Gottesman's code

CHANGELOG.md

@@ -6,8 +6,9 @@
 
 # News
 
-## v0.8.21 - 2024-03-15
+## v0.8.21 - 2024-03-17
 
+- Implemented the Gottesman code family, also known as [[2^j, 2^j - j - 2, 3]] quantum Hamming codes.
 - Bump the `PyQDecoders` dependency, switching to `PythonCall` behind the scenes for reliability.
 - Bump the `LDPCDecoders` dependency.
 

docs/src/references.bib

@@ -1,50 +1,3 @@
-@article{mackay2004sparse,
-  title={Sparse-graph codes for quantum error correction},
-  author={MacKay, David JC and Mitchison, Graeme and McFadden, Paul L},
-  journal={IEEE Transactions on Information Theory},
-  volume={50},
-  number={10},
-  pages={2315--2330},
-  year={2004},
-  publisher={IEEE},
-  doi={10.1109/TIT.2004.834737}
-}
-
-@article{calderbank1998quantum,
-  title={Quantum error correction via codes over GF (4)},
-  author={Calderbank, A Robert and Rains, Eric M and Shor, PM and Sloane, Neil JA},
-  journal={IEEE Transactions on Information Theory},
-  volume={44},
-  number={4},
-  pages={1369--1387},
-  year={1998},
-  publisher={IEEE},
-  doi={10.1109/18.681315}
-}
-
-@inproceedings{steane2007tutorial,
-  title={A tutorial on quantum error correction},
-  author={Steane, Andrew M},
-  booktitle={PROCEEDINGS-INTERNATIONAL SCHOOL OF PHYSICS ENRICO FERMI},
-  volume={162},
-  pages={1},
-  year={2007},
-  organization={IOS Press; Ohmsha; 1999},
-  doi={10.3254/978-1-61499-018-5-1}
-}
-
-@article{wilde2009logical,
-  title={Logical operators of quantum codes},
-  author={Wilde, Mark M},
-  journal={Physical Review A},
-  volume={79},
-  number={6},
-  pages={062322},
-  year={2009},
-  publisher={APS},
-  doi={10.1103/PhysRevA.79.062322}
-}
-
 % The basis of the tableaux algorithm
 
 @inproceedings{gottesman1998heisenberg,
@@ -167,6 +120,18 @@ @article{hein2006entanglement
   year={2006}
 }
 
+@article{wilde2009logical,
+  title={Logical operators of quantum codes},
+  author={Wilde, Mark M},
+  journal={Physical Review A},
+  volume={79},
+  number={6},
+  pages={062322},
+  year={2009},
+  publisher={APS},
+  doi={10.1103/PhysRevA.79.062322}
+}
+
 % Encoding circuits
 
 @article{cleve1997efficient,
@@ -225,3 +190,75 @@ @article{nahum2017quantum
 	author = {Nahum, Adam and Ruhman, Jonathan and Vijay, Sagar and Haah, Jeongwan},
   year = {2017}
 }
+
+% codes
+
+@article{mackay2004sparse,
+  title={Sparse-graph codes for quantum error correction},
+  author={MacKay, David JC and Mitchison, Graeme and McFadden, Paul L},
+  journal={IEEE Transactions on Information Theory},
+  volume={50},
+  number={10},
+  pages={2315--2330},
+  year={2004},
+  publisher={IEEE},
+  doi={10.1109/TIT.2004.834737}
+}
+
+@article{calderbank1998quantum,
+  title={Quantum error correction via codes over GF (4)},
+  author={Calderbank, A Robert and Rains, Eric M and Shor, PM and Sloane, Neil JA},
+  journal={IEEE Transactions on Information Theory},
+  volume={44},
+  number={4},
+  pages={1369--1387},
+  year={1998},
+  publisher={IEEE},
+  doi={10.1109/18.681315}
+}
+
+@inproceedings{steane2007tutorial,
+  title={A tutorial on quantum error correction},
+  author={Steane, Andrew M},
+  booktitle={PROCEEDINGS-INTERNATIONAL SCHOOL OF PHYSICS ENRICO FERMI},
+  volume={162},
+  pages={1},
+  year={2007},
+  organization={IOS Press; Ohmsha; 1999},
+  doi={10.3254/978-1-61499-018-5-1}
+}
+
+@article{yu2013all,
+  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{chao2018quantum,
+  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}
+}
+
+@article{gottesman1996class,
+  title={Class of quantum error-correcting codes saturating the quantum Hamming bound},
+  author={Gottesman, Daniel},
+  journal={Physical Review A},
+  volume={54},
+  number={3},
+  pages={1862},
+  year={1996},
+  publisher={APS}
+}

docs/src/references.md

@@ -28,6 +28,13 @@ For circuit construction routines (for stabilizer measurements for a given code)
 - [grassl2002algorithmic](@cite)
 - [grassl2011variations](@cite)
 
+For code construction routines:
+- [cleve1997efficient](@cite)
+- [gottesman1996class](@cite)
+- [gottesman1997stabilizer](@cite)
+- [yu2013all](@cite)
+- [chao2018quantum](@cite)
+
 # References
 
 ```@bibliography

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")
-
+include("codes/gottesman.jl")
 end #module

src/ecc/codes/gottesman.jl

@@ -0,0 +1,40 @@
+"""The family of `[[2ʲ, 2ʲ - j - 2, 3]]` Gottesman codes, also known as quantum Hamming codes, as described in [Gottesman's 1997 PhD thesis](@cite gottesman1997stabilizer) and in [gottesman1996class](@cite).
+
+You might be interested in consulting [yu2013all](@cite) and [chao2018quantum](@cite) as well.
+
+The ECC Zoo has an [entry for this family](https://errorcorrectionzoo.org/c/quantum_hamming)
+"""
+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 < 21 to remain tractable")
+        new(j)
+    end
+end
+
+code_n(c::Gottesman) = 2^c.j
+
+function parity_checks(c::Gottesman)
+    j = c.j
+    s = j+2
+    n = 2^j
+
+    H = zero(Stabilizer, s, n)
+    for i in 1:n
+        H[1, i] = (true, false)
+        H[2, i] = (false, true)
+    end
+    for i in 0:n-1 # column of H, corresponds to a single qubit error that is detectable)
+        Xⁱ = i
+        Zⁱ = i÷2
+        jeven = j%2 == 0
+        ieven = i%2 == 0
+        if (jeven && ieven) || (!jeven && ieven && i < n÷2) || (!jeven && !ieven && i ≥ n÷2)
+            Zⁱ = ~Zⁱ
+        end
+        for b in 0:j-1 # which check to consider (row of H), also which bit to extract
+            H[s-b,i+1] = isone((Zⁱ>>b)&0x1), isone((Xⁱ>>b)&0x1)
+        end
+    end
+    H
+end

test/runtests.jl

@@ -62,6 +62,7 @@ end
 @doset "ecc_decoder_all_setups"
 @doset "ecc_encoding"
 @doset "ecc_syndromes"
+@doset "ecc_gottesman"
 @doset "precompile"
 @doset "pauliframe"
 @doset "allocations"

test/test_ecc.jl

@@ -9,6 +9,8 @@ codes = [
     Shor9(),
     Perfect5(),
     Cleve8(),
+    Gottesman(3),
+    Gottesman(5),
     CSS([0 1 1 0; 1 1 0 0], [1 1 1 1]),
 ]
 

test/test_ecc_decoder_all_setups.jl

@@ -11,6 +11,9 @@ import LDPCDecoders
         Shor9(),
         Perfect5(),
         Cleve8(),
+        Gottesman(3),
+        #Gottesman(4), bad threshold
+        #Gottesman(5), bad threshold
     ]
 
     noise = 0.001

test/test_ecc_encoding.jl

@@ -15,6 +15,8 @@ using QuantumClifford.ECC
         :(Shor9()),
         :(Perfect5()),
         :(Bitflip3()),
+        :(Gottesman(3)),
+        :(Gottesman(5)),
         :(S"Y_"),
         :(S"Z_"),
         :(S"X_"),

test/test_ecc_gottesman.jl

@@ -0,0 +1,21 @@
+using Test
+using QuantumClifford
+using QuantumClifford: mul_left!
+using QuantumClifford.ECC
+using QuantumClifford.ECC: AbstractECC
+
+
+@testset "Gottesman codes should correct all single-qubit errors" begin
+    for j in 3:12
+        H = parity_checks(Gottesman(j))
+        syndromes = Set([]) # the set automatically removes repeated entries
+        for error_type in (single_x, single_y, single_z)
+            for bit_index in 1:nqubits(H)
+                syndrome = comm(H, error_type(nqubits(H), bit_index))
+                @test any(==(0x1), syndrome) # checking the syndrome is not trivially zero
+                push!(syndromes, syndrome)
+            end
+        end
+        @test length(syndromes) == 3*nqubits(H)
+    end
+end

test/test_ecc_syndromes.jl

@@ -10,6 +10,8 @@ codes = [
     Shor9(),
     Perfect5(),
     Cleve8(),
+    Gottesman(3),
+    Gottesman(5),
     CSS([0 1 1 0; 1 1 0 0], [1 1 1 1]),
     Toric(3,3),
     Toric(3,6),