Zoo: [[2^j, 2^-j-2,3]] Gottesman /Quantum Hamming Code

docs/src/references.bib

@@ -225,3 +225,29 @@ @article{nahum2017quantum
 	author = {Nahum, Adam and Ruhman, Jonathan and Vijay, Sagar and Haah, Jeongwan},
   year = {2017}
 }
+
+% Key details about [[8, 3, 3]] stabilizer construction via alternative approach
+
+@article{6506105,
+  author={Yu, Sixia and Bierbrauer, Jürgen and Dong, Ying and Chen, Qing and Oh, C. H.},
+  journal={IEEE Transactions on Information Theory}, 
+  title={All the Stabilizer Codes of Distance 3}, 
+  year={2013},
+  volume={59},
+  number={8},
+  pages={5179-5185},
+  keywords= {Generators;Frequency modulation;Vectors;Educational institutions;Error correction codes;Physics;Indexes;1-error correcting stabilizer codes;optimal codes;quantum error correction;quantum Hamming bound},
+  doi={10.1109/TIT.2013.2259138}
+}
+
+@article{Chao2017QuantumEC,
+  title={Quantum Error Correction with Only Two Extra Qubits.},
+  author={Rui Chao and Ben W Reichardt},
+  journal={Physical review letters},
+  year={2017},
+  volume={121 5},
+  pages={
+          050502
+        },
+  url={https://api.semanticscholar.org/CorpusID:206314271}
+}

src/ecc/ECC.jl

@@ -19,7 +19,7 @@ export parity_checks, parity_checks_x, parity_checks_z,
     RepCode,
     CSS,
     Shor9, Steane7, Cleve8, Perfect5, Bitflip3,
-    Toric,
+    Toric, Gottesman,
     evaluate_decoder,
     CommutationCheckECCSetup, NaiveSyndromeECCSetup, ShorSyndromeECCSetup,
     TableDecoder,
@@ -334,5 +334,5 @@ include("codes/steanecode.jl")
 include("codes/shorcode.jl")
 include("codes/clevecode.jl")
 include("codes/toric.jl")
-
-end #module
+include("codes/gottesmancode.jl")
+end #module

src/ecc/codes/gottesmancode.jl

@@ -0,0 +1,92 @@
+"""This class implements the family of [[2^j, 2^j - j - 2, 3]] Gottesman codes, also known as [[2^j, 2^j - j - 2, 3]] quantum Hamming codes, as described in Gottesman's 1997 PhD thesis on 'Stabilizer Codes and Quantum Error Correction' [Gottesman1997](@cite).
+
+Key details for the [[8, 3, 3]] code when j = 3 using Stabilizer Formalism:
+- [6506105](@cite) utilizes an explicit construction method for the stabilizer generators. The remaining m generators, apart from Mx and Mz, are constructed using the check matrix [Hm|AmHm], where Hm = [c0, c1, ..., c^2m − 1]. Each column ck (for k = 0, 1, ..., 2^m − 1) represents a binary vector corresponding to the integer k. Also, Am refers to any invertible m × m matrix devoid of fixed points such that Am​.s ≠ 0 and Am.s ≠ s for all s ∈ F₂ᵐ.
+- [Chao2017QuantumEC](@cite) introduces alternative stabilizer generators for the [[8, 3, 3]] code. It identifies permutations effective for all stabilizer generators except for X^⊗8 and Z^⊗8. However, these can be replaced with XXYZIYZI and ZZIXYIXY, respectively. The stabilizer generators are permuted to achieve desired properties, reflecting an alternative approach to constructing the stabilizer code.
+
+Notes:
+- This implementation adopts the Gottesman Notation for the [[8, 3, 3]] stabilizer code as depicted in Table 3.3 on Page 22 on the Gottesman's PhD thesis. Additionally, for j = 4, our [[16, 10, 3]] stabilizer code stemming from QHamming(4) can be cross-verified from Table 8.1 on Page 91 as well [Gottesman1997](@cite).
+- The differences between [6506105](@cite), [Chao2017QuantumEC](@cite), and [Gottesman1997](@cite) primarily lie in the choice of stabilizer generators and their permutations for the [[8, 3, 3]] stabilizer code.
+- The discrepancy in stabilizer generator representations underscores the flexibility within the stabilizer formalism, allowing for various valid choices and permutations, albeit with different notations and implementation details.
+"""
+
+struct Gottesman <: AbstractECC
+    j::Int
+    function Gottesman(j)
+        (j >= 3 && j < 21) || error("In `Gottesman(j)`, `j` must be ≥  3 in order to obtain a valid code and `j` must be < 21 to remain tractable")
+        new(j)
+    end
+end
+
+code_n(c::Gottesman) = 2^c.j
+
+function parity_checks(c::Gottesman)
+    rows = c.j + 2
+    cols = 2^c.j
+
+    Hx = falses(rows,cols)
+    Hz = falses(rows,cols)
+
+    Hx[1, :] .= true
+    Hx[2, :] .= false
+
+    if c.j == 3 
+        for col in 1:cols
+            Hx[3, col] = (col % 8 == 1 || col % 8 == 3 || col % 8 == 6) ? 0 : 1
+        end
+        Hx[3, cols] = Hx[3, cols] == 0 ? 1 : 0
+        for col in 1:cols
+            Hx[4, col] = (col % 4 == 1) || (col % 4 == 3) ? 0 : 1
+        end
+        for a in 1:cols
+            Hx[rows, a] = ((a % 4 == 0) || (a % 4 == 1) ? 0 : 1) ⊻ ((a % 8 == 5) || (a % 8 == 6))
+        end
+        Hx[end, [end-1, end]] .= [0, 1]
+
+    else
+        for a in 1:cols
+            Hx[3, a] = (a == 0) || (a % 2 == 0)
+        end
+        for row in 4:rows - 1
+            for col in 1:cols
+                k = row - 3
+                m = 2^(c.j - k)
+                n = 2^(c.j - k)
+                if (col - 1) % (m + n) < m
+                    if col % 2 == 0
+                        Hx[row, col] = 1
+                    else
+                        Hx[row, col] = 0
+                    end
+                else
+                    if col % 2 == 0
+                        Hx[row, col] = 0
+                    else
+                        Hx[row, col] = 1
+                    end
+                end
+            end
+        end
+
+        for a in 1:cols
+            Hx[rows, a] = (a % 4 == 0) || (a % 4 == 1) ? 0 : 1
+        end
+    end
+
+    Hz[1, :] .= false
+    Hz[2, :] .= true
+
+    for i in 3:rows
+        period = 2^(rows - i)
+        for a in 1:cols
+            Hz[i, a] = div(a - 1, period) % 2 == 1
+        end
+    end
+    extended_Hx = Matrix{Bool}(Hz)
+    extended_Hz = Matrix{Bool}(Hx)
+
+    num_rows = size(Hx, 1)
+
+    fill_array = fill(UInt8(0), num_rows)
+    Stabilizer(fill_array, extended_Hz, extended_Hx)
+end

test/test_ecc.jl

@@ -8,7 +8,7 @@ codes = [
     Steane7(),
     Shor9(),
     Perfect5(),
-    Cleve8(),
+    Cleve8(), Gottesman(3), Gottesman(5),
     CSS([0 1 1 0; 1 1 0 0], [1 1 1 1]),
 ]
 
@@ -76,4 +76,4 @@ end
     @test isdegenerate(Steane7()) == false
     @test isdegenerate(Steane7(), 2) == true
     @test isdegenerate(Bitflip3()) == true
-end
+end

test/test_ecc_decoder_all_setups.jl

@@ -10,7 +10,7 @@ import LDPCDecoders
         Steane7(),
         Shor9(),
         Perfect5(),
-        Cleve8(),
+        Cleve8(), Gottesman(3), Gottesman(5),
     ]
 
     noise = 0.001
@@ -79,11 +79,11 @@ end
 
     for c in codes
         for s in setups
-            e = evaluate_decoder(PyMatchingDecoder(c), s, 10000)
+            e = evaluate_decoder(PyMatchingDecoder(c), s, 100000)
             #@show c
             #@show s
             #@show e
             @assert max(e...) < noise/5
         end
-    end
-end
+    end 
+end

test/test_ecc_encoding.jl

@@ -3,7 +3,6 @@ using QuantumClifford
 using QuantumClifford.ECC
 
 ##
-
 @testset "encoding circuits - compare to algebraic construction of encoded state" begin
     # This test verifies that logical measurements on an encoded state match the physical pre-encoded state.
     # This test skips verifying the permutations of qubits during canonicalization are properly undone,
@@ -15,6 +14,8 @@ using QuantumClifford.ECC
         :(Shor9()),
         :(Perfect5()),
         :(Bitflip3()),
+        :(Gottesman(3)),
+        :(Gottesman(5)),
         :(S"Y_"),
         :(S"Z_"),
         :(S"X_"),
@@ -25,7 +26,6 @@ using QuantumClifford.ECC
         :(Toric(8,8)),
         fill(:(random_stabilizer(5,7)), 100)...
         ]
-
         code = eval(codeexpr)
         if undoperm==false
             # Pre-process the tableau to remove permutations and negative phases.
@@ -55,4 +55,4 @@ using QuantumClifford.ECC
 
         #println("$codeexpr, $(encodedₙ == algebraicₙ)")
     end
-end
+end

test/test_ecc_syndromes.jl

@@ -9,7 +9,7 @@ codes = [
     Steane7(),
     Shor9(),
     Perfect5(),
-    Cleve8(),
+    Cleve8(), Gottesman(3), Gottesman(5),
     CSS([0 1 1 0; 1 1 0 0], [1 1 1 1]),
     Toric(3,3),
     Toric(3,6),
@@ -59,4 +59,4 @@ end
     for c in codes
         pframe_naive_vs_shor_syndrome(c)
     end
-end
+end

test/test_throws.jl

@@ -64,4 +64,4 @@ for m in [sMX,sMZ,sMY,sMRX,sMRZ,sMRY]
     @test_throws ArgumentError m(-1)
     @test_throws ArgumentError m(0,1)
     @test_throws ArgumentError m(-1,0)
-end
+end