Information Quantum Error Correction Threshold

Error correction thresholds

Quantum error correction is the art of protecting fragile quantum information by spreading it across many physical qubits. The fault-tolerance threshold is the physical error rate below which this protection works: above it, errors multiply faster than the code can correct them, and below it, logical errors are suppressed exponentially. The threshold is a property of the code family itself, not of any particular machine, and it is the number that tells engineers whether a given hardware platform can ever reach fault-tolerant computation.

The best-known example is the surface code, the leading candidate for real quantum computers, with a threshold around 1 percent. A close relative, the colour code, sits near 1.7 percent. These numbers look arbitrary at first, but Recognition Science (RS) proposes they are not. The framework's structural prediction is that thresholds land on a phi-ladder, a discrete ladder of values where each rung is the golden ratio phi (about 1.618) times the next, and the two known code families sit on adjacent rungs: the colour code at phi^(-8)/2 and the surface code at phi^(-9)/2, with their ratio 0.017/0.011 ≈ 1.55 matching phi within measurement uncertainty.

The formal module QuantumErrorCorrectionThreshold.lean establishes the ladder's arithmetic with machine-checked proofs. Its central definition is the threshold at rung k: qecThresholdAt(k) = phi^(-k)/2, so higher rungs give lower thresholds. Three theorems follow directly: every threshold is positive, stepping one rung down multiplies the threshold by phi^(-1), and the ratio of adjacent thresholds is exactly phi^(-1). These are not empirical fits; they are consequences of the definition, collected in a certificate structure that bundles the three properties together.

What the module does not do is prove that real quantum codes must sit on this ladder. The surface and colour code values are empirical bench points, measured by the community and compared against the ladder; the ladder itself is a definitional choice, not a derived law. The framework's claim is that the pattern is real and worth testing: if a new code family is found with a threshold near phi^(-7)/2, the ladder gains support, and if a family lands clearly off the ladder, the prediction fails. That is the honest status of the module: it proves the arithmetic of a proposed pattern, and it leaves the physics as a testable hypothesis.

/-- QEC threshold at φ-ladder rung `k` below unity (higher rung = lower threshold). -/
def qecThresholdAt (k : ℕ) : ℝ := phi ^ (-(k : ℤ)) / 2

/-- QEC threshold at φ-ladder rung `k` below unity (higher rung = lower threshold). -/
def qecThresholdAt (k : ℕ) : ℝ := phi ^ (-(k : ℤ)) / 2

/-- QEC threshold at φ-ladder rung `k` below unity (higher rung = lower threshold). -/
def qecThresholdAt (k : ℕ) : ℝ := phi ^ (-(k : ℤ)) / 2

theorem qecThresholdAt_adjacent_ratio (k : ℕ) :
    qecThresholdAt (k + 1) / qecThresholdAt k = phi⁻¹ := by
  rw [qecThresholdAt_succ_ratio]
  field_simp [(qecThresholdAt_pos k).ne']

What this page does not claim

This module does not prove that real quantum codes must sit on the phi-ladder; it proves the arithmetic of the proposed pattern. The surface and colour code threshold values are community measurements, not derived constants within the framework. No claim is made that the phi-ladder is the only possible structure for error correction thresholds.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Information/QuantumErrorCorrectionThreshold.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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