{
  "id": "oai:papers.qnfo.org:planck-fidelity-the-fundamental-leakage-cost-of-binary-truncation-of-bosonic-lad",
  "slug": "planck-fidelity-the-fundamental-leakage-cost-of-binary-truncation-of-bosonic-lad",
  "title": "Planck Fidelity: The Fundamental Leakage Cost of Binary Truncation of Bosonic Ladders",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "Superconducting transmon qubits are weakly anharmonic oscillators whose spectra form discrete but unbounded ladders of energy eigenstates; the binary register n ∈ \\0,1\\ is imposed on this ladder by truncation, not by physics. Because every physical drive contains the raising operator a^†, whose matrix elements √n+1 never vanish, any finite-time gate populates levels outside the computational subspace. We formalize this as the Planck Loss L_P = 1 - Tr(P_qubit|ψ⟩⟨ψ|) and conjecture that L_P > 0 for any finite-time gate, independent of materials quality — a fundamental cost of base-2 encoding on a base-d physical object. We derive the harmonic-limit second-order leakage L_P^(2) = θ⁴/2, which evaluates to $3.0440341$ for a π-pulse and therefore signals perturbative breakdown; with anharmonicity we obtain the first-order bound L₂ = 8(Ω/|α|)² sin²(α T_g/2), giving a gate-averaged leakage of 1.0 × 10^-2 at Ω/|α| = 0.05, and the exact three-level peak P₂,max = 2r²/(1+2r²) = 1.96 × 10^-2 at $r = 0.1$. We compute the radix-truncation entropy S_Base(12) = ln 6 = 1.7917595 nats $= 2.5849625$ bits, a discarded-information fraction of 72.1\\% of the ladder's ln 12 content. The results reframe the qubit as a radix approximation and motivate qudit (base-d) architectures and a falsifiable leakage-floor test programme.",
  "abstract_markdown": "Superconducting transmon qubits are weakly anharmonic oscillators whose spectra form discrete but unbounded ladders of energy eigenstates; the binary register $n \\in \\{0,1\\}$ is imposed on this ladder by truncation, not by physics. Because every physical drive contains the raising operator $a^\\dagger$, whose matrix elements $\\sqrt{n+1}$ never vanish, any finite-time gate populates levels outside the computational subspace. We formalize this as the Planck Loss $L_P = 1 - \\mathrm{Tr}(P_{\\mathrm{qubit}}|\\psi\\rangle\\langle\\psi|)$ and conjecture that $L_P > 0$ for any finite-time gate, independent of materials quality — a fundamental cost of base-2 encoding on a base-$d$ physical object. We derive the harmonic-limit second-order leakage $L_P^{(2)} = \\theta^4/2$, which evaluates to $3.0440341$ for a $\\pi$-pulse and therefore signals perturbative breakdown; with anharmonicity we obtain the first-order bound $L_2 = 8(\\Omega/|\\alpha|)^2 \\sin^2(\\alpha T_g/2)$, giving a gate-averaged leakage of $1.0 \\times 10^{-2}$ at $\\Omega/|\\alpha| = 0.05$, and the exact three-level peak $P_{2,\\max} = 2r^2/(1+2r^2) = 1.96 \\times 10^{-2}$ at $r = 0.1$. We compute the radix-truncation entropy $S_{\\mathrm{Base}}(12) = \\ln 6 = 1.7917595$ nats $= 2.5849625$ bits, a discarded-information fraction of $72.1\\%$ of the ladder's $\\ln 12$ content. The results reframe the qubit as a radix approximation and motivate qudit (base-$d$) architectures and a falsifiable leakage-floor test programme.",
  "date": "2026-10-10",
  "first_published": "2026-10-10",
  "last_revised": "2026-10-15T03:45:29Z",
  "modified": "2026-10-15T03:45:29Z",
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  "url": "https://papers.qnfo.org/papers/planck-fidelity-the-fundamental-leakage-cost-of-binary-truncation-of-bosonic-lad",
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  "version_url": "https://papers.qnfo.org/papers/planck-fidelity-the-fundamental-leakage-cost-of-binary-truncation-of-bosonic-lad/v/2.0.0",
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    "creators": [
      {
        "name": "Rowan Brad Quni-Gudzinas",
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    "titles": [
      {
        "title": "Planck Fidelity: The Fundamental Leakage Cost of Binary Truncation of Bosonic Ladders"
      }
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    "publicationYear": "2026",
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        "date": "2026-10-10",
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        "description": "Superconducting transmon qubits are weakly anharmonic oscillators whose spectra form discrete but unbounded ladders of energy eigenstates; the binary register n ∈ \\0,1\\ is imposed on this ladder by truncation, not by physics. Because every physical drive contains the raising operator a^†, whose matrix elements √n+1 never vanish, any finite-time gate populates levels outside the computational subspace. We formalize this as the Planck Loss L_P = 1 - Tr(P_qubit|ψ⟩⟨ψ|) and conjecture that L_P > 0 for any finite-time gate, independent of materials quality — a fundamental cost of base-2 encoding on a base-d physical object. We derive the harmonic-limit second-order leakage L_P^(2) = θ⁴/2, which evaluates to $3.0440341$ for a π-pulse and therefore signals perturbative breakdown; with anharmonicity we obtain the first-order bound L₂ = 8(Ω/|α|)² sin²(α T_g/2), giving a gate-averaged leakage of 1.0 × 10^-2 at Ω/|α| = 0.05, and the exact three-level peak P₂,max = 2r²/(1+2r²) = 1.96 × 10^-2 at $r = 0.1$. We compute the radix-truncation entropy S_Base(12) = ln 6 = 1.7917595 nats $= 2.5849625$ bits, a discarded-information fraction of 72.1\\% of the ladder's ln 12 content. The results reframe the qubit as a radix approximation and motivate qudit (base-d) architectures and a falsifiable leakage-floor test programme.",
        "descriptionType": "Abstract"
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