SOC Observation Code

Description

SOC stands for SOC Observation Code, and is composed of two C programs:

• One, SOCgen, to generate SOC graphs (here, SOC stands for Siblings-on-Cycles),
• and another, SOCadmissible, to verify the admissibility of these graphs as quantum causal structures.

These programs are used in support of Conjecture 1 in the article Admissible Causal Structures and Correlations, arXiv:2210.12796 [quant-ph].

Installation

First, clone this repository, and then simply run

$ cd soc-observation-code/
$ make

This compiles the two programs as SOCgen and SOCadmissible.

Usage

To display help and exit, run the respective program without command-line arguments.

SOCgen

$ ./SOCgen 
Usage: ./SOCgen -n <order> [-r <num> ] [FILTER ...]
       -n <order>     Generate SOCs with `order' connected nodes
       -r <num>       Pick directed graphs at random, and exit after having found `num' SOCs

[FILTER]              Consider only simple directed graphs ...
       -c             ... that are cyclic (i.e., not DAGs)
       --no-sink      ... without sink nodes (this logically implies -c)
       --no-source    ... without source nodes (also this logically implies -c)

This program prints the found SOCs as adjacency matrices to stdout.
To exclude (some) of the isomorphic SOCs, it uses a degree-order filter.

SOCadmissible

$ ./SOCadmissible 
Usage: ./SOCadmissible <filename> [<startline> [<endline> | +<count>]]
       <filename>     File name with adjacency matrices of simple directed graphs
       <startline>    Verify graphs starting from line `startline'
       <endline>      Verify graphs up to and including line `endline'
       +<count>       Verify `count' number of graphs

[FILE FORMAT]
       Each line in `filename' must contain the adjacency matrix of a simple directed graph in the format
         {{a00,a01,...},{a10,a11,...},...}  where aij=1 if and only if the graph has the edge i -> j
       The file `filename' may contain graphs with different order (number of vertices)

This program verifies the admissibility of simple directed graphs.

Example

To generate all SOCs with three nodes, and save them in the file 3.soc, run:

$ ./SOCgen -n 3 > 3.soc
Generating SOCs with 3 nodes
100.00%    64/64 (found=6 at rate 64.00 graphs/s in 0 seconds)
Found 6 SOCs at rate 64.00 graphs/s in 0 seconds

The admissibility of these graphs can then be checked by running:

$ ./SOCadmissible 3.soc 
Verifying the admissibility of 6 graphs in the file `3.soc' (line 1 to line 6)
100.00%    6/6 (verification rate 6.00 graphs/s in 0 seconds; current line 6)
These graphs are admissible

The SOCs generated can easily be displayed with Mathematica using the following:

SOCs = DirectedGraph[AdjacencyGraph[#]] & /@ ToExpression[Import["./3.soc", "List"]];
SOCs = DeleteDuplicatesBy[SOCs, CanonicalGraph];
SOCs

Limitations

In SOCgen, each simple directed graph is represented by a 64bit unsigned integer: This integer is interpreted as a vector of bits, where each bit specifies the absence or presence of a directed edge from one node to another. Since we consider simple directed graphs only (no self-loops), there are n(n-1) possible directed edges, where n is the number of nodes. This means that the largest number of nodes possible is limited by n=8.

While the SOCs generated by SOCgen satisfy some degree-order (see function isdegreeordered(...) in SOCgen.c), SOCgen does not perform graph-isomorphism tests, and may output multiple isomorphic graphs.

License

GPL-3.0-or-later