The Two-Level Lie: The Transmon Is Not a Qubit — And the Entire Field Knows It

In the paper's words

v2.0 adds Part II: The Approximation Entropy — a meta-mathematical framework quantifying the irreducible cost of translating continuous bosonic physics to discrete digital computation. Introduces Rosetta Constant, Trotter Wall, and Fractal Wall theorem with a falsifiable calorimetry experimental protocol.

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

Numbers it reports

1. Table 1 reproduces the key results. $E_J / E_C$ $\alpha_r$ Resolvable levels $d$ "Qubit" fraction Harmonic oscillator fraction 53 6.2% 4 6.2% 93.8% 100 3.5% 6 3.5% 96.5% 200 2.5% 9 2.5% 97.5% 325 1.9% 12 1.9% 98.1% At the highest $E_J / E_C$ — the "best" transmon by conventional metrics — the device resolves 12 levels, and the fraction of its Hamiltonian that distinguishes it from a harmonic oscillator is $1.9\%$.

   from 3.1 The Wang et al. (2024) Benchmark

2. The second is contradicted by 15 years of leakage measurement.

   from 7. What Would Refute This?

3. At $E_J / E_C = 325$ — the highest ratio reported by Wang et al.

   from 1. The Confession

4. (2020) demonstrated that optimal control achieves better leakage suppression than DRAG — but at the cost of reduced gate speed.

   from 3.4 The DRAG Pulse: A Suppression Mechanism Masquerading as a Gate

5. The minimum entropy generated by this projection is given by the logarithm of the ratio of accessible phase-space volumes: $$S_A = k_B \ln\left(\frac{\text{Vol}(\mathcal{H}_B)}{\text{Vol}(\mathcal{H}_C)}\right)$$ For a bosonic mode with $d$ resolvable levels compressed to a 2-level subspace: $$S_A = k_B \ln\left(\frac{d}{2}\right)$$ At $E_J/E_C = 325$, where $d = 12$: $$S_A = k_B \ln(6) \approx 1.79 \, k_B$$ This is the irreducible entropy generated by the act of treating a 12-level system as a 2-level system — before any gate error, decoherence, or environmental coupling.

   from 10.1 Definition

6. This occurs when: $$d \approx \frac{1}{\alpha_r}$$ yielding: $$S_A = k_B \ln\left(\frac{1}{2\alpha_r}\right) \approx k_B \ln(1/\alpha_r) - k_B \ln 2$$ Define Rosetta's Constant $R$ as the thermodynamic cost per translation operation: $$R \equiv k_B T \ln(1/\alpha_r)$$ For state-of-the-art transmons at $T = 15 \, \text{mK}$ and $\alpha_r = 1.9\%$: $$R \approx k_B \cdot (0.015 \, \text{K}) \cdot \ln(52.6) \approx 4.0 \, k_B T \approx 8.3 \times 10^{-25} \, \text{J}$$ This is the heat that must be dissipated to the dilution refrigerator per gate operation , purely from the act of projecting a 12-level bosonic mode into a 2-level computational subspace.

   from 10.2 Rosetta's Constant

How it is done

1. It is a bosonic mode actively coerced into two-level behavior through suppression techniques whose very existence is proof of the coercion.

   from 1. The Confession

2. (2024) fabricated an array of transmon qubits spanning $E_J / E_C$ from $53$ to $325$.

   from 3.1 The Wang et al. (2024) Benchmark

3. Compression of this kind is lossy by definition.

   from 9.1 The Categorical Divide

4. The cosine function on the real line cannot be uniformly approximated by polynomials beyond a certain bound without exponential error growth in the complex plane.

   from 9.2 Why $\cos(\phi)$ Resists Algebra

What would falsify it

1. Falsifiability is a condition of the QNFO research program.

   from 7. What Would Refute This?

2. This paper's central claim would be disconfirmed if: A transmon architecture achieves $E_J / E_C \gt 500$ with $\alpha_r \gt 10\%$, violating the $1/\sqrt{E_J / E_C}$ scaling derived from the cosine potential.

   from 7. What Would Refute This?

3. It makes a specific, falsifiable, experimental prediction.

   from 12.2 An Experimental Test: The Calorimetry Protocol

Where the claim stops

1. The physics does not care what we call things.

   from 8. Conclusion

2. The transmon — the dominant superconducting qubit architecture, absorbing the majority of the field's approximately $ 35 billion in global investment — is not a two-level quantum system.

   from 1. The Confession

3. The transmon that best approximates a qubit most resembles a harmonic oscillator.

   from 1. The Confession

Where to read

1. 1. The Confession381 words · 2 min
2. 2. The Transmon from First Principles258 words · 1 min
3. 3. The Evidence from Experiment500 words · 2 min
4. 4. The Inescapable Trade-Off186 words · 1 min
5. 5. What the Field Knows (And Says It Knows)156 words · 1 min
6. 6. Implications270 words · 1 min
7. 7. What Would Refute This?155 words · 1 min
8. 8. Conclusion149 words · 1 min
9. 9. The Translation Gap632 words · 3 min
10. 10. Approximation Entropy414 words · 2 min
11. 11. The Trotter Wall353 words · 2 min