Planck Fidelity: The Fundamental Leakage Cost of Binary Truncation of Bosonic Ladders

In the paper's words

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.

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

Numbers it reports

1. The design philosophy — extract all the information the quantized field carries, not a conventionally truncated subset — is the exact analogue of our thesis that a register should match the physical Hilbert-space dimension rather than a chosen binary slice of it. [2] Planck Early Results: The Planck mission records that the Planck satellite was launched on 14 May 2009, has been surveying the sky stably and continuously since 13 August 2009, carries 74 detectors sensitive to frequencies between approximately 25 and 1000 GHz, and that after its first year the major elements of the satellite's performance exceeded their original technical requirements.

   from 2. Background and Related Work

2. We use it as the factual anchor for the mission whose name we borrow, and for the demonstration that a quantized physical channel can be operated continuously against its full design capacity rather than a nominal subset. [3] The Planck Catalogue of Compact Sources (PCCS) describes nine single-frequency catalogues of compact Galactic and extragalactic sources detected over the entire sky in the first 15 months of operations, covering 30–857 GHz, with 90% completeness at 180 mJy in the best channel.

   from 2. Background and Related Work

3. The catalogue's distinction between confirmed clusters and candidates illustrates the general accounting problem of separating registered signal from registered-but-unvalidated excess, structurally the same bookkeeping as separating computational population from leakage population: both require that the unvalidated excess be counted and reported, not discarded. [5] The statistical study of extragalactic radio sources uses the Early Release Compact Source Catalogue to measure number counts $\mathrm{d}N/\mathrm{d}S$ at 30, 44, 70, 100, 143 and 217 GHz, finding counts at 30, 44 and 70 GHz in good agreement with WMAP at nearby frequencies.

   from 2. Background and Related Work

4. 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 $r = 0.1$.

   from Abstract

5. 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.

   from Abstract

6. A first-order leakage bound $L_2 = 8(\Omega/|\alpha|)^2 \sin^2(\alpha T_g/2)$ and an exact three-level leakage formula, evaluated with explicit arithmetic at stated operating points (Section 4.2).

   from 1. Introduction

How it is done

1. 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 $r = 0.1$.

   from Abstract

2. A first-order leakage bound $L_2 = 8(\Omega/|\alpha|)^2 \sin^2(\alpha T_g/2)$ and an exact three-level leakage formula, evaluated with explicit arithmetic at stated operating points (Section 4.2).

   from 1. Introduction

3. We use it as the factual anchor for the mission whose name we borrow, and for the demonstration that a quantized physical channel can be operated continuously against its full design capacity rather than a nominal subset. [3] The Planck Catalogue of Compact Sources (PCCS) describes nine single-frequency catalogues of compact Galactic and extragalactic sources detected over the entire sky in the first 15 months of operations, covering 30–857 GHz, with 90% completeness at 180 mJy in the best channel.

   from 2. Background and Related Work

4. We model the transmon as a weakly anharmonic oscillator, $$H_0 = \hbar\,\omega\, a^\dagger a + \frac{\hbar\,\alpha}{2}\, a^\dagger a^\dagger a\, a,$$ where $\omega$ is the ladder spacing, $\alpha$ the anharmonicity (the detuning of the $|1\rangle \leftrightarrow |2\rangle$ transition from $\omega$), and $a$, $a^\dagger$ the ladder operators satisfying $a^\dagger|n\rangle = \sqrt{n+1}\,|n+1\rangle$.

   from 3.1 Model

What would falsify it

1. The conjecture of Section 3.2 is falsified if a finite-time gate on a physical transmon achieves $L_P = 0$ exactly.

   from 4.4 Falsifiability threshold for the conjecture $L_P \gt 0$

2. Operationally, since any measurement has finite resolution $\delta L$, the conjecture is tested by the chain: Under assumptions (A1)–(A2), the detuned three-level subsystem sustains strictly positive leakage $L_2 = 8(\Omega/|\alpha|)^2 \sin^2(\alpha T_g/2) \gt 0$ for any nonzero drive over a finite time, except at the measure-zero gate timings where $\sin^2(\alpha T_g/2) = 0$, which [9] notes can be nulled by shaped envelopes only at isolated parameter points.

   from 4.4 Falsifiability threshold for the conjecture $L_P \gt 0$

3. The results reframe the qubit as a radix approximation and motivate qudit (base-$d$) architectures and a falsifiable leakage-floor test programme.

   from Abstract

Where the claim stops

1. The single observation that would falsify the main result is a finite-time gate on a well-characterized transmon that achieves leakage strictly zero up to arbitrarily small materials-limited error floors — i.e., if pulse shaping can drive $L_P$ to zero faster than all other error channels and hold it there across a continuum of gate timings, the conjecture fails.

   from 6. Discussion

2. We agree that this is the weakest point of the paper: our $1.0 \times 10^{-2}$ gate-averaged figure is a model estimate under assumptions (A1)–(A3), and the honest claim is the structural one — the coupling $\sqrt{n+1}\,\varepsilon(t)$ never vanishes — not the magnitude.

   from 6. Discussion

3. The transmon is best regarded as a qudit; the qubit is an approximation whose entropy cost we have made explicit.

   from 8. Conclusion

Where to read

1. Abstract202 words · 1 min
2. 1. Introduction661 words · 3 min
3. 2. Background and Related Work1,205 words · 5 min
4. 3. Methods538 words · 2 min
5. 4. Analysis1,007 words · 4 min
6. 5. Results356 words · 2 min
7. 6. Discussion554 words · 2 min
8. 7. Practical Implications and Engineering Pathways400 words · 2 min
9. 8. Conclusion175 words · 1 min
10. Appendix A. Divergence report277 words · 1 min
11. Appendix B. Claim attribution471 words · 2 min
12. Provenance and AI Involvement225 words · 1 min