noncliff: Inner Product between two GeneralizedStabilizers

ext/QuantumCliffordQOpticsExt/QuantumCliffordQOpticsExt.jl

@@ -1,11 +1,12 @@
 module QuantumCliffordQOpticsExt
 
 using QuantumClifford
-import QuantumClifford: mul_left!, mul_right!
+import QuantumClifford: mul_left!, mul_right!, dot
 using QuantumOpticsBase
 using Graphs
 using DocStringExtensions
 import QuantumOpticsBase: Ket, Operator
+using LinearAlgebra
 
 const _b2 = SpinBasis(1//2)
 const _l0 = spinup(_b2)
@@ -234,6 +235,41 @@ function Operator(c::CliffordOperator)
     cliff_to_unitary(c)
 end
 
+"""
+The inner product of two [`GeneralizedStabilizer`](@ref) states, `sm₁` and `sm₂`.
+
+```jldoctest
+julia> using QuantumOpticsBase; using LinearAlgebra; # hide
+
+julia> sm = GeneralizedStabilizer(S"X")
+A mixture ∑ ϕᵢⱼ Pᵢ ρ Pⱼ† where ρ is
+𝒟ℯ𝓈𝓉𝒶𝒷
++ Z
+𝒮𝓉𝒶𝒷
++ X
+with ϕᵢⱼ | Pᵢ | Pⱼ:
+ 1.0+0.0im | + _ | + _
+
+julia> apply!(sm, pcT)
+A mixture ∑ ϕᵢⱼ Pᵢ ρ Pⱼ† where ρ is
+𝒟ℯ𝓈𝓉𝒶𝒷
++ Z
+𝒮𝓉𝒶𝒷
++ X
+with ϕᵢⱼ | Pᵢ | Pⱼ:
+ 0.0+0.353553im | + _ | + Z
+ 0.0-0.353553im | + Z | + _
+ 0.853553+0.0im | + _ | + _
+ 0.146447+0.0im | + Z | + Z
+
+julia> dot(sm, sm)
+0.9999999999999994
+```
+"""
+function LinearAlgebra.dot(sm₁::GeneralizedStabilizer, sm₂::GeneralizedStabilizer)
+    return real(tr(Operator(sm₁)' * Operator(sm₂)))
+end
+
 end
 
 ##

src/nonclifford.jl

@@ -462,3 +462,16 @@ const pcT = UnitaryPauliChannel(
     (I, Z),
     ((1+exp(im*π/4))/2, (1-exp(im*π/4))/2)
 )
+
+##
+# QuantumOpticsBaseExt methods
+##
+
+function LinearAlgebra.dot(args...)
+    ext = Base.get_extension(QuantumClifford, :QuantumCliffordOpticsBase)
+    if isnothing(ext)
+        throw("The `LinearAlgebra.dot` depends on the package `QuantumOpticsBase` but you have not installed or imported it yet. Immediately after you import `QuantumOpticsBase`, the `LinearAlgebra.dot` will be available.")
+    end
+    return ext.LinearAlgebra.dot(args...)
+end
+

test/test_nonclifford_quantumoptics.jl

@@ -82,6 +82,7 @@ end
         apply!(sm, embed(n, i, pcT))
         smcopy = copy(sm)
         @test smcopy == sm
+        @test dot(sm, smcopy) ≈ 1
         nc = embed(n, rand(1:n), pcT)
         @test copy(nc) == nc
     end