Surface code (QuantumSavory#246)

---------

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

CHANGELOG.md

@@ -5,13 +5,11 @@
 
 # News
 
-## v0.9.2 - 2024-03-22
-
-- Implemented the classical Reed-Muller code
-
-## v0.9.1 - 2024-03-20
+## v0.9.1 - dev
 
 - Implemented `iscss` function to identify whether a given code is known to be a CSS (Calderbank-Shor-Steane) code.
+- Implemented the classical Reed-Muller code
+- Added the surface code to the ECC module.
 
 ## v0.9.0 - 2024-03-19
 

docs/src/references.bib

@@ -263,6 +263,34 @@ @article{gottesman1996class
   publisher={APS}
 }
 
+@article{kitaev2003fault,
+	title = {Fault-tolerant quantum computation by anyons},
+	volume = {303},
+	issn = {00034916},
+	url = {https://linkinghub.elsevier.com/retrieve/pii/S0003491602000180},
+	doi = {10.1016/S0003-4916(02)00018-0},
+	abstract = {A two-dimensional quantum system with anyonic excitations can be considered as a quantum computer. Unitary transformations can be performed by moving the excitations around each other. Measurements can be performed by joining excitations in pairs and observing the result of fusion. Such computation is fault-tolerant by its physical nature.},
+	pages = {2--30},
+	number = {1},
+	journaltitle = {Annals of Physics},
+	shortjournal = {Annals of Physics},
+	author = {Kitaev, A.Yu.}
+}
+
+@article{fowler2012surface,
+	title = {Surface codes: Towards practical large-scale quantum computation},
+	volume = {86},
+	issn = {1050-2947, 1094-1622},
+	url = {https://link.aps.org/doi/10.1103/PhysRevA.86.032324},
+	doi = {10.1103/PhysRevA.86.032324},
+	shorttitle = {Surface codes},
+	pages = {032324},
+	number = {3},
+	journaltitle = {Physical Review A},
+	shortjournal = {Phys. Rev. A},
+	author = {Fowler, Austin G. and Mariantoni, Matteo and Martinis, John M. and Cleland, Andrew N.},
+}
+
 @article{muller1954application,
   title={Application of Boolean algebra to switching circuit design and to error detection},
   author={Muller, David E},
@@ -308,4 +336,4 @@ @book{djordjevic2021quantum
   author={Djordjevic, Ivan B},
   year={2021},
   publisher={Academic Press}
-}
+}

docs/src/references.md

@@ -34,6 +34,8 @@ For quantum code construction routines:
 - [gottesman1997stabilizer](@cite)
 - [yu2013all](@cite)
 - [chao2018quantum](@cite)
+- [kitaev2003fault](@cite)
+- [fowler2012surface](@cite)
 
 For classical code construction routines:
 - [muller1954application](@cite)

src/ecc/ECC.jl

@@ -19,7 +19,7 @@ export parity_checks, parity_checks_x, parity_checks_z, iscss,
     RepCode,
     CSS,
     Shor9, Steane7, Cleve8, Perfect5, Bitflip3,
-    Toric, Gottesman,
+    Toric, Gottesman, Surface,
     evaluate_decoder,
     CommutationCheckECCSetup, NaiveSyndromeECCSetup, ShorSyndromeECCSetup,
     TableDecoder,
@@ -345,5 +345,6 @@ include("codes/shorcode.jl")
 include("codes/clevecode.jl")
 include("codes/toric.jl")
 include("codes/gottesman.jl")
+include("codes/surface.jl")
 include("codes/classical/reedmuller.jl")
 end #module

src/ecc/codes/surface.jl

@@ -0,0 +1,60 @@
+"""The planar surface code refers to the code [kitaev2003fault](@cite) in a 2D lattice with open boundaries.
+
+Illustration of a 3×2 surface code, where qubits are located on the edges:
+
+```
+|---1--(Z)--2---|---3---|
+|  (X)  7       8       o
+|---4---|---5---|---6---|
+|       o       o       o
+|       |       |       |
+```
+
+The surface code has open boundary conditions, unlike the toric code. To this end, we remove qubits (denoted by "o") and parity checks on the right and bottom sides.
+
+Faces like `(1,4,7)` have X checks, and crosses like `(1,2,7)` have Z checks. Due to the removal of the bottom and right sides, we have some 3-qubit checks on the boundaries.
+
+```jldoctest
+julia> parity_checks(Surface(3,2))
++ X__X__X_
++ _X__X_XX
++ __X__X_X
++ ZZ____Z_
++ _ZZ____Z
++ ___ZZ_Z_
++ ____ZZ_Z
+```
+
+More information can be seen in [fowler2012surface](@cite).
+"""
+struct Surface <: AbstractECC
+    dx::Int
+    dz::Int
+end
+
+function iscss(::Type{Surface})
+    return true
+end
+
+code_n(c::Surface) = 2*c.dx*c.dz - c.dx -c.dz + 1
+
+function parity_checks_xz(c::Surface)
+    tc = Toric(c.dx, c.dz)
+    hx, hz = parity_checks_xz(tc)
+    n = code_n(tc)
+    nchecks = c.dx*c.dz - 1
+    # remove qubits on the right and bottom sides from the toric code
+    removed_qubit_indices = vcat(
+        c.dx*(c.dz+1):c.dx:c.dx*(2*c.dz-1), # right side
+        c.dx*(2*c.dz-1)+1:2*c.dx*c.dz) # bottom side
+    qubit_indices = setdiff(1:n, removed_qubit_indices)
+    # also remove the checks on the right and bottom sides
+    x_check_indices = setdiff(1:nchecks, nchecks-c.dx+2:nchecks)
+    z_check_indices = setdiff(1:nchecks, c.dx:c.dx:nchecks)
+    hx[x_check_indices,qubit_indices], hz[z_check_indices,qubit_indices]
+end
+
+parity_checks_x(c::Surface) = parity_checks_xz(c)[1]
+parity_checks_z(c::Surface) = parity_checks_xz(c)[2]
+
+parity_checks(c::Surface) = parity_checks(CSS(parity_checks_xz(c)...))

src/ecc/codes/toric.jl

@@ -1,4 +1,29 @@
-"""The Toric code."""
+"""The Toric code [kitaev2003fault](@cite).
+
+Illustration of a 2x2 toric code, where qubits are located on the edges:
+
+```
+|--1-(Z)-2--|
+| (X) 5     6
+|--3--|--4--|
+|     7     8
+|     |     |
+```
+
+It is important to note that the toric code has periodic boundary conditions, which means that the top and bottom sides are essentially glued together, as are the left and right sides.
+
+Faces like `(1,3,5,6)` have X checks, and crosses like `(1,2,5,7)` have Z checks.
+
+```jldoctest
+julia> parity_checks(Toric(2,2))
++ X_X_XX__
++ _X_XXX__
++ X_X___XX
++ ZZ__Z_Z_
++ ZZ___Z_Z
++ __ZZZ_Z_
+```
+"""
 struct Toric <: AbstractECC
     dx::Int
     dz::Int

test/test_ecc_decoder_all_setups.jl

@@ -11,7 +11,7 @@ import LDPCDecoders
         Shor9(),
         Perfect5(),
         Cleve8(),
-        Gottesman(3),
+        Gottesman(3)
         #Gottesman(4), bad threshold
         #Gottesman(5), bad threshold
     ]
@@ -76,7 +76,9 @@ import LDPCDecoders
 @testset "matching decoder, good as long as column weight of the code is limited" begin
     codes = [
         Toric(8,8),
-        Toric(9,9)
+        Toric(9,9),
+        Surface(8,8),
+        Surface(9,9)
     ]
 
     noise = 0.01

test/test_ecc_encoding.jl

@@ -25,6 +25,10 @@ using QuantumClifford.ECC
         :(Toric(3,6)),
         :(Toric(6,4)),
         :(Toric(8,8)),
+        :(Surface(3,3)),
+        :(Surface(3,6)),
+        :(Surface(6,4)),
+        :(Surface(8,8)),
         fill(:(random_stabilizer(5,7)), 100)...
         ]
 

test/test_ecc_iscss.jl

@@ -3,6 +3,18 @@ using QuantumClifford
 using QuantumClifford.ECC
 using QuantumClifford.ECC: AbstractECC
 
+known_all_codes = [
+    Shor9(),
+    Steane7(),
+    Gottesman(3),
+    Cleve8(),
+    Perfect5(),
+    Toric(8,8),
+    Surface(8,8),
+    CSS([0 1 1 0; 1 1 0 0], [1 1 1 1]),
+    Bitflip3()
+]
+
 function is_css_matrix(H)
     nrows, ncols = size(H)
     for i in 1:nrows
@@ -17,8 +29,6 @@ function is_css_matrix(H)
     return true
 end
 
-known_all_codes = [Shor9(), Steane7(), Gottesman(3), Cleve8(), Perfect5(), Toric(8,8), CSS([0 1 1 0; 1 1 0 0], [1 1 1 1]), Bitflip3()]
-
 @testset "is CSS" begin
     for code in known_all_codes
         H = parity_checks(code)

test/test_ecc_syndromes.jl

@@ -17,6 +17,8 @@ codes = [
     Toric(3,6),
     Toric(6,4),
     Toric(8,8),
+    Surface(4,6),
+    Surface(8,8)
 ]
 
 ##