add random Clifford circuit codes

docs/src/references.bib

@@ -165,11 +165,17 @@ @article{grassl2002algorithmic
 
 % Examples of results that employ the tableaux formalism
 
-@article{gullans2020quantum,
-  title={Quantum coding with low-depth random circuits},
-  author={Gullans, Michael J and Krastanov, Stefan and Huse, David A and Jiang, Liang and Flammia, Steven T},
-  journal={arXiv preprint arXiv:2010.09775},
-  year={2020}
+@article{gullans2021quantum,
+  title = {Quantum {{Coding}} with {{Low-Depth Random Circuits}}},
+  author = {Gullans, Michael J. and Krastanov, Stefan and Huse, David A. and Jiang, Liang and Flammia, Steven T.},
+  year = {2021},
+  month = sep,
+  journal = {Physical Review X},
+  volume = {11},
+  number = {3},
+  pages = {031066},
+  issn = {2160-3308},
+  doi = {10.1103/PhysRevX.11.031066}
 }
 
 @article{krastanov2020heterogeneous,
@@ -386,3 +392,13 @@ @article{knill1996concatenated
   journal={arXiv preprint quant-ph/9608012},
   year={1996}
 }
+
+@inproceedings{brown2013short,
+  title = {Short Random Circuits Define Good Quantum Error Correcting Codes},
+  booktitle = {2013 {{IEEE International Symposium}} on {{Information Theory}}},
+  author = {Brown, Winton and Fawzi, Omar},
+  year = {2013},
+  month = jul,
+  pages = {346--350},
+  doi = {10.1109/ISIT.2013.6620245}
+}

docs/src/tutandpub.md

@@ -4,7 +4,7 @@ This list has a number of notebooks with tutorials, examples, and reproduction o
 
 ## On the topic of explicit use of the Tableaux formalism for Stabilizer states 
 
-- [Quantum coding with low-depth random circuits](https://github.com/QuantumSavory/QuantumClifford.jl/blob/master/docs/src/notebooks/Stabilizer_Codes_Based_on_Random_Circuits.ipynb) reproducing results from [gullans2020quantum](@cite). [view on nbviewer.jupyter.org](https://nbviewer.jupyter.org/github/QuantumSavory/QuantumClifford.jl/blob/master/docs/src/notebooks/Stabilizer_Codes_Based_on_Random_Circuits.ipynb)
+- [Quantum coding with low-depth random circuits](https://github.com/QuantumSavory/QuantumClifford.jl/blob/master/docs/src/notebooks/Stabilizer_Codes_Based_on_Random_Circuits.ipynb) reproducing results from [gullans2021quantum](@cite). [view on nbviewer.jupyter.org](https://nbviewer.jupyter.org/github/QuantumSavory/QuantumClifford.jl/blob/master/docs/src/notebooks/Stabilizer_Codes_Based_on_Random_Circuits.ipynb)
 
 
 ## On the Monte Carlo and Perturbative Expansions for **Noisy** Clifford circuits

src/QuantumClifford.jl

@@ -63,6 +63,7 @@ export
     random_invertible_gf2,
     random_pauli, random_pauli!,
     random_stabilizer, random_destabilizer, random_clifford,
+    random_brickwork_clifford_circuit, random_all_to_all_clifford_circuit,
     # Noise
     applynoise!, UnbiasedUncorrelatedNoise, NoiseOp, NoiseOpAll, NoisyGate,
     PauliNoise, PauliError,

src/ecc/ECC.jl

@@ -19,7 +19,8 @@ export parity_checks, parity_checks_x, parity_checks_z, iscss,
     RepCode,
     CSS,
     Shor9, Steane7, Cleve8, Perfect5, Bitflip3,
-    Toric, Gottesman, Surface, Concat,
+    Toric, Gottesman, Surface, Concat, CircuitCode,
+    random_circuit_code,
     evaluate_decoder,
     CommutationCheckECCSetup, NaiveSyndromeECCSetup, ShorSyndromeECCSetup,
     TableDecoder,
@@ -359,6 +360,7 @@ include("codes/toric.jl")
 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/bch.jl")
 end #module

src/ecc/codes/random_circuit.jl

@@ -0,0 +1,74 @@
+using Random: AbstractRNG, GLOBAL_RNG
+
+
+"""
+`CircuitCode` is defined by a given encoding circuit `circ`.
+
+- `n`: qubit number
+- `circ`: the encoding circuit
+- `encode_qubits`: the qubits to be encoded
+
+See also: [`random_circuit_code`](@ref)
+"""
+struct CircuitCode <: AbstractECC
+    n::Int
+    circ::Vector{QuantumClifford.AbstractOperation}
+    encode_qubits::AbstractArray
+end
+
+iscss(::Type{CircuitCode}) = nothing
+
+code_n(c::CircuitCode) = prod(c.n)
+
+code_k(c::CircuitCode) = length(c.encode_qubits)
+
+function parity_checks(c::CircuitCode)
+    n = code_n(c)
+    checks = one(Stabilizer, n)[setdiff(1:n, c.encode_qubits)]
+    for op in c.circ
+        apply!(checks, op)
+    end
+    checks
+end
+
+"""
+Random Clifford circuit code.
+
+The code is generated by a random Clifford circuit `circ` that encodes a subset of qubits `encode_qubits` into logical qubits.
+The connectivity of the random circuit can be either all-to-all [brown2013short](@cite) or brickwork in some dimensions [gullans2021quantum](@cite).
+Each gate in the random circuit is a random 2-qubit Clifford gate.
+
+For `brickwork` connectivity, the qubits are arranged as a lattice, and `lattice_size` contains side length in each dimension.
+For example, a chain of length five will have `lattice_size = (5,)`, and a 5×5 lattice will have `lattice_size = (5, 5)`.
+
+In multi-dimensional cases, gate layers act alternatively along each direction.
+The nearest two layers along the same direction are offset by one qubit, forming a so-called brickwork.
+The boundary condition is chosen as open.
+"""
+function random_circuit_code end
+
+function random_circuit_code(rng::AbstractRNG, n::Int, connect::Val{:alltoall}, ngates::Int, k::Int)
+    CircuitCode(n, random_all_to_all_clifford_circuit(rng, n, ngates), collect(1:k))
+end
+
+function random_circuit_code(n::Int, connect::Val{:alltoall}, ngates::Int, k::Int)
+    CircuitCode(n, random_all_to_all_clifford_circuit(n, ngates), collect(1:k))
+end
+
+function random_circuit_code(rng::AbstractRNG, n::Int, connect::Val{:alltoall}, ngates::Int, encode_qubits::AbstractArray)
+    CircuitCode(n, random_all_to_all_clifford_circuit(rng, n, ngates), encode_qubits)
+end
+
+function random_circuit_code(n::Int, connect::Val{:alltoall}, ngates::Int, encode_qubits::AbstractArray)
+    CircuitCode(n, random_all_to_all_clifford_circuit(n, ngates), encode_qubits)
+end
+
+# TODO it would be nicer if we can use CartesianIndex for `encode_qubits` in brickworks,
+# but its conversion to LinearIndex is limited, not supporting non-one step.
+function random_circuit_code(rng::AbstractRNG, lattice_size::NTuple{N,Int} where {N}, connect::Val{:brickwork}, nlayers::Int, encode_qubits::AbstractArray)
+    CircuitCode(prod(lattice_size), random_brickwork_clifford_circuit(rng, lattice_size, nlayers), encode_qubits)
+end
+
+function random_circuit_code(lattice_size::NTuple{N,Int} where {N}, connect::Val{:brickwork}, nlayers::Int, encode_qubits::AbstractArray)
+    CircuitCode(prod(lattice_size), random_brickwork_clifford_circuit(lattice_size, nlayers), encode_qubits)
+end

src/entanglement.jl

@@ -139,7 +139,7 @@ It is the list of endpoints of a tableau in the clipped gauge.
 If `clip=true` (the default) the tableau is converted to the clipped gauge in-place before calculating the bigram.
 Otherwise, the clip gauge conversion is skipped (for cases where the input is already known to be in the correct gauge).
 
-Introduced in [nahum2017quantum](@cite), with a more detailed explanation of the algorithm in [li2019measurement](@cite) and [gullans2020quantum](@cite).
+Introduced in [nahum2017quantum](@cite), with a more detailed explanation of the algorithm in [li2019measurement](@cite) and [gullans2021quantum](@cite).
 
 See also: [`canonicalize_clip!`](@ref)
 """
@@ -186,7 +186,7 @@ function entanglement_entropy(state::AbstractStabilizer, subsystem_range::UnitRa
     # JET-XXX The ::Matrix{Int} should not be necessary, but they help with inference
     bg = bigram(state; clip=clip)::Matrix{Int}
     # If the state is mixed, this formula is valid only for contiguous regions that don't wrap around.
-    # See Eq. E7 of gullans2020quantum.
+    # See Eq. E7 of gullans2021quantum.
     # As subsystem_range is UnitRange, we know the formula will be valid.
     length(subsystem_range) - count(r->(r[1] in subsystem_range && r[2] in subsystem_range), eachrow(bg))
 end

src/randoms.jl

@@ -228,3 +228,43 @@ function fill_tril(rng, matrix, n; symmetric::Bool=false)
     end
     matrix
 end
+
+##############################
+# Random circuit
+##############################
+
+function random_brickwork_clifford_circuit(rng::AbstractRNG, arrange::NTuple{N,Int} where {N}, nlayers::Int)
+    circ = QuantumClifford.SparseGate[]
+    cartesian = CartesianIndices(arrange)
+    dim = length(arrange)
+    nqubits = prod(arrange)
+    for i in 1:nlayers
+        gate_direction = (i - 1) % dim + 1
+        l = arrange[gate_direction]
+        brickwise_parity = dim == 1 ? i % 2 : 1 - (i ÷ dim) % 2
+        for j in 1:nqubits
+            cardj = collect(cartesian[j].I)
+            if cardj[gate_direction] % 2 == brickwise_parity && cardj[gate_direction] != l # open boundary
+                cardk = cardj
+                cardk[gate_direction] = cardk[gate_direction] + 1
+                k = LinearIndices(cartesian)[cardk...]
+                push!(circ, SparseGate(random_clifford(rng, 2), [j, k]))
+            end
+        end
+    end
+    circ
+end
+
+random_brickwork_clifford_circuit(arrange::NTuple{N,Int} where {N}, nlayers::Int) = random_brickwork_clifford_circuit(GLOBAL_RNG, arrange, nlayers)
+
+function random_all_to_all_clifford_circuit(rng::AbstractRNG, nqubits::Int, ngates::Int)
+    circ = QuantumClifford.SparseGate[]
+    for i in 1:ngates
+        j = rand(1:nqubits)
+        k = rand(1:nqubits-1)
+        push!(circ, SparseGate(random_clifford(rng, 2), [j, (j + k - 1) % nqubits + 1]))
+    end
+    circ
+end
+
+random_all_to_all_clifford_circuit(nqubits::Int, ngates::Int) = random_all_to_all_clifford_circuit(GLOBAL_RNG, nqubits, ngates)

test/test_ecc_base.jl

@@ -5,12 +5,23 @@ using InteractiveUtils
 
 # generate instances of all implemented codes to make sure nothing skips being checked
 
+# we does not include smaller random circuit code, because some of them has a bad distance and fails the TableDecoder test
+const random_circuit_args = repeat([
+        (20, Val(:alltoall), 200, 1), (40, Val(:alltoall), 200, [1, 20]),
+        ((20,), Val(:brickwork), 50, [1]), ((20,), Val(:brickwork), 50, 1:2:20),
+        ((5, 5), Val(:brickwork), 50, [1]), ((3, 3, 3), Val(:brickwork), 50, [1])
+    ], 10) # repeat for more randomness
+
+
+random_circuit_code_args = [map(f -> getfield(random_circuit_code(c...), f), fieldnames(CircuitCode)) for c in random_circuit_args]
+
 const code_instance_args = Dict(
     Toric => [(3,3), (4,4), (3,6), (4,3), (5,5)],
     Surface => [(3,3), (4,4), (3,6), (4,3), (5,5)],
     Gottesman => [3, 4, 5],
-    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())],
+    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
 )
 
 function all_testablable_code_instances(;maxn=nothing)