Information Quantum Error Correction Threshold Qec Threshold At Succ Ratio

The threshold ratio

In quantum computing, the quantum error correction threshold is the physical error rate below which a quantum code can suppress logical errors exponentially. Above the threshold, errors accumulate faster than they can be corrected; below it, the code wins. The surface code, a leading candidate for real hardware, has a threshold near one percent, while the colour code, a more complex family, sits near 1.7 percent. The ratio between those two measured thresholds is about 1.55, very close to the golden ratio φ ≈ 1.618.

The Recognition Science framework models these thresholds as sitting on a ladder of values defined by powers of φ. Its machine-checked library of formal theorems defines a function qecThresholdAt(k) = φ^(-k)/2 for any natural number k, and proves a theorem about what happens when you move from one rung to the next. The theorem qecThresholdAt_succ_ratio establishes that the threshold at rung k+1 equals the threshold at rung k multiplied by φ^(-1). In plain terms: each step down the ladder divides the threshold by φ, so adjacent thresholds always stand in the same ratio, exactly φ.

The proof is short and mechanical. It unfolds the definition of qecThresholdAt, rewrites the exponent using the fact that -(k+1) = -k + (-1), and applies the rule for multiplying powers of the same base. The library also derives the equivalent statement that the quotient of adjacent thresholds equals φ^(-1), and packages both facts with the positivity of every threshold into a certificate structure. The certificate is a single object that bundles the three properties: every threshold is positive, the one-step ratio holds, and the adjacent ratio holds.

What the theorem does not claim is that real quantum hardware obeys this ladder. The surface code and colour code values are measurements, not consequences of the theorem. The theorem is a statement about a defined mathematical function, and the match with experiment is an empirical observation, not a proof. The framework's claim is that the structure of thresholds, if it follows the φ-ladder, has this exact ratio property; whether nature chooses that structure is a separate question.

theorem qecThresholdAt_succ_ratio (k : ℕ) :
    qecThresholdAt (k + 1) = qecThresholdAt k * phi⁻¹ := by
  unfold qecThresholdAt
  have hphi_ne := Constants.phi_ne_zero
  have : phi ^ (-((k : ℤ) + 1)) = phi ^ (-(k : ℤ)) * phi⁻¹ := by
    rw [show (-((k : ℤ) + 1)) = -(k : ℤ) + (-1 : ℤ) by ring]
    rw [zpow_add₀ hphi_ne]; simp
  have hcast : ((k + 1 : ℕ) : ℤ) = (k : ℤ) + 1 := by push_cast; ring
  rw [hcast, this]; ring

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

/-- QEC threshold certificate. -/
def qecThresholdCert : QECThresholdCert where
  threshold_pos := qecThresholdAt_pos
  one_step_ratio := qecThresholdAt_succ_ratio
  adjacent_ratio := qecThresholdAt_adjacent_ratio

What this page does not claim

The theorem does not prove that any real quantum code has a threshold equal to φ^(-k)/2. The theorem does not derive the measured values of the surface code or colour code thresholds. The theorem does not establish that the golden ratio is the only possible ratio between adjacent 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)

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