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QuantumExpanders is a
pkg> add https://github.com/QuantumSavory/QuantumExpanders.jl.gitTo update, just type up in the package mode.
The library provides the following methods to construct explicit instances of Quantum Tanner codes.
graph TD
QuantumTannerCodes["Quantum Tanner Codes"] --> RandomMethods["Random Methods"]
QuantumTannerCodes --> DeterministicMethods["Deterministic Methods"]
subgraph "Random construction"
RandomMethods --> RandomQuantumTannerCode["`random_quantum_Tanner_code`"]
end
subgraph "Deterministic construction"
DeterministicMethods --> QuantumTannerCode["`QuantumTannerCode`"]
DeterministicMethods --> GeneralizedQuantumTannerCode["`GeneralizedQuantumTannerCode`"]
end
random_quantum_Tanner_codeconstructs a quantum CSS code by instantiating two classical Tanner codes, 𝒞ᶻ and 𝒞ˣ, on the graphs 𝒢₀□ and 𝒢₁□ of a left-right Cayley complex leverrier2022quantum. This complex is generated from a group G, and two generating sets A and B of sizes Δ_A and Δ_B, which, need not satisfy the Total No-Conjugacy condition because of quadripartite construction of LRCC. The code's qubits bijectively correspond to the squares Q of the complex, with theedge_*_idxoutput providing the essential mappings between qubit indices, graph edges in 𝒢₀□ and 𝒢₁□, and the local coordinate sets A×B at each vertex. The Z-parity checks of the quantum code are defined as the generators of 𝒞ᶻ = T(𝒢₀□, (C_A⊗C_B)⊥), enforcing local constraints from the dual tensor code at each vertex of V₀. Similarly, the X-parity checks are generators of 𝒞ˣ = T(𝒢₁□, (C_A⊥⊗C_B⊥)⊥), enforced at vertices of V₁. When the Cayley graphs Cay(G,A) and Cay(G,B) are Ramanujan and the component codes C_A and C_B are randomly chosen with robust dual tensor properties, this construction produces an asymptotically good quantum LDPC code with parameters [[n, Θ(n), Θ(n)]]. The QT code implementation provides a simplified variant of the Panteleev-Kalachev quantum LDPC codes panteleev2022asymptoticallygoodquantumlocally and is related to the locally testable code of dinur2022locally.
Here is the novel [[360, 61, (3, 10)]] quantum Tanner code constructed from Morgenstern Ramanujan graphs
for even prime power q.
julia> l = 1; i = 2;
julia> q = 2^l
2
julia> Δ = q+1
3
julia> SL₂, B = morgenstern_generators(l, i)
[ Info: |SL₂(𝔽(4))| = 60
(SL(2,4), Oscar.MatrixGroupElem{Nemo.FqFieldElem, Nemo.FqMatrix}[[o+1 o+1; 1 o+1], [o+1 1; o+1 o+1], [o+1 o; o o+1]])
julia> A = alternative_morgenstern_generators(B, FirstOnly())
4-element Vector{Oscar.MatrixGroupElem{Nemo.FqFieldElem, Nemo.FqMatrix}}:
[0 1; 1 o+1]
[o+1 1; 1 0]
[o+1 o+1; o 0]
[0 o+1; o o+1]
julia> rng = MersenneTwister(892529278);
julia> hx, hz = random_quantum_Tanner_code(0.75, SL₂, A, B, rng=rng);
(length(group), length(A), length(B)) = (60, 4, 3)
length(group) * length(A) * length(B) = 720
[ Info: |V₀| = |V₁| = |G| = 60
[ Info: |E_A| = Δ|G| = 240, |E_B| = Δ|G| = 180
[ Info: |Q| = Δ²|G|/2 = 360
Hᴬ = [1 1 1 0]
Hᴮ = [0 1 1; 1 1 0]
Cᴬ = [1 1 0 0; 1 0 1 0; 0 0 0 1]
Cᴮ = [1 1 1]
size(Cˣ) = (3, 12)
size(Cᶻ) = (2, 12)
r1 = rank(𝒞ˣ) = 179
r2 = rank(𝒞ᶻ) = 120
julia> c = CSS(hx, hz);
julia> import JuMP; import HiGHS;
julia> code_n(c), code_k(c)
(360, 61)
julia> distance(c, DistanceMIPAlgorithm(solver = name ,logical_operator_type = :Z,time_limit = 900)), distance(c, DistanceMIPAlgorithm(solver = name ,logical_operator_type = :X,time_limit = 900))
(3, 10)QuantumSavory.jl is developed by many volunteers, managed at Prof. Krastanov's lab at University of Massachusetts Amherst.
The development effort is supported by The NSF Engineering and Research Center for Quantum Networks, and by NSF Grant 2346089 "Research Infrastructure: CIRC: New: Full-stack Codesign Tools for Quantum Hardware".