T0-28: Quantum Error Correction Theory - Formal Framework
1. Mathematical Foundation
1.1 Zeckendorf Code Space
Definition 1.1.1 (Zeckendorf Codeword Space):
C_φ = {c ∈ {0,1}^n : ∀i, c_i·c_{i+1} = 0 ∧ c ∈ Z_valid}
where Z_valid denotes valid Zeckendorf representations.
Definition 1.1.2 (φ-Hamming Weight):
wt_φ(x) = Σ_{i: x_i=1} F_i
where F_i is the ith Fibonacci number.
Definition 1.1.3 (φ-Distance Metric):
d_φ(x,y) = wt_φ(x ⊕ y) + λ·δ_11(x ⊕ y)
where λ > 0 is the constraint violation penalty.
1.2 Error Operators
Definition 1.2.1 (Single-Bit Error):
E_i|c⟩ = |c ⊕ e_i⟩
where e_i has bit 1 only at position i.
Definition 1.2.2 (Syndrome Function):
S: {0,1}^n → {0,1}^{n-k}
S(r) = rH^T mod 2
2. Core Theorems
2.1 Error Detection Theorems
Theorem 2.1.1 (Immediate Detection):
∀c ∈ C_φ, ∀i: E_i(c) ∉ C_φ ⟹ detectable
Proof:
1. Let c ∈ C_φ with c_i·c_{i+1} = 0 ∀i
2. Apply E_j: c' = c ⊕ e_j
3. If c_j = 0 → 1 and (c_{j-1} = 1 ∨ c_{j+1} = 1)
4. Then c'_j·c'_{j-1} = 1 or c'_j·c'_{j+1} = 1
5. Therefore c' ∉ C_φ, error detected □
Theorem 2.1.2 (Minimum Distance):
d_min(C_φ) = min{F_n : F_n ≥ 3}
2.2 Error Correction Capability
Theorem 2.2.1 (Fibonacci Error Correction Bound):
t_max = ⌊(F_n - 1)/2⌋
where n is code length following Fibonacci sequence.
Proof:
1. By sphere packing: Σ_{i=0}^t C(n,i) ≤ 2^{n-k}
2. For Zeckendorf codes: |C_φ| = F_{n+2}
3. Minimum distance d = F_n by construction
4. Error correction: t = ⌊(d-1)/2⌋
5. Substitute: t = ⌊(F_n - 1)/2⌋
6. Asymptotically: t ~ φ^n/(2√5) □
3. Fibonacci Code Construction
3.1 Systematic Construction
Definition 3.1.1 (Generator Matrix):
G_φ = [I_k | P]
P_{ij} = 1 iff j-th Fibonacci number appears in i-th check equation
Definition 3.1.2 (Parity Check Matrix):
H_φ = [P^T | I_{n-k}]
Constraint: GH^T = 0 (mod 2)
3.2 Optimal Code Parameters
Theorem 3.2.1 (Fibonacci Code Family):
∃ [F_m, F_{m-2}, F_{m-1}]_φ codes ∀m ≥ 3
Proof by Construction:
Base: [2,1,1]_φ repetition code
Induction: Given [F_m, F_{m-2}, F_{m-1}]_φ
Construct: C_{m+1} = C_m × C_{m-1} with No-11 constraint
Dimension: dim(C_{m+1}) = F_{m-2} + F_{m-3} = F_{m-1}
Distance: d_{m+1} = F_m □
4. Quantum Error Correction
4.1 Quantum Fibonacci Codes
Definition 4.1.1 (Quantum Code Space):
C_Q = span{|c⟩ : c ∈ C_φ}
Definition 4.1.2 (Stabilizer Generators):
g_i = X^{a_i}Z^{b_i}
where (a_i, b_i) ∈ Z_2^n satisfy No-11 constraint
4.2 Knill-Laflamme Conditions
Theorem 4.2.1 (φ-Modified KL Condition):
⟨ψ|E_i^†E_j|ψ⟩ = δ_{ij}c_{ij}
∀ wt_φ(E_i), wt_φ(E_j) ≤ t
Proof:
1. Error space restricted by No-11: dim = F_{n+2}
2. Correctable errors: {E : wt_φ(E) ≤ t}
3. Orthogonality in φ-metric ensures correction
4. Syndrome measurement projects to error subspace □
5. Information Recovery
5.1 Entropy Analysis
Theorem 5.1.1 (Entropy Increase in Recovery):
H(corrected) = H(original) + ΔH_detection + ΔH_correction
where ΔH_detection ≥ log φ, ΔH_correction > 0
Proof:
1. Detection requires measurement: ΔH_d ≥ log(syndrome space)
2. Syndrome space has φ-structure: dim ~ φ^k
3. Correction is irreversible: ΔH_c > 0
4. Total: ΔH ≥ log φ > 0 □
5.2 Asymptotic Optimality
Theorem 5.2.1 (Approach to Shannon Limit):
lim_{n→∞} R_φ = 1 - H_2(p)
where p is channel error probability.
Proof:
1. Cascaded construction with density evolution
2. Error rate evolution: p_{i+1} = f_φ(p_i)
3. Fixed point analysis: p* = 0 stable
4. Convergence rate: |p_{i+1}| ≤ |p_i|/φ
5. Shannon capacity achieved asymptotically □
6. Holographic Correspondence
6.1 AdS/CFT Error Correction
Theorem 6.1.1 (Bulk-Boundary Equivalence):
C_bulk^{(n+1)} ≅ C_boundary^{(n)}
under φ-structured holographic map.
Proof:
1. Bulk code: [[n,k,d]]_φ quantum Fibonacci code
2. Boundary code: Topological Fibonacci anyon code
3. Error operators map: E_bulk ↔ Wilson_boundary
4. Correction capability preserved: t_bulk = t_boundary
5. No-11 constraint is holographic invariant □
7. Formal Properties
7.1 Completeness
Theorem 7.1.1 (Error Correction Completeness):
∀ε ∈ Errors, |ε| ≤ t: ∃! correction algorithm
7.2 Consistency with A1
Theorem 7.2.1 (Entropy Increase Necessity):
Error correction ⟹ ΔH > 0
consistent with A1 axiom.
8. Algorithmic Complexity
Theorem 8.1 (Decoding Complexity):
T_decode = O(n log φ) = O(0.694n)
Proof:
1. Syndrome computation: O(n)
2. Syndrome table lookup: O(log φ^k) = O(k log φ)
3. Error localization: O(n)
4. Correction application: O(1)
5. Total: O(n log φ) □
References
- A1: Self-referential systems increase entropy
- T0-3: Zeckendorf constraint emergence
- T0-17: Information entropy encoding
- T0-18: Quantum state emergence
- T0-27: Fluctuation theorem