From cd74b92b7cde5e462f868103f76844305b1a0da5 Mon Sep 17 00:00:00 2001
From: Feroz Ahmad <feroz.ahmad.email@gmail.com>
Date: Tue, 5 Nov 2024 20:28:39 +0500
Subject: [PATCH] =?UTF-8?q?noncliff:=20introducing=20inverse=20sparsity=20?=
 =?UTF-8?q?`=CE=9B(=CF=87)`=20and=20`=CE=9B(=CF=95=E1=B5=A2=E2=B1=BC)`=20(?=
 =?UTF-8?q?#376)?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

Co-authored-by: Stefan Krastanov <github.acc@krastanov.org>
---
 src/nonclifford.jl       | 36 ++++++++++++++++++++++++++++++++++++
 test/test_nonclifford.jl | 17 ++++++++++++++++-
 2 files changed, 52 insertions(+), 1 deletion(-)

diff --git a/src/nonclifford.jl b/src/nonclifford.jl
index 7ab5c3dd0..524085025 100644
--- a/src/nonclifford.jl
+++ b/src/nonclifford.jl
@@ -418,6 +418,42 @@ 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)
+"""
+function invsparsity end
+
+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
 ##
diff --git a/test/test_nonclifford.jl b/test/test_nonclifford.jl
index c4ff7c658..b26239de8 100644
--- a/test/test_nonclifford.jl
+++ b/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,))