Corollary 3: Distance-Balanced Family of Quantum Codes #
There exists an explicit construction of a family of [[N, K, D]] LDPC quantum codes
with K ∈ Θ(N^{4/5}) and D ∈ Ω(N^{3/5}).
Construction #
- Start with the
[[N₀, K₀, D_{X,0}, D_{Z,0}]]LDPC subsystem CSS codes from Thm_15 withK₀ ∈ Θ(N₀^{2/3}),D_{X,0} ∈ Ω(N₀^{1/3}),D_{Z,0} ∈ Θ(N₀). - Choose classical codes of length
n_c ∈ Θ(N₀^{2/3})via Gilbert–Varshamov (Thm_10) withk_c ∈ Θ(n_c)andd_c ∈ Θ(n_c). - Apply the distance balancing procedure of Evra–Kaufman–Zemor [EKZ, Thm 1],
producing a CSS code with
N = N₀ · n_c,K = K₀ · k_c,D_X ≥ D_{X,0} · d_c,D_Z ≥ D_{Z,0}. - Compute:
N ∈ Θ(N₀^{5/3}),K ∈ Θ(N₀^{4/3}) = Θ(N^{4/5}),D ∈ Ω(N₀) = Ω(N^{3/5}).
Main Results #
ekz_distance_balancing— axiom for the EKZ distance balancing proceduredistanceBalancedFamily— the main corollary
Distance-Balanced Parameters #
The distance-balanced code length is N = N₀ · n_c where N₀ = codeLength vp
and n_c = q² is the classical code length.
The classical code length n_c = q², which is in Θ(N₀^{2/3}).
Equations
Instances For
The balanced code length N = N₀ · n_c = codeLength(vp) · q².
Equations
Instances For
EKZ Distance Balancing Axiom #
The distance balancing procedure of Evra–Kaufman–Zemor [EKZ22] takes a subsystem CSS code and a classical code, and produces a (non-subsystem) CSS code with balanced distances. We axiomatize this as a single well-known external theorem.
The output of the EKZ distance balancing procedure for a valid prime q.
This bundles the CSS code produced by the procedure together with its LDPC property
and the parameter bounds guaranteed by [EKZ22, Theorem 1].
- rX : ℕ
Number of X-type check rows of the balanced code.
- rZ : ℕ
Number of Z-type check rows of the balanced code.
- code : CSSCode (balancedCodeLength vp) self.rX self.rZ
The CSS code produced by the EKZ distance balancing procedure. Its block length is
N = N₀ · n_c = codeLength(vp) · q². - w : ℕ
Weight bound for the LDPC property of the balanced code.
- isLDPC_HX : HasBoundedWeight self.code.HX self.w
The X-parity check matrix has bounded weight.
- isLDPC_HZT : HasBoundedWeight self.code.HZT self.w
The Z-parity check matrix has bounded weight.
- hK : self.code.logicalQubits ≥ (ExplicitFamilyQuantumCodes.balanced_product_construction vp).code.logicalQubits * (classicalCodeLength vp / 2 + 1)
The number of logical qubits satisfies
K ≥ K₀ · k_cwhereK₀is the subsystem code's logical qubit count andk_c > n_c/2is the classical code dimension from Gilbert–Varshamov (Thm_10). - hDX : self.code.dX ≥ (ExplicitFamilyQuantumCodes.balanced_product_construction vp).code.dX * (classicalCodeLength vp / 10)
The X-distance satisfies
D_X ≥ D_{X,0} · d_cwhereD_{X,0}is the subsystem code's X-distance andd_c ≥ δ·n_cwithδ = 1/10. - hK_upper : self.code.logicalQubits ≤ (ExplicitFamilyQuantumCodes.balanced_product_construction vp).code.logicalQubits * classicalCodeLength vp
Upper bound on K:
K ≤ K₀ · n_c. SinceK = K₀ · k_candk_c ≤ n_c, this follows from the EKZ construction. The Z-distance is preserved:
D_Z ≥ D_{Z,0}.
Instances For
Evra–Kaufman–Zemor Distance Balancing ([EKZ22, Theorem 1]; Evra, Kaufman, Zemor, "Decodable quantum LDPC codes beyond the √n distance barrier using high-dimensional expanders", FOCS 2022).
Given the subsystem CSS code from Theorem 15 (with parameters
[[N₀, K₀, D_{X,0}, D_{Z,0}]]) and a classical [n_c, k_c, d_c]-code from
Gilbert–Varshamov (Theorem 10), the EKZ distance balancing procedure produces a
(non-subsystem) CSS code with:
- Block length
N = N₀ · n_c - Logical qubits
K ≥ K₀ · k_c - X-distance
D_X ≥ D_{X,0} · d_c - Z-distance
D_Z ≥ D_{Z,0} - LDPC property preserved (weight bounded by a constant).
Satisfiability witness: the EKZ axiom's premise (a valid prime) is satisfiable.
The balanced code for a given valid prime.
Equations
Instances For
Arithmetic Helper Lemmas #
For a valid prime, q² / 2 + 1 ≥ 1.
For a valid prime (q ≥ 41), q² / 2 + 1 ≥ q² / 4.
For a valid prime (q ≥ 41), q² / 10 ≥ q² / 20.
N Bounds: N ∈ Θ(q⁵) #
The balanced code length N = codeLength(vp) · q² satisfies N ∈ Θ(q⁵) since
codeLength(vp) ∈ Θ(q³).
balancedCodeLength expressed in terms of codeLength and classicalCodeLength.
N lower bound: N = codeLength · q² ≥ c · q⁵ for a constant c > 0.
N upper bound: N = codeLength · q² ≤ C · q⁵ for a constant C > 0.
K Bounds: K ∈ Θ(q⁴) #
The logical qubit count K of the balanced code satisfies K ∈ Θ(q⁴).
The lower bound uses K ≥ K₀ · k_c from the EKZ axiom, combined with
K₀ ∈ Θ(q²) from Thm_15 and k_c ∈ Θ(q²) from GV.
K lower bound: K ≥ c · q⁴ for a constant c > 0.
K upper bound: K ≤ C · q⁴ for a constant C > 0.
Uses hK_upper: K ≤ K₀ · n_c from the EKZ axiom, combined with
K₀ ≤ C₀(q²-1) from Thm_15 and n_c = q².
Distance Bounds: D_X, D_Z ∈ Ω(q³) #
D_X lower bound: DX ≥ c · q³ for a constant c > 0.
D_Z lower bound: DZ ≥ c · q³ for a constant c > 0.
N Unbounded #
The family is infinite: for any M, there exists a valid prime with
the balanced code length exceeding M.
Main Theorem #
Corollary 3: Distance-Balanced Family of Quantum Codes.
There exists an explicit construction of a family of [[N, K, D]] LDPC quantum codes
(Def_5) with K ∈ Θ(N^{4/5}) and D = min(D_X, D_Z) ∈ Ω(N^{3/5}).
The proof combines Theorem 15 (explicit family with unbalanced distances) with the distance balancing procedure of Evra–Kaufman–Zemor [EKZ22, Theorem 1].
The balanced code for each valid prime vp is (distanceBalancing vp).code, a CSS
code of length N = balancedCodeLength vp. Since N ∈ Θ(q⁵), we have
q ∈ Θ(N^{1/5}), giving K ∈ Θ(q⁴) = Θ(N^{4/5}) and
D ∈ Ω(q³) = Ω(N^{3/5}).