## NoncommutativeGraphs.jl

Noncommutative graphs in Julia
Author dstahlke
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2 Years Ago
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December 2020

# NoncommitativeGraphs.jl - Non-commutative graphs in Julia

This package provides support for non-commutative graphs, a quantum analogue of graphs, as defined in Duan, Severini, Winter, Zero-error communication via quantum channels, non-commutative graphs and a quantum Lovasz theta function, arXiv:1002.2514.

Aside from a data structure for holding such graphs, we provide functions for computing the weighted Lovasz theta function as defined in Stahlke, Weighted theta functions for non-commutative graphs, arXiv:2101.00162.

## Example

```julia> using NoncommutativeGraphs, Random

julia> Random.seed!(0);

julia> sig = [3 2; 2 3]; # S₀ algebra is M₃⊗I₂ ⊕ M₂⊗I₃

julia> S = random_S0Graph(sig)
S0Graph{S0=[3 2; 2 3] S=Subspace{ComplexF64} size (12, 12) dim 83}

julia> S.S0 # vertex C*-algebra
Subspace{ComplexF64} size (12, 12) dim 13

julia> T = complement(S) # T = perp(S) + S₀
S0Graph{S0=[3 2; 2 3] S=Subspace{ComplexF64} size (12, 12) dim 74}

julia> W = randn(ComplexF64, S.n, S.n); W = W' * W; # random weight operator

julia> opt1 = dsw(S, W, eps=1e-7).λ # compute weighted theta
133.57806623525727

julia> opt2 = dsw_via_complement(complement(S), W, eps=1e-7).λ # compute weighted θ via the complement graph, using theorem 29 of arxiv:2101.00162.
133.57806730600717

julia> abs(opt1 - opt2) / abs(opt1 + opt2) < 1e-6
true```

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