add BP-OSD (#22)

Co-authored-by: Stefan Krastanov <[email protected]>

CHANGELOG.md

@@ -0,0 +1,7 @@
+# News
+
+## v0.3.2 - 2024-11-15
+
+- Add a (still unoptimized) implementation of a BP OSD decoder.
+
+## Older - before 2021-10-28 unrecorded

Project.toml

@@ -1,7 +1,7 @@
 name = "LDPCDecoders"
 uuid = "3c486d74-64b9-4c60-8b1a-13a564e77efb"
 authors = ["Krishna Praneet Gudipaty", "Stefan Krastanov", "QuantumSavory contributors"]
-version = "0.3.1"
+version = "0.3.2"
 
 [deps]
 DelimitedFiles = "8bb1440f-4735-579b-a4ab-409b98df4dab"

src/LDPCDecoders.jl

@@ -12,6 +12,7 @@ using RowEchelon
 export
     decode!, batchdecode!,
     BeliefPropagationDecoder,
+    BeliefPropagationOSDDecoder,
     BitFlipDecoder
 
 include("generator.jl")
@@ -22,7 +23,9 @@ include("parity_generator.jl")
 
 include("decoders/abstract_decoder.jl")
 include("decoders/belief_propagation.jl")
+include("decoders/belief_propagation_osd.jl")
 include("decoders/iterative_bitflip.jl")
+
 include("syndrome_bp_decoder.jl")
 include("syndrome_simulator.jl")
 include("syndrome_it_decoder.jl")

src/decoders/belief_propagation.jl

@@ -44,7 +44,7 @@ struct BeliefPropagationDecoder <: AbstractDecoder
   scratch::BeliefPropagationScratchSpace
 end
 
-function BeliefPropagationDecoder(H, per::Float64, max_iters::Int)
+function BeliefPropagationDecoder(H::Union{SparseArrays.SparseMatrixCSC{Bool,Int}, BitMatrix}, per::Float64, max_iters::Int)
   s, n = size(H)
   sparse_H = sparse(H)
   sparse_HT = sparse(H')
@@ -108,7 +108,7 @@ true
 function decode!(decoder::BeliefPropagationDecoder, syndrome::AbstractVector) # TODO check if casting to bitarrays helps with performance -- if it does, set up warnings to the user for cases where they have not done the casting
   reset!(decoder)
   rows::Vector{Int} = rowvals(decoder.sparse_H);
-  rowsT::Vector{Int} = rowvals(decoder.sparse_HT); 
+  rowsT::Vector{Int} = rowvals(decoder.sparse_HT);
   setup = decoder.scratch
 
   for j in 1:decoder.n
@@ -138,7 +138,7 @@ function decode!(decoder::BeliefPropagationDecoder, syndrome::AbstractVector) #
 
     for j in 1:decoder.n
       temp::Float64 = setup.channel_probs[j] / (1 - setup.channel_probs[j])
-      
+
       for k in nzrange(decoder.sparse_H, j)
         setup.bit_2_check[rows[k],j] = temp
         temp *= setup.check_2_bit[rows[k],j]
@@ -166,7 +166,7 @@ function decode!(decoder::BeliefPropagationDecoder, syndrome::AbstractVector) #
 
     syndrome_decoded = (decoder.sparse_H * setup.err) .% 2
     if all(syndrome_decoded .== syndrome)
-      converged = true 
+      converged = true
       break # Break if converged
     end
   end

src/decoders/belief_propagation_osd.jl

@@ -0,0 +1,113 @@
+struct BeliefPropagationOSDDecoder <: AbstractDecoder
+    """A belief propagation decoder as a subroutine"""
+    bp_decoder::BeliefPropagationDecoder
+    """Dense form of the parity check matrix"""
+    H::BitMatrix
+    """The order of OSD; defaulted to be 0 in the constructor"""
+    osd_order::Int
+end
+
+function BeliefPropagationOSDDecoder(H::BitMatrix, per::Float64, max_iters::Int; osd_order::Int=0)
+    bp_decoder = BeliefPropagationDecoder(H, per, max_iters)
+    return BeliefPropagationOSDDecoder(bp_decoder, H, osd_order)
+end
+
+function rowswap!(H::BitMatrix, i, j)
+    @inbounds H[i, :], H[j, :] = H[j, :], H[i, :] # TODO This could be further optimized?
+end
+
+function decode!(decoder::BeliefPropagationOSDDecoder, syndrome::AbstractVector)
+    # use BP to get hard and soft decisions
+    bp_err, converged = decode!(decoder.bp_decoder, syndrome) # hard decisions
+    bp_log_probabs = decoder.bp_decoder.scratch.log_probabs # soft decisions
+    bp_probabs = exp.(bp_log_probabs)
+    # sort columns by reliability, less reliable columns first
+    sort_by_reliability = sortperm(max.(bp_probabs, 1 .- bp_probabs), rev=true)
+    H_sorted = decoder.H[:, sort_by_reliability]
+    bp_err_sorted = bp_err[sort_by_reliability]
+    # TODO an optimized version of OSD can be implemented when osd_order = 0, see Algorithm 2 in	https://doi.org/10.22331/q-2021-11-22-585
+    err = osd(H_sorted, syndrome, bp_err_sorted, decoder.osd_order)
+    return err[invperm(sort_by_reliability)], converged # also return whether BP is converged
+end
+
+function osd(H, syndrome, bp_err, osd_order)
+    m, n = size(H)
+    # diagnolize the submatrix corresponding to independent columns via Gaussian elimination
+    # first obtain the row canonical form
+    # and find least reliable indices, i.e., the first r pivot columns (assume H is rearranged by reliability)
+    least_reliable_rows = [] # row indices of pivot elements
+    least_reliable_cols = [] # column indices of pivot elements
+    r = 0 # compute rank of H
+    i, j = 1, 1
+    s = copy(syndrome) # transform syndrome along with H in Gaussian elimination
+
+    while i <= m && j <= n
+        k = findfirst(H[i:end, j])
+        if isnothing(k) # not an independent column
+            j += 1
+        else
+            if k > 1
+                ii = i + k - 1 # the first row after `i` with 1 in column `j`
+                rowswap!(H, i, ii) # TODO For optimization: Is this swap necessary? We may just track the row index
+                s[i], s[ii] = s[ii], s[i]
+            end
+            for ii in i+1:m
+                if H[ii, j]
+                    H[ii, :] .⊻= H[i, :]
+                    s[ii] ⊻= s[i]
+                end
+            end
+            push!(least_reliable_rows, i)
+            push!(least_reliable_cols, j)
+            i += 1
+            j += 1
+            r += 1
+        end
+    end
+
+    # then obtain a diagonal submatrix on the least reliable part
+    for (i, j) in zip(reverse(least_reliable_rows), reverse(least_reliable_cols))
+        for ii in 1:i-1
+            if H[ii, j]
+                H[ii, :] .⊻= H[i, :]
+                s[ii] ⊻= s[i]
+            end
+        end
+    end
+
+    if osd_order > n - r
+        @warn "The order of OSD $osd_order is greater than the size of the information set $(n-r). We set osd_order = $(n-r)."
+        osd_order = n - r
+    end
+
+    best_err = copy(bp_err)
+    err = Bool.(copy(bp_err)) # TODO why error is in Float in BP?
+    most_reliable_cols = setdiff(1:n, least_reliable_cols)
+    min_weight = n + 1
+
+    for x in 0:2^osd_order-1
+        # first compute the `most_reliable_cols` part of errors
+        # try all possible errors on the first `osd_order` bits within `most_reliable_cols`
+        if x != 0
+            trial_err = BitArray([x >> i & 1 for i in 0:osd_order-1])
+            for j in 1:osd_order
+                err[most_reliable_cols[j]] = trial_err[j]
+            end
+        end
+        # then based on the `most_reliable_cols` part of errors, compute the `least_reliable_cols` part of errors
+        for (i, j) in zip(least_reliable_rows, least_reliable_cols)
+            err[j] = s[i]
+            for k in most_reliable_cols
+                err[j] ⊻= H[i, k] * err[k]
+            end
+        end
+        weight = sum(err) # This weight is set for depolarizing noise
+        # TODO More generally, it should be a function depending on the noise model
+        if weight < min_weight
+            min_weight = weight
+            best_err = copy(err)
+        end
+    end
+
+    return best_err
+end

test/runtests.jl

@@ -30,6 +30,7 @@ println("ENV[\"PYTHON\"] = \"$(get(ENV,"PYTHON",nothing))\"")
 
 @doset "oldtests"
 @doset "bp_decoder"
+@doset "bposd_decoder"
 @doset "bf_decoder"
 
 VERSION >= v"1.10" && @doset "doctests"

test/test_bposd_decoder.jl

@@ -0,0 +1,53 @@
+using Test
+using LDPCDecoders
+
+@testset "test_bposd_decoder.jl" begin
+
+  """Test for BP-OSD decoder"""
+  function test_bposd_decoder()
+    H = LDPCDecoders.parity_check_matrix(1000, 10, 9)
+    per = 0.01
+    err = rand(1000) .< per
+    syn = (H * err) .% 2
+
+    bposd = BeliefPropagationOSDDecoder(H, per, 100)
+    guess, success = decode!(bposd, syn)
+
+    return guess == err
+  end
+
+  """Test high order OSD"""
+  function test_bposd_decoder_high_order()
+    H = LDPCDecoders.parity_check_matrix(1000, 10, 9)
+    per = 0.01
+    err = rand(1000) .< per
+    syn = (H * err) .% 2
+
+    orders = 2:5
+    succ = true
+    for osd_order in orders
+      bposd = BeliefPropagationOSDDecoder(H, per, 100; osd_order=osd_order)
+      guess, success = decode!(bposd, syn)
+      succ = succ & (guess == err)
+    end
+
+    return succ
+  end
+
+  """Test for BP-OSD decoder with large error rate. Even if the decoding is not accurate, OSD will still ensure consistency between guess and syndromes."""
+  function test_bposd_decoder_large_error_rate()
+    H = LDPCDecoders.parity_check_matrix(1000, 10, 9)
+    per = 0.2
+    err = rand(1000) .< per
+    syn = (H * err) .% 2
+
+    bposd = BeliefPropagationOSDDecoder(H, per, 100)
+    guess, success = decode!(bposd, syn)
+
+    return syn == (H * guess) .% 2
+  end
+
+  @test test_bposd_decoder()
+  @test test_bposd_decoder_high_order()
+  @test test_bposd_decoder_large_error_rate()
+end