Leakage as the Price of Binary Truncation: A Scaling Law for Transmon Leakage, Removal Capacity, and Its Ontological Reading

In the paper's words

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|.

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

Numbers it reports

1. 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.

   from Abstract

2. At $E_{J}/E_{C}=325$, the highest ratio reported for a functional transmon, the device supports $d=12$ resolvable energy levels and its relative anharmonicity $\alpha_{r}\equiv|\alpha|/\omega_{01}$ is $1.9\%$; the same source documents the approximately $ 35 billion in global investment and proposes a falsifiable calorimetry protocol: compare heat dissipated in free evolution of a transmon prepared in $|1\rangle$ for 100 ns against heat dissipated by a DRAG-calibrated gate sequence implementing the same identity rotation, isolating the translation cost of the digital map from the computation cost [9].

   from 2. Background and Related Work

3. This paper derives, from the binary-truncation leakage analysis of bosonic ladders [8], an operative scaling law for transmon leakage, $\overline{L}_{2}=4(\Omega/|\alpha|)^{2}$, evaluates it at an illustrative operating point with fully shown arithmetic, and converts it into an architectural requirement: holding steady-state leakage at $10^{-4}$ under a 50 ns gate cycle demands a removal capacity of $2\times10^{9}\ \mathrm{s}^{-1}$ per qubit, exceeding the fastest measured passive leakage hopping rate in tunable-coupler transmon arrays by a factor of 200 [4].

   from 8. Conclusion

4. 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 $\Omega/|\alpha|$.

   from Abstract

5. 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 $\chi\approx3$.

   from Abstract

6. The contributions are: A restatement of the binary-truncation leakage bound $L_{2}=8(\Omega/|\alpha|)^{2}\sin^{2}(\alpha T_{g}/2)$ and the exact three-level result $P_{2,\max}=2r^{2}/(1+2r^{2})$ as the operative scaling law for transmon leakage, with numerical evaluation at illustrative parameters (Sections 3 and 4.1, from the derivations of [8]).

   from 1. Introduction

How it is done

1. The contributions are: A restatement of the binary-truncation leakage bound $L_{2}=8(\Omega/|\alpha|)^{2}\sin^{2}(\alpha T_{g}/2)$ and the exact three-level result $P_{2,\max}=2r^{2}/(1+2r^{2})$ as the operative scaling law for transmon leakage, with numerical evaluation at illustrative parameters (Sections 3 and 4.1, from the derivations of [8]).

   from 1. Introduction

2. At $E_{J}/E_{C}=325$, the highest ratio reported for a functional transmon, the device supports $d=12$ resolvable energy levels and its relative anharmonicity $\alpha_{r}\equiv|\alpha|/\omega_{01}$ is $1.9\%$; the same source documents the approximately $ 35 billion in global investment and proposes a falsifiable calorimetry protocol: compare heat dissipated in free evolution of a transmon prepared in $|1\rangle$ for 100 ns against heat dissipated by a DRAG-calibrated gate sequence implementing the same identity rotation, isolating the translation cost of the digital map from the computation cost [9].

   from 2. Background and Related Work

3. We proceed in five steps, using only formulas already derived in [8] and measured rates from [4], plus explicitly stated assumptions where the sources are silent.

   from 3. Methods

4. This is our own assessment from the deficit arithmetic: the measured hopping rates of [4] bound passive transport, and no passive rate in that measured range reaches the required $\Gamma_{\mathrm{rem}}$.

   from 4.4 Removal-capacity deficit against measured hopping rates

What would falsify it

1. 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.

   from Abstract

2. A falsifiable platform-ordering prediction — bosonic leakage burden exceeds fermionic — with explicit falsification conditions (Sections 5 and 6).

   from 1. Introduction

3. At $E_{J}/E_{C}=325$, the highest ratio reported for a functional transmon, the device supports $d=12$ resolvable energy levels and its relative anharmonicity $\alpha_{r}\equiv|\alpha|/\omega_{01}$ is $1.9\%$; the same source documents the approximately $ 35 billion in global investment and proposes a falsifiable calorimetry protocol: compare heat dissipated in free evolution of a transmon prepared in $|1\rangle$ for 100 ns against heat dissipated by a DRAG-calibrated gate sequence implementing the same identity rotation, isolating the translation cost of the digital map from the computation cost [9].

   from 2. Background and Related Work

Where the claim stops

1. What the result establishes and what it does not.

   from 6. Discussion

2. Each link carries its own assumptions, and the chain is only as strong as them.

   from 6. Discussion

3. (ii) A measurement showing leakage production in operating transmons orders of magnitude below $2\times10^{5}\ \mathrm{s}^{-1}$ at comparable gate rates, which would mean the illustrative parameters misrepresent practice by more than the assumed linear sensitivity tolerates.

   from 6. Discussion

Where to read

1. Abstract235 words · 1 min
2. 1. Introduction530 words · 2 min
3. 2. Background and Related Work1,053 words · 5 min
4. 3. Methods392 words · 2 min
5. 4. Analysis798 words · 3 min
6. 5. Results397 words · 2 min
7. 6. Discussion765 words · 3 min
8. 7. Practical Implications and Engineering Pathways699 words · 3 min
9. 8. Conclusion273 words · 1 min
10. Provenance and AI Involvement225 words · 1 min