Physics Superconducting Qubit From Jcost

The qubit ladder

A superconducting qubit is a tiny electrical circuit that behaves as a two-level quantum system, the basic unit of information in many quantum computers. The most common type, the transmon, uses a Josephson junction, a thin insulating barrier between two superconductors, to create an energy difference between two quantum states. Its performance is measured by coherence times, T₁ and T₂, which describe how long the qubit retains its quantum information before noise destroys it. Longer coherence times mean more reliable quantum computations, so improving them is a central engineering goal.

In the Recognition Science framework, the cost of recognition, a forced quantity derived from five plain conditions on how a ledger records events, is J(x) = (x + 1/x)/2 - 1. This cost function forces the golden ratio φ = (1 + √5)/2 ≈ 1.618 as the unique self-similar scaling. The framework models superconducting qubits by placing them on a ladder of coherence times: the coherence time at rung k is φ^k, so each step up the ladder multiplies the coherence time by φ. This is a definitional model, not a measured law, but it makes a sharp prediction: coherence times should improve in discrete golden-ratio steps.

The module also counts five canonical superconducting qubit types: transmon, fluxonium, capacitively shunted flux, quantum dot hybrid, and spin qubit. In the framework this count is not arbitrary; it matches a structural dimension that the cost function forces. The machine-checked library of formal theorems proves that the ratio of coherence times at consecutive rungs is exactly φ, and that the number of qubit types is exactly five. These are theorems about the model's internal consistency, not empirical measurements of real devices.

The practical consequence is a design target. If the framework's model is right, engineers should aim for transmon coherence times that sit at φ-ladder positions, where the model predicts optimal anharmonicity and decoherence suppression. A real transmon with T₂ = 100 microseconds would, at the next rung, be expected to reach about 162 microseconds. This is a falsifiable prediction: if coherence times never cluster near φ multiples, the model fails.

noncomputable def coherenceAtRung (k : ℕ) : ℝ := phi ^ k

theorem coherenceRatio (k : ℕ) :
    coherenceAtRung (k + 1) / coherenceAtRung k = phi := by
  unfold coherenceAtRung
  have hpos := pow_pos phi_pos k
  rw [pow_succ, div_eq_iff hpos.ne']
  ring

theorem qubitTypeCount : Fintype.card SuperconductingQubitType = 5 := by decide

What this page does not claim

The φ-ladder is a measured property of real superconducting qubits; it is a definitional model with a falsifiable prediction. The framework derives the specific materials or fabrication processes that achieve longer coherence times. The five-type count is a theorem about the framework's model, not an empirical classification of all possible qubit designs.

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/Physics/SuperconductingQubitFromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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