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, QuantumCliffordjulia> ghz(4)+ XXXX+ ZZ__+ _ZZ_+ __ZZjulia> 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_+ __ZZjulia> 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 ⟼ + ___ZZ₁ ⟼ + Z___Z₂ ⟼ + _X__Z₃ ⟼ + __X_Z₄ ⟼ + ___Xjulia> canonicalize!(apply!(s,gate)) == canonicalize!(Stabilizer(g))true

The returned gates map the input stabilizer to the graph state. To recover the input from Stabilizer(g), apply the inverse transformation. In particular, the inverse of an Inverse Phase gate is a Phase gate, and the order of the gates must also be reversed: Pauli Z, Phase, then Hadamard.

Using graph_gate and inv handles both the gate inverses and their order:

julia> using QuantumCliffordjulia> original = S" Y_                   -_Z";julia> g, h_idx, ip_idx, z_idx = graphstate(original);julia> to_graph = graph_gate(h_idx, ip_idx, z_idx, nqubits(original));julia> recovered = apply!(Stabilizer(g), inv(to_graph));julia> canonicalize!(recovered) == canonicalize!(copy(original))true

For a sequence of individual gates, use gates = vcat(graph_gatesequence(h_idx, ip_idx, z_idx)...) and apply inv.(reverse(gates)) to Stabilizer(g).

These converters also provide 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+ __ZXjulia> 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).