introducing inverse sparsity Λ(χ)

src/QuantumClifford.jl

@@ -81,7 +81,7 @@ export
     # petrajectories
     petrajectories, applybranches,
     # nonclifford
-    GeneralizedStabilizer, UnitaryPauliChannel, PauliChannel, pcT,
+    GeneralizedStabilizer, UnitaryPauliChannel, PauliChannel, pcT, invsparsity,
     # makie plotting -- defined only when extension is loaded
     stabilizerplot, stabilizerplot_axis,
     # sum types

src/nonclifford.jl

@@ -298,6 +298,46 @@ function rowdecompose(pauli,state::Union{MixedDestabilizer, Destabilizer})
     return p+pauli.phase[], b, c
 end
 
+"""
+Calculates the number of non-zero elements in the density matrix `χ`
+of a [`GeneralizedStabilizer`](@ref), representing the inverse sparsity
+of `χ`. It provides a measure of the state's complexity, with bounds
+`Λ(χ) ≤ 4ⁿ`.
+
+```jldoctest
+julia> sm = GeneralizedStabilizer(S"-ZY XZ")
+A mixture ∑ ϕᵢⱼ Pᵢ ρ Pⱼ† where ρ is
+𝒟ℯ𝓈𝓉𝒶𝒷
++ Z_
++ _Z
+𝒮𝓉𝒶𝒷
++ XZ
+- ZY
+with ϕᵢⱼ | Pᵢ | Pⱼ:
+ 1.0+0.0im | + __ | + __
+
+julia> apply!(sm, embed(2, 1, pcT))
+A mixture ∑ ϕᵢⱼ Pᵢ ρ Pⱼ† where ρ is
+𝒟ℯ𝓈𝓉𝒶𝒷
++ Z_
++ _Z
+𝒮𝓉𝒶𝒷
++ XZ
+- ZY
+with ϕᵢⱼ | Pᵢ | Pⱼ:
+ 0.5+0.0im | + __ | + __
+ 0.0+0.5im | + __ | + Z_
+ 0.0-0.5im | + Z_ | + __
+ 0.5+0.0im | + Z_ | + Z_
+
+julia> invsparsity(sm)
+4
+```
+
+See also: [`GeneralizedStabilizer`](@ref)
+"""
+invsparsity(sm::GeneralizedStabilizer) = count(!iszero, values(sm.destabweights))
+
 """A Pauli channel datastructure, mainly for use with [`GeneralizedStabilizer`](@ref).
 
 More convenient to use than [`PauliChannel`](@ref) when you know your Pauli channel is unitary.

test/test_nonclifford.jl

@@ -1,5 +1,5 @@
 using QuantumClifford
-using QuantumClifford: GeneralizedStabilizer, rowdecompose, PauliChannel, mul_left!, mul_right!
+using QuantumClifford: GeneralizedStabilizer, rowdecompose, PauliChannel, invsparsity, mul_left!, mul_right!
 using Test
 using InteractiveUtils
 using Random
@@ -39,6 +39,17 @@ end
 
 ##
 
+@testset "Inverse sparsity Λ(χ) ≤ 4ⁿ" begin
+    for n in 1:5
+        s = random_stabilizer(n)
+        gs = GeneralizedStabilizer(s)
+        for i in 1:rand(1:4)
+            apply!(gs, embed(n, i, pcT))
+        end
+        @test invsparsity(gs) <= 4^n
+    end
+end
+
 @test_throws ArgumentError GeneralizedStabilizer(S"XX")
 @test_throws ArgumentError PauliChannel(((P"X", P"Z"), (P"X", P"ZZ")), (1,2))
 @test_throws ArgumentError PauliChannel(((P"X", P"Z"), (P"X", P"Z")), (1,))