Protecting quantum states against loss

This work is licensed under CC BY 4.0 .

Requirements

All scripts run on Python 3.7 or newer. The following packages are required:

• numpy
• scipy
• mosek (+ license)
• sympy for the multinomial_coefficients_iterator function
• psutil for the exemplary command line programs, only used to get the physical CPU count

Content

The Optimizer...py files define a class Optimizer that is responsible for carrying out the optimization. The filename indicates the problem this corresponds to; this is detailed in the file header.

We additionally provide three command line applications:

• Main.py runs the optimizations for given parameters d, s, and r. Various configuration options are available, see the help.

  The output is stored in a subfolder named <d> <s> <r> according to the parameters; the files are named <pdist> <f> <type>.dat, where <pdist> is the distillation success probability and <f> the optimized fidelity. If <type> is choi, it contains the vectorized upper triangle of the Choi state of the distillation map (in the Dicke basis). If <type> is rho, it contains the vectorized upper triangle of the input density matrix (in the Dicke basis). If the --erasure parameter is specified, the optimization is done in the full computational basis and rho is replaced by psi; the file then contains the full input state vector. Additionally, the subfolder is suffixed with erasure.

• MainAllR.py runs the optimizations for given parameters d, s, and ptrans. Various configuration options are available, see the help.

  This program uses the data created by Main.py (without the --erasure option) as initial points and therefore can only be run afterwards. It explores the possibility to use different maps for various r values.

  The output is stored in a subfolder named <ptrans> <d> <s>; the files are named <ptot> <f> <type>.dat, where <ptot> now is the total success probability.

• MainStepwise.py runs the optimization using the iterative basis selection approach introduced in (Thesis link yet to come). Arguments are as before d, s, and r, as well as pdist. The search starts with two kets; this choice can be overwritten specifying --min=<larger number>.

Both applications are parallelized and by default use either the environment variable SLURM_NTASK (if the SLURM manager is used) or the number of physical cores available. This can be configured using the --workers option.

Generating the numerical data

The data that underlies the paper can be generated by appropriate calls of the Main.py program. Note that for qubits and low dimensions, this can easily be done on a personal computer, but qudits and larger values of s and r require substantial memory and time resources.

The actual commands may therefore depend on the scheduler. For example, with SLURM, a shell script Main.sh may be created that looks as follows:

#!/bin/sh
# potentially set up environment
python Main.py $1 ${SLURM_ARRAY_TASK_ID} $2

Then, a call such as sbatch --ntasks=<multithreading level> --array=2-10 --time=<time constraint> Main.sh 2 1 will queue jobs that create the folders for every configuration 2 2 1 until 2 10 1 with the corresponding numerical data. Note that the main application will automatically determine the number of threads from the appropriate SLURM environment variable. If a different scheduler is used, this value must be passed in the optional parameter -workers=<multithreading level>.

Using polynomial optimization for bounds

The Julia file qeccPolynomial.jl contains code to construct the polynomial optimization problems associated with this problem. Usage instructions are at the bottom of the file.