Why Cutting a Quantum Ladder in Half Has a Hidden Cost

In the paper's words

The transmon is a bosonic oscillator whose spectrum is a discrete but unbounded ladder of energy eigenstates; encoding it as a qubit, i.e. restricting the computational space to n ∈ \0,1\, is a radix choice rather than a physical necessity. We formalize the conjecture, originating in the QNFO "Rosetta" corpus, that any finite-time gate driven through the raising operator a^† on such a ladder incurs a strictly positive "Planck Loss" L_P = 1 - Tr(P_qubit |ψ⟩⟨ψ|), a leakage floor independent of materials and control engineering. We give two complementary derivations: an exact detuned-Rabi solution in the three-level truncation, and a first-order perturbative bound L₂ = 8(Ω/|α|)² sin²(α T_g/2).

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. Reference [1], The Scientific Programme of Planck , states that for 40 years the CMB has been the most important source of information about the geometry and contents of the Universe, that only a small fraction of the available information had been extracted to date, and that Planck — the third space CMB mission after COBE and WMAP — is designed to extract essentially all of the information in the CMB temperature anisotropies.

   from 2. Background and Related Work

2. Reference [5] exploits the Early Release Compact Source Catalogue to measure number counts $\mathrm{d}N/\mathrm{d}S$ of extragalactic radio sources at 30, 44, 70, 100, 143 and 217 GHz, noting that the full-sky nature of the catalogue extends the measurement to the rarest and brightest sources; counting the population of each rung of a brightness ladder is structurally the exercise we propose for occupation numbers $n$ of a bosonic ladder.

   from 2. Background and Related Work

3. We quantify the truncation entropy $S_{\mathrm{Base}} = \ln(d/2) = 1.791759$ nats for $d = 12$ resolvable levels, showing binary encoding uses $\approx 27.9\%$ of the ladder's Hilbert-space information content.

   from Abstract

4. This observation, and the conjecture that the resulting leakage is a fundamental cost of the radix choice rather than an avoidable imperfection, is due to the QNFO "Rosetta" corpus [9], [10], which argues that the transmon is properly a qudit with $d \sim 12$ resolvable levels and that binary truncation of it constitutes a "radix error," not a physics error.

   from 1. Introduction

5. Reference [9], Project Rosetta: The Approximation Entropy & The Fractal Limits of Digital Physics — v2.0 (https://papers.qnfo.org/papers/rosetta-fractal-math/), is the direct source of the central conjecture.

   from 2. Background and Related Work

6. Its summary states that v2.0 corrects v1.0: the transmon spectrum is discrete (Planck quantization), so binary truncation is a radix error, not a physics error; the transmon is a qudit with $d \sim 12$, not a qubit; and the base-truncation entropy is $S_{\mathrm{Base}} = \ln(d/2) \sim 1.79$ nats.

   from 2. Background and Related Work

How it is done

1. We give two complementary derivations: an exact detuned-Rabi solution in the three-level truncation, and a first-order perturbative bound $L_2 = 8(\Omega/|\alpha|)^2 \sin^2(\alpha T_g/2)$.

   from Abstract

2. A qubit is, by definition, a two-dimensional quantum system.

   from 1. Introduction

3. Reference [10], The Two-Level Lie: The Transmon Is Not a Qubit — And the Entire Field Knows It (https://papers.qnfo.org/papers/the-two-level-lie/), supplies the framing: its v2.0 adds "Part II: The Approximation Entropy," described as a meta-mathematical framework quantifying the irreducible cost of translating continuous bosonic physics to discrete digital computation, and introduces a "Rosetta Constant," a "Trotter Wall," and a "Fractal Wall theorem" together with a falsifiable calorimetry experimental proposal.

   from 2. Background and Related Work

4. We model the transmon as a weakly anharmonic bosonic oscillator with Hamiltonian ($\hbar = 1$) $$H_0 = \omega\, a^\dagger a + \frac{\alpha}{2}\, a^\dagger a^\dagger a\, a,$$ where $a^\dagger|n\rangle = \sqrt{n+1}\,|n+1\rangle$, $\omega$ is the $|0\rangle \to |1\rangle$ transition frequency, and $\alpha$ is the anharmonicity.

   from 3.1 Device model

What would falsify it

1. We propose a Planck Fidelity benchmark, state falsification conditions, and argue the results motivate base-$d$ qudit architectures.

   from Abstract

2. Reference [10], The Two-Level Lie: The Transmon Is Not a Qubit — And the Entire Field Knows It (https://papers.qnfo.org/papers/the-two-level-lie/), supplies the framing: its v2.0 adds "Part II: The Approximation Entropy," described as a meta-mathematical framework quantifying the irreducible cost of translating continuous bosonic physics to discrete digital computation, and introduces a "Rosetta Constant," a "Trotter Wall," and a "Fractal Wall theorem" together with a falsifiable calorimetry experimental proposal.

   from 2. Background and Related Work

3. The supplied summary gives no further detail on these objects, so we do not use them quantitatively; we note only that the falsifiability orientation of [10] matches the falsification criteria we state in Section 6.

   from 2. Background and Related Work

Where the claim stops

1. The strict-positivity claim of Section 3.5 is a model-class result: it holds for finite-time, finite-amplitude drives of the stated $a^\dagger$-form under the rotating-wave approximation, not for all conceivable Hamiltonians, and shaped envelopes can null the first-order leakage at isolated, measure-zero parameter points.

   from 6. Discussion

2. The sequence-level survival $S_{1000} \approx 0.6212$ assumes independent gates and no leakage return or reset; any leakage-reduction unitary or post-selection invalidates it.

   from 6. Discussion

3. The worst-case value $L_P^{(\mathrm{wc})} = 5/6$ is a scenario under a uniform-population assumption, not a physical prediction.

   from 6. Discussion

Where to read

1. Abstract188 words · 1 min
2. 1. Introduction449 words · 2 min
3. 2. Background and Related Work892 words · 4 min
4. 3. Methods627 words · 3 min
5. 4. Analysis614 words · 3 min
6. 5. Results246 words · 1 min
7. 6. Discussion324 words · 1 min
8. 7. Conclusion153 words · 1 min