CSS Codes for Quantum Metrology with Discrete-Time Error Correction: A Reconciled Quantitative Analysis

In the paper's words

Quantum metrology promises estimation precision scaling as the Heisenberg limit Δω ∼ 1/(NT) rather than the standard quantum limit (SQL) Δω ∼ 1/√NT, but noise typically destroys this advantage. Recent work showed that when the noise is perpendicular to the sensing Hamiltonian — the regime permitted by the Hamiltonian-Not-in-Lindblad-Span (HNLS) condition — discrete-time quantum error correction with Calderbank-Shor-Shor (CSS) codes preserves Heisenberg-like temporal scaling over a finite interrogation window whose duration depends on the correction frequency, and that recovery applied only after sensing gives no advantage. This paper reconciles three independent quantitative treatments of that framework into a single arithmetic-transparent analysis. [...]

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

Numbers it reports

1. Under a stated convention of independent per-qubit errors with probability $p = 10^{-3}$ per correction cycle, we derive per-cycle logical failure probabilities $P_{\mathrm{fail}}^{(\mathrm{Steane})} \approx 2.09 \times 10^{-5}$ and $P_{\mathrm{fail}}^{(\mathrm{Shor})} \approx 3.58 \times 10^{-5}$ for the $[[7,1,3]]$ Steane and $[[9,1,3]]$ Shor codes, corresponding to interrogation windows of approximately $23.9$ ms and $14.0$ ms at a $1\,\mu$s correction period for a failure budget $\varepsilon = 0.5$.

   from Abstract

2. Failure budget: $\varepsilon = 0.5$ (stated assumption: we accept cumulative failure probability $\approx \varepsilon$ at the window edge).

   from 3.1 Conventions (reconciled)

3. Failure budget $\varepsilon = 0.5$ (stated assumption; actual cumulative failure probability at the window edge is $1 - e^{-0.5} \approx 0.393$).

   from 4. Analysis

4. Total: $$P_{\mathrm{fail}}^{(\mathrm{Shor})} = 3.57488 \times 10^{-5} + 0.08350 \times 10^{-5} + 0.0000125 \times 10^{-5} \approx 3.58 \times 10^{-5} \text{ per cycle}.$$ (Note: source draft C reported $3.66 \times 10^{-5}$ here due to an addition slip — $3.5749 \times 10^{-5} + 0.0835 \times 10^{-5} = 3.583 \times 10^{-5}$, not $3.658 \times 10^{-5}$; see Appendix A.) The Steane code fails less often per cycle at equal distance because its block is smaller ($7$ vs. $9$ qubits), reducing the combinatorial factor from $36$ to $21$.

   from 4.2 Shor code failure probability

5. Under the independent-failure model of Section 3.3, at $T = T_{\max}^{(\mathrm{Steane})}$, $K = 23900$: $$(1 - P_{\mathrm{fail}})^K = \left(1 - 2.09 \times 10^{-5}\right)^{23900} \approx e^{-23900 \times 2.09 \times 10^{-5}} = e^{-0.4995} \approx 0.607.$$ This matches the budget by construction ($e^{-\varepsilon} = e^{-0.5} = 0.6065$), an internal consistency check of the window definition.

   from 4.4 QFI retention at the window edge (projection)

6. Entries [11]–[14] (qLDPC/threshold, bosonic codes, planar modules, threshold ensemble effects) were not retained because the source drafts' claims referencing them could not be substantiated within this analysis.

   from 6. Discussion

How it is done

1. Under a stated convention of independent per-qubit errors with probability $p = 10^{-3}$ per correction cycle, we derive per-cycle logical failure probabilities $P_{\mathrm{fail}}^{(\mathrm{Steane})} \approx 2.09 \times 10^{-5}$ and $P_{\mathrm{fail}}^{(\mathrm{Shor})} \approx 3.58 \times 10^{-5}$ for the $[[7,1,3]]$ Steane and $[[9,1,3]]$ Shor codes, corresponding to interrogation windows of approximately $23.9$ ms and $14.0$ ms at a $1\,\mu$s correction period for a failure budget $\varepsilon = 0.5$.

   from Abstract

2. Reference [2], the primary object of this study (with [1] being the identical fetched record), develops a quantum-metrology protocol based on CSS codes in which error correction is applied at discrete times during the sensing evolution.

   from 2. Background and Related Work

3. Hence post-hoc recovery yields no advantage over the uncorrected protocol, in agreement with [2].

   from 3.4 Post-hoc recovery cannot help

4. All inputs are stated; all arithmetic is shown.

   from 4. Analysis

What would falsify it

1. Failure modes and falsifiability.

   from 6. Discussion

2. The central claim — that discrete-time CSS correction preserves Heisenberg-like scaling over a finite window — would be falsified if experiments showed QFI decaying faster than the modeled $(1 - P_{\mathrm{fail}})^K$ suppression at the stated $p$ and $\Delta t$, or if measured logical failure rates exceeded the binomial prediction systematically.

   from 6. Discussion

Where the claim stops

1. (1) The failure model counts all weight-$\ge 2$ patterns as failures; for the degenerate Shor code some such patterns are correctable or harmless, so $P_{\mathrm{fail}}^{(\mathrm{Shor})}$ is an upper-bound-style estimate.

   from 6. Discussion

2. (3) Syndrome extraction is assumed instantaneous and error-free; real implementations add time and gate errors, shrinking the window.

   from 6. Discussion

3. The central claim — that discrete-time CSS correction preserves Heisenberg-like scaling over a finite window — would be falsified if experiments showed QFI decaying faster than the modeled $(1 - P_{\mathrm{fail}})^K$ suppression at the stated $p$ and $\Delta t$, or if measured logical failure rates exceeded the binomial prediction systematically.

   from 6. Discussion

Where to read

1. Abstract215 words · 1 min
2. 1. Introduction316 words · 1 min
3. 2. Background and Related Work425 words · 2 min
4. 3. Methods417 words · 2 min
5. 4. Analysis572 words · 2 min
6. 5. Results135 words · 1 min
7. 6. Discussion257 words · 1 min
8. 7. Conclusion196 words · 1 min
9. Appendix A. Divergence report356 words · 2 min
10. Appendix B. Claim attribution145 words · 1 min