Quantum Error Correction
Physical Qubits
Noisy, error-prone quantum states with finite coherence time. Each physical qubit degrades without active correction.
|ψ⟩_physical → decoherence
Structural
Logical Qubits
Protected quantum states encoded across multiple physical qubits via surface code. Higher fidelity than constituent parts through error correction.
|ψ⟩_logical = f(|ψ₁⟩, |ψ₂⟩, ..., |ψₙ⟩)
Structural
Surface Code Distance
Scale of the error correction code. Distance-3/5/7 requires 9/25/49 physical qubits per logical qubit. Question: does adding more improve or degrade fidelity?
d = 3,5,7 → n = d²
Structural
Below-Threshold Operation
Willow's achievement: error rates decrease as code distance increases. The system can now scale without quality degradation.
ε(d=7) < ε(d=5) < ε(d=3)
Threshold
Error Correction Cycles
Continuous active correction at 1.1 μs cycle time. System must detect and correct errors faster than they accumulate.
t_cycle = 1.1 μs
t_decoder = 63 μs
Temporal
Decoherence / Noise
Environmental coupling induces errors. Thermal noise, electromagnetic interference, material defects. Fundamental constraint on system operation.
dρ/dt = -i[H,ρ] + L(ρ)
Temporal
Real-Time Decoder
Processing error syndromes and applying corrections within cycle time. If latency exceeds cycle time, errors compound uncontrolled.
correction_rate > error_rate
Temporal
Threshold Crossing
Critical boundary where correction outpaces noise. Above threshold: system collapses. Below threshold: system can scale to useful computation.
p_physical < p_threshold
Threshold
HIR Framework
Individual Agents
People/nodes subject to pressure P_t with finite capacity. Prone to misalignment under stress. Each individual degrades without active correction.
S_t = A_t B_t - P_t
Structural
Collective Alignment States
Protected by mutual reinforcement (B_t synergy). Stronger than individual components when properly structured. Requires continuous accountability loops (A_t).
U_t = A_t B_t (1 + g_G G_t) Fint_t
Structural
Network Structure Scale
Carrier density × internalization (C_t × Fint_t). Question: does growth strengthen or dilute the collective?
C_{t+1} = C_t + α E_t Ξ_t U_t (1-C_t) - δ_C C_t
Structural
Regenerative Scaling
As C_t grows, collective strength must increase faster than degradation. Synergy term k captures pairwise reinforcement making whole stronger than sum.
B_t = H_t + I_t + R_t + k(H_t I_t + H_t R_t + I_t R_t)
Threshold
Accountability Loop Speed
A_t must activate frequently enough that misalignments are caught before compounding into dogma K_t or irreversible degradation D_t.
ΔD_t = -β U_t C_t L_t R_{s,t} E_t Θ_t
Temporal
Pressure P_t
Work exhaustion W_t, financial strain F_t, interaction effects. These couple to individuals and induce misalignment. Fundamental stressor on system.
P_t = w_W W_t + w_F F_t + w_WF W_t F_t
Temporal
Restorative Capacity R_{s,t}
Can infrastructure detect and address degradation before it becomes irreversible? If response time exceeds degradation rate, collapse accelerates.
response_capacity > degradation_rate
Temporal
Irreversibility Boundary
When GROWTH_t exceeds maximum correction capacity for all feasible interventions, system enters irreversible collapse. Below threshold: correction can scale.
GROWTH_t < β U_t C_t L_t R_{s,t} E_t Θ_t
Threshold