{
  "id": "oai:papers.qnfo.org:leakage-as-the-price-of-binary-truncation-a-scaling-law-for-transmon-leakage-rem",
  "slug": "leakage-as-the-price-of-binary-truncation-a-scaling-law-for-transmon-leakage-rem",
  "title": "Leakage as the Price of Binary Truncation: A Scaling Law for Transmon Leakage, Removal Capacity, and Its Ontological Reading",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "Superconducting transmon processors lose population from the computational \\|0⟩,|1⟩\\ subspace into higher levels of an anharmonic oscillator, and this leakage is normally treated as an engineering defect to be suppressed pulse by pulse. The gap addressed here is conceptual and quantitative: no existing framework states how much leakage a gate must produce as a consequence of encoding a bosonic ladder into two levels, nor what removal capacity that production rate demands of a processor architecture. We adopt the perturbative and exact three-level leakage formulas of the Planck-Fidelity analysis of binary truncation of bosonic ladders, insert illustrative transmon parameters (f_Ω=20 MHz, f_α=200 MHz, Ω/|α|=0.05), and derive a per-gate time-averaged leakage L₂=0.01 with quadratic scaling in Ω/|α|. Combining this production rate with an assumed 50 ns gate cycle gives a leakage production rate of 2×10⁵ s^-1, so holding steady-state leakage at 10^-4 requires a removal capacity Γ_rem=2×10⁹ s^-1, which exceeds the fastest leakage hopping rate reported for tunable-coupler transmon arrays (∼10 MHz) by a factor of 200. For a state-of-the-art d=12 transmon the binary map discards a fraction f_disc=0.721057 of the Hilbert-space information content, and a labeled projection with an assumed proportionality constant gives a surface-code overhead factor χ≈3. The residual leakage is interpreted as the empirical cost of projecting a continuous bosonic degree of freedom onto a discrete bit, a falsifiable platform-ordering prediction is stated, and the removal-capacity deficit is shown to justify dedicated leakage-reduction hardware.",
  "abstract_markdown": "Superconducting transmon processors lose population from the computational $\\{|0\\rangle,|1\\rangle\\}$ subspace into higher levels of an anharmonic oscillator, and this leakage is normally treated as an engineering defect to be suppressed pulse by pulse. The gap addressed here is conceptual and quantitative: no existing framework states how much leakage a gate must produce as a consequence of encoding a bosonic ladder into two levels, nor what removal capacity that production rate demands of a processor architecture. We adopt the perturbative and exact three-level leakage formulas of the Planck-Fidelity analysis of binary truncation of bosonic ladders, insert illustrative transmon parameters ($f_{\\Omega}=20$ MHz, $f_{\\alpha}=200$ MHz, $\\Omega/|\\alpha|=0.05$), and derive a per-gate time-averaged leakage $\\overline{L}_{2}=0.01$ with quadratic scaling in $\\Omega/|\\alpha|$. Combining this production rate with an assumed 50 ns gate cycle gives a leakage production rate of $2\\times10^{5}\\ \\mathrm{s}^{-1}$, so holding steady-state leakage at $10^{-4}$ requires a removal capacity $\\Gamma_{\\mathrm{rem}}=2\\times10^{9}\\ \\mathrm{s}^{-1}$, which exceeds the fastest leakage hopping rate reported for tunable-coupler transmon arrays ($\\sim10$ MHz) by a factor of 200. For a state-of-the-art $d=12$ transmon the binary map discards a fraction $f_{\\mathrm{disc}}=0.721057$ of the Hilbert-space information content, and a labeled projection with an assumed proportionality constant gives a surface-code overhead factor $\\chi\\approx3$. The residual leakage is interpreted as the empirical cost of projecting a continuous bosonic degree of freedom onto a discrete bit, a falsifiable platform-ordering prediction is stated, and the removal-capacity deficit is shown to justify dedicated leakage-reduction hardware.",
  "date": "2026-10-17",
  "first_published": "2026-10-17",
  "last_revised": "2026-10-17T00:00:00Z",
  "modified": "2026-10-17T00:00:00Z",
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  "url": "https://papers.qnfo.org/papers/leakage-as-the-price-of-binary-truncation-a-scaling-law-for-transmon-leakage-rem",
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        "name": "Rowan Brad Quni-Gudzinas",
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        "title": "Leakage as the Price of Binary Truncation: A Scaling Law for Transmon Leakage, Removal Capacity, and Its Ontological Reading"
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        "description": "Superconducting transmon processors lose population from the computational \\|0⟩,|1⟩\\ subspace into higher levels of an anharmonic oscillator, and this leakage is normally treated as an engineering defect to be suppressed pulse by pulse. The gap addressed here is conceptual and quantitative: no existing framework states how much leakage a gate must produce as a consequence of encoding a bosonic ladder into two levels, nor what removal capacity that production rate demands of a processor architecture. We adopt the perturbative and exact three-level leakage formulas of the Planck-Fidelity analysis of binary truncation of bosonic ladders, insert illustrative transmon parameters (f_Ω=20 MHz, f_α=200 MHz, Ω/|α|=0.05), and derive a per-gate time-averaged leakage L₂=0.01 with quadratic scaling in Ω/|α|. Combining this production rate with an assumed 50 ns gate cycle gives a leakage production rate of 2×10⁵ s^-1, so holding steady-state leakage at 10^-4 requires a removal capacity Γ_rem=2×10⁹ s^-1, which exceeds the fastest leakage hopping rate reported for tunable-coupler transmon arrays (∼10 MHz) by a factor of 200. For a state-of-the-art d=12 transmon the binary map discards a fraction f_disc=0.721057 of the Hilbert-space information content, and a labeled projection with an assumed proportionality constant gives a surface-code overhead factor χ≈3. The residual leakage is interpreted as the empirical cost of projecting a continuous bosonic degree of freedom onto a discrete bit, a falsifiable platform-ordering prediction is stated, and the removal-capacity deficit is shown to justify dedicated leakage-reduction hardware.",
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