This repository contains code for the efficient computation scheme introduced in the manuscript [1]. This code can be used to compute phase-space representations of spin systems, such as symmetric qubits states. Using a set of precalculated transformation kernels, our approach computes Fourier decompositions of phase spaces and applies fast Fourier transforms to them. We provide example code in scripting languages MATLAB, Mathematica and Python which reads in precalculated kernels and computes phase spaces via fast Fourier transforms.

Precalculated kernels are provided in Calculated\Kernels up to dimension 30 (up to 29 qubits in a symmetric state). Kernels are also provided up to dimension 120 (up to 119 qubits in a symmetric state) as an additional compressed asset in the release Kernels_dim120.

For larger dimensions the scripts build.sh and build.bat can be used to compile and run the source code located in src, as discussed below. The C code src/Precalculate_Kernel.c is used to calculate and store the kernels which will be used by the MATLAB, Mathematica and Python scripts.

A simple example is provided in Matlab/example.m which calculates the Wigner function of a Schrödinger cat state, the well-knwon GHZ state, for a given dimension. This program uses the precalculated kernels located in Calculated/Kernels. Note that if the desired dimension is larger than 30, then the corresponding kernel needs to be downloaded from the release Kernels_dim120 and extracted to Calculated/Kernels. Alternatively, run the script build.sh or build.bat to compute the kernels with our C code.

Run the example by entering the directory Matlab/ and type in the command line matlab -r "run ./example.m" or in the Matlab terminal run ./example.m. This will open a plot of the GHZ state as shown below.

A simple example as a Mathematica Notebook is provided in Mathematica/example.nb which calculates the Wigner function of a Schrödinger cat state, the well-knwon GHZ state, for a given dimension. This program uses the precalculated kernels located in Calculated/Kernels. Note that if the desired dimension is larger than 30, then the corresponding kernel needs to be downloaded from the release Kernels_dim120 and extracted to Calculated/Kernels. Alternatively, run the script build.sh or build.bat to compute the kernels with our C code.

Run the example by openeing the file Mathematica/example.nb in Mathematica and by selecting "Evaluate Notebook" in the "Evaluation" drop-down menu. This will generate a plot of the GHZ state as shown below.

A simple example is provided in Python/example.py which calculates the Wigner function of a Schrödinger cat state, the well-knwon GHZ state, for a given dimension. This program uses the precalculated kernels located in Calculated/Kernels. Note that if the desired dimension is larger than 30, then the corresponding kernel needs to be downloaded from the release Kernels_dim120 and extracted to Calculated/Kernels. Alternatively, run the script build.sh or build.bat to compute the kernels with our C code.

Run the example by entering the directory Python/ and type in the command line python ./example.py. This will create a plot of the GHZ state and save it in the file "plot_D10.png" as shown below.

The source code in src contains the following two programs wirtten in C. These can be compiled and ran by build.sh and build.bat.

• on UNIX or macOS run ./build.sh -- gcc needs to be installed
• on Windows run ./build.bat -- MinGW needs to be installed which provides gcc for Windows

After successful compilations and calculations, the precalculated kernels should be located in Calculated/Kernels.

The function

• void PrecalculateKernel(complex* parity, complex* u, complex* ptilde, int Ndim)

in src/Precalculate_Kernel.c calculates and stores the kernel coefficients K_{\lambda}^{lm} required for calculating the Fourier series representation of phase-space functions. It requires both precalulated parity operators and eigenvectors of the rotation operator, which are provided up to dimension 120 in Calculated/Parity and in Calculated/Eigenvectors. We furthermore provide these up to dimension 500 in the releases Parity_Op_Dim500 and Eigenvectors_dim500.

The example provided in src/Precalculate_Kernel.c calculates the kernels for Wigner functions (s=0) up to dimension 120 and stores them in Calculated/Kernels. These resulting precalculated kernels are used by src/EfficientCalculation.c and by the Matlab, Mathematica and Python codes.

The function

• void CalcPSrepresentationL(complex* rho, complex* matrL, complex* PSrepr, complex* prod, int l, int Ndim)

in src/EfficientCalculation.c calculates the Fourier series decomposition of the phase-space function F_{\rho} representing the density matrix \rho for a fixed index l. It requires the precalculated transformation kernel as the coefficients K_{\lambda}^{lm}.

The example provided in src/EfficientCalculation.c calculates Fourier decomposition coefficients of Wigner functions W_{\rho} up to dimension 120 using the kernels initially preculculated by src/Precalculate_Kernel.c.

We provide further source code for precalculating the following for arbitrary large dimensions. Note that these depend on external libraries, such as LAPACK, etc.

• computing eigenvectors of the rotation operator via src/Precalculate_Eigenvectors/Precalculate_Eigenvectors.c
• computing tensor operators via src/Precalculate_Tensor_Operators/Tensor_Operator.cpp
• computing parity operators via src/Precalculate_Parity/Precalc_Parity.cpp

[1] Fast computation of spherical phase-space functions of quantum many-body states, Bálint Koczor, Robert Zeier and Steffen J. Glaser (2020) ArXiv preprint, arXiv:2008.06481