Graph States

Conversion to and from graph states is possible.

Consider a GHZ state:

ghz(4)

+ XXXX
+ ZZ__
+ _ZZ_
+ __ZZ

It can be converted to a graph state with graphstate

graphstate(ghz(4))[1]

Notice that the initial GHZ state was not in the typical graph state form. We can see that explicitly by converting back and forth between the two forms:

julia> using Graphs, QuantumClifford

julia> ghz(4)
+ XXXX
+ ZZ__
+ _ZZ_
+ __ZZ

julia> Stabilizer(Graph(ghz(4)))
+ XZZZ
+ ZX__
+ Z_X_
+ Z__X

There is a set of single-qubit operations that can convert any stabilizer tableau into a state representable as a graph. These transformations are performed implicitly by the Graph constructor when converting from a Stabilizer. If you need the explicit transformation you can use the graphstate function that specifies which qubits require a Hadamard, Inverse Phase, or Phase Flip gate. The graph_gatesequence or graph_gate helper functions can be used to generate the exact operations:

julia> s = ghz(4)
+ XXXX
+ ZZ__
+ _ZZ_
+ __ZZ

julia> g, h_idx, ip_idx, z_idx = graphstate(s);

julia> gate = graph_gate(h_idx, ip_idx, z_idx, nqubits(s))
X₁ ⟼ + X___
X₂ ⟼ + _Z__
X₃ ⟼ + __Z_
X₄ ⟼ + ___Z
Z₁ ⟼ + Z___
Z₂ ⟼ + _X__
Z₃ ⟼ + __X_
Z₄ ⟼ + ___X

julia> canonicalize!(apply!(s,gate)) == canonicalize!(Stabilizer(g))
true

These converters also provides for a convenient way to create graph and cluster states, by using the helper constructors provided in Graphs.jl.

julia> Stabilizer(grid([4,1])) # Linear cluster state
+ XZ__
+ ZXZ_
+ _ZXZ
+ __ZX

julia> Stabilizer(grid([2,2])) # Small 2D cluster state
+ XZZ_
+ ZX_Z
+ Z_XZ
+ _ZZX

Graphs are represented with the Graphs.jl package and plotting can be done both in Plots.jl and Makie.jl (with GraphMakie).