introducing inverse sparsity Λ(χ) and Λ(ϕᵢⱼ)

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

@@ -356,6 +356,40 @@ nqubits(pc::UnitaryPauliChannel) = nqubits(pc.paulis[1])
 
 apply!(state::GeneralizedStabilizer, gate::UnitaryPauliChannel; prune_threshold=1e-10) = apply!(state, gate.paulichannel; prune_threshold)
 
+"""
+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"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) |> invsparsity
+4
+```
+
+Similarly, it calculates the number of non-zero elements in the density
+matrix `ϕᵢⱼ`​ of a PauliChannel, providing a measure of the channel
+complexity.
+
+```jldoctest
+julia> invsparsity(pcT)
+4
+```
+
+See also: [`GeneralizedStabilizer`](@ref)
+"""
+invsparsity(sm::GeneralizedStabilizer) = count(!iszero, values(sm.destabweights::DefaultDict{Tuple{BitVector, BitVector}, ComplexF64, ComplexF64}))
+invsparsity(gate::AbstractPauliChannel) = count(!iszero, values(gate.paulichannel.weights::Vector{ComplexF64}))
+
 ##
 # Predefined Pauli Channels
 ##

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 +39,21 @@ end
 
 ##
 
+@testset "Inverse sparsity" 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
+        # Λ(χ) ≤ 4ⁿ
+        @test invsparsity(gs) <= 4^n
+        channels = [embed(n, i, pcT) for i in 1:rand(1:4)]
+        # Λ(ϕᵢⱼ) ≤ 4ⁿ
+        @test all(invsparsity(channel) <= 4^n for channel in channels)
+    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,))