[[2ᵐ - 1, 1, 3]] quantum Reed-Muller code (#302)

--------- Co-authored-by: Fe-r-oz <[email protected]> Co-authored-by: Stefan Krastanov <[email protected]> Co-authored-by: Stefan Krastanov <[email protected]>
QuantumSavory · Sep 27, 2024 · f420834 · f420834
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diff --git a/CHANGELOG.md b/CHANGELOG.md
@@ -8,6 +8,7 @@
 ## v0.9.11-dev
 
 - `hcat` of Tableaux objects
+- `QuantumReedMuller` codes added to the ECC module
 
 ## v0.9.10 - 2024-09-26
 

diff --git a/docs/src/references.bib b/docs/src/references.bib
@@ -454,3 +454,36 @@ @article{raveendran2022finite
   issn = {2521-327X},
   doi = {10.22331/q-2022-07-20-767},
 }
+
+@article{steane1999quantum,
+  title={Quantum reed-muller codes},
+  author={Steane, Andrew M},
+  journal={IEEE Transactions on Information Theory},
+  volume={45},
+  number={5},
+  pages={1701--1703},
+  year={1999},
+  publisher={IEEE}
+}
+
+@article{campbell2012magic,
+  title={Magic-state distillation in all prime dimensions using quantum reed-muller codes},
+  author={Campbell, Earl T and Anwar, Hussain and Browne, Dan E},
+  journal={Physical Review X},
+  volume={2},
+  number={4},
+  pages={041021},
+  year={2012},
+  publisher={APS}
+}
+
+@article{anderson2014fault,
+  title={Fault-tolerant conversion between the steane and reed-muller quantum codes},
+  author={Anderson, Jonas T and Duclos-Cianci, Guillaume and Poulin, David},
+  journal={Physical review letters},
+  volume={113},
+  number={8},
+  pages={080501},
+  year={2014},
+  publisher={APS}
+}
diff --git a/docs/src/references.md b/docs/src/references.md
@@ -37,6 +37,9 @@ For quantum code construction routines:
 - [kitaev2003fault](@cite)
 - [fowler2012surface](@cite)
 - [knill1996concatenated](@cite)
+- [steane1999quantum](@cite)
+- [campbell2012magic](@cite)
+- [anderson2014fault](@cite)
 
 For classical code construction routines:
 - [muller1954application](@cite)

diff --git a/src/ecc/ECC.jl b/src/ecc/ECC.jl
@@ -20,7 +20,7 @@ export parity_checks, parity_checks_x, parity_checks_z, iscss,
     RepCode, LiftedCode,
     CSS,
     Shor9, Steane7, Cleve8, Perfect5, Bitflip3,
-    Toric, Gottesman, Surface, Concat, CircuitCode,
+    Toric, Gottesman, Surface, Concat, CircuitCode, QuantumReedMuller,
     LPCode, two_block_group_algebra_codes, generalized_bicycle_codes, bicycle_codes,
     random_brickwork_circuit_code, random_all_to_all_circuit_code,
     evaluate_decoder,
@@ -376,10 +376,10 @@ include("codes/gottesman.jl")
 include("codes/surface.jl")
 include("codes/concat.jl")
 include("codes/random_circuit.jl")
-
 include("codes/classical/reedmuller.jl")
 include("codes/classical/recursivereedmuller.jl")
 include("codes/classical/bch.jl")
+include("codes/quantumreedmuller.jl")
 
 # qLDPC
 include("codes/classical/lifted.jl")

diff --git a/src/ecc/codes/quantumreedmuller.jl b/src/ecc/codes/quantumreedmuller.jl
@@ -0,0 +1,41 @@
+"""
+The family of `[[2ᵐ - 1, 1, 3]]` CSS Quantum-Reed-Muller codes, as discovered by Steane in his 1999 paper [steane1999quantum](@cite).
+
+Quantum codes are constructed from shortened Reed-Muller codes `RM(1, m)`, by removing the first row and column of the generator matrix `Gₘ`. Similarly, we can define truncated dual codes `RM(m - 2, m)` using the generator matrix `Hₘ` [anderson2014fault](@cite). The quantum Reed-Muller codes `QRM(m)` derived from `RM(1, m)` are CSS codes. 
+
+Given that the stabilizers of the quantum code are defined through the generator matrix of the classical code, the minimum distance of the quantum code corresponds to the minimum distance of the dual classical code, which is `d = 3`, thus it can correct any single qubit error. Since one stabilizer from the original and one from the dual code are removed in the truncation process, the code parameters are `[[2ᵐ - 1, 1, 3]]`.
+
+You might be interested in consulting [anderson2014fault](@cite) and [campbell2012magic](@cite) as well.
+
+The ECC Zoo has an [entry for this family](https://errorcorrectionzoo.org/c/quantum_reed_muller).
+"""
+struct QuantumReedMuller <: AbstractECC
+    m::Int
+    function QuantumReedMuller(m)
+        if  m < 3
+            throw(DomainError("Invalid parameters: m must be bigger than 2 in order to have a valid code."))
+        end
+        new(m)
+    end
+end
+
+function iscss(::Type{QuantumReedMuller})
+    return true
+end
+
+function parity_checks(c::QuantumReedMuller)
+    RM₁₋ₘ = generator(RecursiveReedMuller(1, c.m))
+    RM₍ₘ₋₂₎₋ₘ₎ = generator(RecursiveReedMuller(c.m-2, c.m))
+    QRM = CSS(RM₁₋ₘ[2:end, 2:end], RM₍ₘ₋₂₎₋ₘ₎[2:end, 2:end])
+    Stabilizer(QRM)
+end
+
+code_n(c::QuantumReedMuller) = 2^c.m - 1
+
+code_k(c::QuantumReedMuller) = 1
+
+distance(c::QuantumReedMuller) = 3
+
+parity_checks_x(c::QuantumReedMuller) = stab_to_gf2(parity_checks(QuantumReedMuller(c.m)))[1:c.m, 1:end÷2]
+
+parity_checks_z(c::QuantumReedMuller) = stab_to_gf2(parity_checks(QuantumReedMuller(c.m)))[end-(code_n(c::QuantumReedMuller)-2-c.m):end, end÷2+1:end]
diff --git a/test/test_ecc_base.jl b/test/test_ecc_base.jl
@@ -63,7 +63,8 @@ const code_instance_args = Dict(
     CSS => (c -> (parity_checks_x(c), parity_checks_z(c))).([Shor9(), Steane7(), Toric(4, 4)]),
     Concat => [(Perfect5(), Perfect5()), (Perfect5(), Steane7()), (Steane7(), Cleve8()), (Toric(2, 2), Shor9())],
     CircuitCode => random_circuit_code_args,
-    LPCode => (c -> (c.A, c.B)).(vcat(LP04, LP118, test_gb_codes, other_lifted_product_codes))
+    LPCode => (c -> (c.A, c.B)).(vcat(LP04, LP118, test_gb_codes, other_lifted_product_codes)),
+    QuantumReedMuller => [3, 4, 5]
 )
 
 function all_testablable_code_instances(;maxn=nothing)

diff --git a/test/test_ecc_decoder_all_setups.jl b/test/test_ecc_decoder_all_setups.jl
@@ -60,6 +60,33 @@
         end
     end
 
+    @testset "BitFlipDecoder decoder, good for sparse codes" begin
+        codes = [
+                 QuantumReedMuller(3),
+                 QuantumReedMuller(4)
+                ]
+
+        noise = 0.001
+
+        setups = [
+                  CommutationCheckECCSetup(noise),
+                  NaiveSyndromeECCSetup(noise, 0),
+                  ShorSyndromeECCSetup(noise, 0),
+                 ]
+
+        for c in codes
+            for s in setups
+                for d in [c->BitFlipDecoder(c, maxiter=10)]
+                    e = evaluate_decoder(d(c), s, 100000)
+                    #@show c
+                    #@show s
+                    #@show e
+                    @assert max(e...) < noise/4
+                end
+            end
+        end
+    end
+
     ##
 
     using Test

diff --git a/test/test_ecc_quantumreedmuller.jl b/test/test_ecc_quantumreedmuller.jl
@@ -0,0 +1,50 @@
+@testitem "Quantum Reed-Muller" begin
+    using Test
+    using Nemo: echelon_form, matrix, GF
+    using LinearAlgebra
+    using QuantumClifford
+    using QuantumClifford: canonicalize!, Stabilizer, stab_to_gf2
+    using QuantumClifford.ECC
+    using QuantumClifford.ECC: AbstractECC, QuantumReedMuller, Steane7, CSS
+
+    function designed_distance(mat)
+        dist = 3
+        for row in eachrow(mat)
+            count = sum(row)
+            if count < dist
+                return false
+            end
+        end
+        return true
+    end
+
+    @testset "Test QuantumReedMuller(r,m) properties" begin
+        for m in 3:10
+            stab = parity_checks(QuantumReedMuller(m))
+            H = stab_to_gf2(stab)
+            @test designed_distance(H) == true
+            # QuantumReedMuller(3) is the Steane7 code.
+            @test canonicalize!(parity_checks(Steane7())) == parity_checks(QuantumReedMuller(3))
+            @test code_n(QuantumReedMuller(m)) == 2^m - 1
+            @test code_k(QuantumReedMuller(m)) == 1
+            @test distance(QuantumReedMuller(m)) == 3
+            @test H == stab_to_gf2(parity_checks(CSS(parity_checks_x(QuantumReedMuller(m)), parity_checks_z(QuantumReedMuller(m)))))
+            # [[15,1,3]] qrm code from table 1 of https://arxiv.org/pdf/1705.0010
+            qrm₁₅₁₃ = S"ZIZIZIZIZIZIZIZ
+                        IZZIIZZIIZZIIZZ
+                        IIIZZZZIIIIZZZZ
+                        IIIIIIIZZZZZZZZ
+                        IIZIIIZIIIZIIIZ
+                        IIIIZIZIIIIIZIZ
+                        IIIIIZZIIIIIIZZ
+                        IIIIIIIIIZZIIZZ
+                        IIIIIIIIIIIZZZZ
+                        IIIIIIIIZIZIZIZ
+                        XIXIXIXIXIXIXIX
+                        IXXIIXXIIXXIIXX
+                        IIIXXXXIIIIXXXX
+                        IIIIIIIXXXXXXXX"
+            @test canonicalize!(parity_checks(qrm₁₅₁₃)) == canonicalize!(parity_checks(QuantumReedMuller(4)))
+        end
+    end
+end