] SG Schirmer, EA Jonckheere, FC Langbein. Design of Feedback Control Laws for Information Transfer in Spintronics Networks. IEEE Trans Automatic Control, 63(8):2523-2536, 2018. [DOI:10.1109/TAC.2017.2777187] [arXiv:1607.05294] [PDF:paper] [Details]

[3] EA Jonckheere, SG Schirmer, FC Langbein. Jonckheere-Terpstra test for nonclassical error versus log-sensitivity relationship of quantum spin network controllers. Int J Robust and Nonlinear Control, 28(6):2383-2403, 2018. [DOI:10.1002/rnc.4022] [arXiv:1612.02784] [PDF:paper] [Details]

This repository is mirrored at https://github.com/qyber-black/Data-Static-Bias-Controllers-XX-Spin-Rings.

Versions

Version 2.0.1

S O'Neil, FC Langbein, SG Schirmer. Static Bias Controllers for XX Spin-1/2 Rings, Version 2.0.1. Data set, 2023. [Github: https://github.com/qyber-black/Data-Static-Bias-Controllers-XX-Spin-Rings/releases/tag/v2.0.1] [Qyber\black: https://qyber.black/spinnet/data-static-bias-controllers-xx-spin-rings/-/tree/v2.0.0]

Update of t and dt controllers for spin rings N=1..20. This has been created/checked for completeness by running matlab/generate_all.m

Version 2.0.0

S O'Neil, FC Langbein, SG Schirmer. Static Bias Controllers for XX Spin-1/2 Rings, Version 2.0.0. Data set, 2022. [Github: https://github.com/qyber-black/Data-Static-Bias-Controllers-XX-Spin-Rings/releases/tag/v2.0.0] [Qyber\black: https://qyber.black/spinnet/data-static-bias-controllers-xx-spin-rings/-/tree/v2.0.0]

Published here:

FC Langbein, S O'Neil, SG Schirmer. Energy landscape controllers for quantum state transfer in spin-1/2 networks with ring topology, Version 1, Research Directions, Cambridge Open Engage, Dec 02, 2022. [DOI:10.33774/coe-2022-35xgg]

Version 1.0.0

FC Langbein, SG Schirmer, EA Jonckheere. Static Bias Controllers for XX Spin-1/2 Rings, Version 1.0.0. Data set, 2022. [Github: https://github.com/qyber-black/Data-Static-Bias-Controllers-XX-Spin-Rings/releases/tag/v1.0.0] [Qyber\black: https://qyber.black/spinnet/data-static-bias-controllers-xx-spin-rings/-/tree/v1.0.0]

The original version has been published as

FC Langbein, SG Schirmer, EA Jonckheere. Static Bias Controllers for XX Spin-1/2 Rings. Data set, figshare, 3rd July 2016. [DOI:10.6084/m9.figshare.3485240.v1] [Details]

and is also available as [FeedbackControlLaws20160703.zip].

Data files

The controller dataset is contained in the directory data. A summary of the controller dataset containing the core information is available as a csv file which can be downloaded with the published data set at:

https://www.cambridge.org/engage/coe/article-details/63c9eda607308a0e4c9f89a9?show=item

The controller data files, including details about the optimization parameters and additional information, are available in Matlab mat v7.3 format (created with Matlab 2021b) in data/data_bias_control. The naming convention is

data_bias_control_D-N-M.mat for transfer
data_localization_N.mat for localization on spin 1

where

D - indicates type of readout at target time:
  t  - controllers for readout at specific time,
  dt - controllers for average readout over time window
  dtp - controllers subject to readout time uncertainty
N - number of spins in the network from 3 to 20+;
M - information transfer from spin |1> to |M> for M = 2,...,ceil(N/2).

Each data file contains the following variables/structures. Info is a structure containing information about the optimization options. The information of interest is mostly in Info.args which has subfields:

obj - matspinnet object (if available)
in  - input spin (usually 1 for ring)
out - output spin (between 1 and N)
T   - constraints on transfer time (upper limit), NaN if none
B   - constraints on bias field strength (upper limit), NaN if none
readout - vector [0.1000*pi 0 0]
min_err - optional error threshold such as 0.0100
repeats - number optimization repeats
symm    - 1 to impose symmetry constraints, 0 no constraints
initT   - initial range of transfer times, e.g., [500 5000]
bias_init - initial bias values
noise   - 0 if none added

Results is a cell array of optimisation results, with each cell containing the following fields:

err       - error/infidelity of controller
exec_time - time it took to find controller with L-BFGS optimiser, on single core Xeon E7 2.7GHz
exec_flag - exit flag of matlab optimiser
output    - output of matlab optimiser
bias      - vector of biases on spins
init_bias - initial value for bias
time      - readout time at spin M (with time window time+/-info.dt/2)
init_time - initial value for time

In addition there are two variables

best    - index for results of best controller
fastest - index for results of fastest controller (with error smaller than info.fastest_min_err).

Figures

The data is visualized in the following figures in PNG and Matlab figure files (generated with Matlab 2021b) and saved in the /figures/ directory

bias_D-N-M.{fig,png}

These figures reside in the figures/core directory. These depict results for optimizing the information transfer probability from spin 1 to M for a ring of N spins with XXZ coupling in the single excitation subspace over the spatial biases and time. The left column shows the biases and evolution (in blue vs. the natural evolution in red) giving the best fidelity at time T or over a window T +/- dt/2 and the best error. The middle columns shows the fastest solution found with a fidelity greater than fastest_min at time T. The right column shows the overall solutions found by repeated optimization, plotting the time vs. the logarithm of the infidelity and a histogram of the logarithm of the infidelity. The bottom row shows the eigenstructure of the best and the fastest solution and their symmetries, with the eigenvectors being the columns of the matrices (in cyan, green and red rows indicate the |in> and |out> state resp.) and the corresponding eigenvalues at the bottom (in purple).

fastest_D.{fig.png}

Also located in the figures/core directory, these display the shortest times achieved for instantaneous transition fidelities greater than 0.999 for D=t or 0.99 for D=dt for rings of size N = 3, ..., 20 and transitions from 1 to M = 2, ..., ceil(N/2). Note that for some transitions no solution with high fidelity were found, so no fastest results are reported (see data files). The color of the bars indicate the infidelity of the fastest solution.

sensitivity_D-N-M.{fig,png}

Logarithm of transfer probability (red) and logarithm of sensitivity (blue) to Hamiltonian uncertainty only (coupling uncertainty), ordered by increasing infidelity from left to right for the 1 -> M controllers of an N-ring. These figures are located in the figures/sensitivity directory.

data_bias_control_D-N-M-robustness.{fig,png} or data_localization_D-M-robustness.{fig,png}

These figures are located in the figures/robustness directory and display a plot of the error versus log-sensitivity to Hamiltonian and controller perturbations for each of the controllers included in the data/data_bias_control directory from N = 3:20 and M = {t,dt} .

log_sens_figure_D-N-M_S=P.fig

These figures are located in the figures/log-sensitivity/figures directory and depict log-sensitivity versus error for the entire controller set in a given transfer 1 -> M and for a particular perturbation P. For P less than or equal N, the perturbations are to the bias fields. For P between N+1 and 2N, the perturbations are to the spin couplings (Hamiltonian perturbations).

log_sens_composite_D-N-M.{fig,png}

Located in the figures/log-sensitivity/composite directory, these figures provide a log-log plot of the log-sensitivity of the fidelty error to Hamiltonian perturbations and controller perturbations versus the fidelity error. The data in these plots is filtered for controllers with a fidelity greater than 0.9. The blue crosses depict the norm of the N perturbations to the couplings, and the red asteriks depcit the norm of the N bias-field pertrubations.

log_sens_composite_dt_N-M_comparison{fig,png}

Located in /figures/log-sensitivity/comparison/ , these figures display a log-log plot of the norm of the log-sensitivity, log-sensitivity to controller uncertainty, and log-sensitivity to Hamiltonian uncertainty versus error to provide a visual depiction of the correlation data

Dataset generation

The dataset was generated using the create_data_set.m function in the the directory matlab. This data was generated using https://qyber.black/spinnet/code-matspinnet and the matlab functions in the directory matlab.

Analysis

The following routines are available in the matlab directory to calculate (log-)sensitivity, analyze and visualize the data.

generate_all.m

This script calls on the create_data_set.m routine and routines below to produce the folder hierarchy, results, and figures in the published data set. If desired, the user can update the range and type of controller sets to use the production in each routine individually.

script_calc_robustness_all.m

This is the main routine for calculating robustness data (sensitivity and log-sensitivity) from the controller data files in the data/data_bias_control directory and in accordance with [3]. If the data/data_bias_control folder is not empty, the routine requires no input. It calls on calc_sensitivity.m for the computations and saves the files as data_bias_control_D-N-M-robustness.mat and data_localization_D-N-robustness.mat in the results/data_bias_control_robustness directory.

calc_sensitivity.m

This routine calculates the sensitivity and log-sensitivity to both Hamiltonian and controller perturbations. It takes as input the data file provided by script_calc_robustness_all.m and calls on the bias_sensitivity.m routine from the MatSpinNet package to calculate the sensitivity in accordance with [3]. It then uses this data to compute the log-sensitivity (without scaling by the nominal value of the uncertain parameter). It saves all computed robustness data in a variable ”robustness” and sends to script_calc_robustness_all.m for saving.

script_plot_log_sensitivity_all.m

This script is the main routine for plotting of robustness data from the results in the results/data_bias_control_robustness directory. It requires no input and is designed to run from the /matlab directory. It calls the plot_logSens_vs_error.m routine to generate plots from the data in results/data_bias_control_robustness directory. It saves the plots as MATLAB figures and accompanying .png files in the figures/robustness directory with filenames data_bias_control_D-N-M-robustness{fig.png} or data_localization_D-N-robustness{fig.png}.

plot_logSen_vs_error.m

This script produces a scatter plot of log-sensitivity to Hamiltonian and controller perturbations versus error based on the data files in the results/data_bias_control_robustness directory. The script is designed to run based on a call from the parent script_plot_log_sensitivity_all.m function.

analyze_log_sens_{t,dt}

This script takes as input the controller data in the data_bias_control_D_N-M.mat files and the robustness data from the calc_sensitivity.m routine saved in the data_bias_control_D_N-M-robustness or data_localization_D-N-robustness files and computes the correlation between the error and log-sensitivity along with figures depicting the relation. The correlation data is saved as Corrleation_Data.xlsx in the /results/log_sens_results/ directory. The figures are saved as log_sens_composite_dt_N-M_comparison{fig,png} in the figures/log-sensitivity/comparison/ directory.

calc_log_sens.m

This is the main routine for calculation of the log-sensitivity of the error to Hamiltonian and bias field perturbations using an updated analytical formula of the Bloch-transformed system. This computation scales the quotient of the sensitivity and error by the nominal value of the uncertain parameter. It requires no user input but takes the controller and transfer time data from the data_bias_control_D-N-M.ma or data_localisation_D-N.mat data files and computes the fidelity error, sensitivity of the error to parameter variation, and log-sensitivity of the error to parameter variations. On output the routine saves the data as log_sens_data_D-N-M in the results/log_sens_results directory. It also creates and saves .fig files for the log-sensitivity versus error in the figures/log-sensitivity/figures directory. Finally, it also creates .fig and .png plots of the log-sensitivity versus fidelity error for high-fidelity controllers (those with and error less than 0.1) and saves them in the figures/log-sensitivity/composite directory.

calc_error_dt.m

This is an optional file for producing the fidelity error of time-windowed controllers using a formula that corrects for the normalization of the sinc argument by MATLAB.

compare_err.m

An optional analysis file for computing checking the fidelity error computations for instantaneous read-out controllers.