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Zero-lift checking-time fits versus physical decay rate

PDF p. 16: Last paragraph; PDF p. 17: Fig. S5.2(a)–(b) and caption

Statement

In the finite equal-rate rigid-rotor benchmark, set the degeneracy lift $Δ$ to zero and let $γ > 0$ be the common instantaneous downward rate of an excited magnetic basis state. For any one of the four specified single-manifold codes and its transpose recovery, the checking time at target $F_{0} \in (0, 1]$ obeys $τ_{γ} (F_{0}) = τ_{1} (F_{0}) / γ$ , including the value $+ \infty$ with the usual convention. Here the subscript $1$ denotes the same model with rate scale one. This is an exact rescaling of the specified model, not a determination of the crossing constant.

For $F_{0} = 0.9999$ , the article reports $τ \approx 1.3 \times 10^{- 4} / γ$ for the minimal code, $τ \approx 2.0 \times 10^{- 4} / γ$ for the detection code, $τ \approx 0.020 / γ$ for Aydin–Barg, and $τ \approx 0.026 / γ$ for Æ. Among these reported values Æ is largest. The article calls the last two coefficients numerical fits and supplies no fit uncertainties in the cited caption. It gives the minimal-code relation $1 - F_{t r} = 3 p_{d e c a y} / 4$ and the resulting $τ = - γ^{- 1} \log [1 - 4 (1 - F_{0}) / 3]$ , and the detection-code approximation $1 - F_{t r} ≃ γ t / 2$ , yielding $τ ≃ 2 (1 - F_{0}) / γ$ . These baseline formulas and the fit coefficients are reported source results, not independently verified calculations here.

The checking interval therefore scales inversely with the modeled physical rate when the same first-crossing target and other model parameters are retained. This statement alone does not establish a repeated-cycle performance guarantee.

Facts & Assumptions

[F1]

The checking time is the infimum of nonnegative times at which the transpose-recovered entanglement fidelity is strictly below the target; an empty crossing set gives $+ \infty$ . The fidelity is the overlap with a maximally entangled logical-reference state after encoding, noise and recovery. Operational checking time

[F2]

In the benchmark, the finite physical space is a direct sum of integer-spin blocks $J = 0, \dots, N$ . The phenomenological lift is $H_{Δ} = Δ \sum_{J, m} J m ∣ J, m ⟩ ⟨ J, m ∣$ on these finitely many complete blocks; here $Δ = 0$ , so $H_{Δ} = 0$ . Its unlifted Hamiltonian is scalar on each block, and each downward jump maps the $J$ block into the $J - 1$ block and has the form $L_{δ} = \sqrt{γ} A_{δ}$ , where $A_{δ}$ is independent of $γ$ . The jump model, its finite-emitter, bath, spectral and other stated physical hypotheses, and the equal-rate parameter choice are assumed, together with the finite-emitter and physical hypotheses of the lift postulate. Equal-rate rigid-rotor benchmark parameters Phenomenological angular-momentum degeneracy lift Canonical angular-momentum downward jumps

[F3]

Each selected code is contained in one block: the minimal code in $J = 1$ , the detection code in $J = 2$ , Aydin–Barg in $J = 4$ , and Æ in $J = 5$ . Choose $N \geq 5$ so that all four comparisons are represented. Minimal single-manifold encoding J equals two detection encoding J equals four Aydin–Barg encoding J equals five absorption-emission code

[F4]

The finite numerical protocol exponentiates the GKSL generator and evaluates the transpose-recovery fidelity for an isometric code from the channel's Kraus operators and their code-compressed products. It specifies exact target operations, not a certified floating-point result. Lindbladian exponentiation and Choi Kraus extraction

[F5]

For every finite complex matrix $B$ and scalar $z \in C$ , the matrix exponential is the operator-norm convergent series $e^{z B} = \sum_{k = 0}^{\infty} z^{k} B^{k} / k!$ , with $B^{0} = I$ . No Hermiticity assumption on $B$ is required. Finite-dimensional matrix exponential

[F6]

For the benchmark jump matrices, $\sum_{δ} L_{δ}^{†} L_{δ} = γ \sum_{J = 1}^{N} Π_{J}$ . Thus every excited magnetic basis state has total instantaneous downward rate $γ$ , and the ground block has rate zero. This is a conditional result under the benchmark's full physical, spectral and choice hypotheses. Normalized jumps give a scalar total decay operator

[F7]

Put $C (X) = N (V X V^{†})$ and $Q = C (I_{H_{L}}) = N (P)$ . The logical form of the support-normalized transpose map is $Y \mapsto \sum_{a} A_{a}^{†} Q^{- 1 / 2} Y Q^{- 1 / 2} A_{a}$ , where $A_{a} = E_{a} V$ and $(E_{a})$ is any finite Kraus family of $N$ . Its completion has no effect after noise on code-supported inputs. Transpose (Petz) recovery on a code

[A1]

Assume the Axiom of Choice, Countable Choice and Dependent Choice carried by the benchmark, lift and numerical interfaces, as well as their stated physical and spectral hypotheses. The Axiom of Choice The Axiom of Countable Choice (AC_ω) The axiom of dependent choice: a relation in which every element is related to something admits an ℕ-indexed chain Equal-rate rigid-rotor benchmark parameters Phenomenological angular-momentum degeneracy lift Lindbladian exponentiation and Choi Kraus extraction

Proof

Given: One of the four code isometries in [F3], the zero-lift model and assumptions in [F2] and [A1], and a target $F_{0} \in (0, 1]$ .

Goal: Establish the exact inverse-rate scaling of its first-crossing time and distinguish it from the attributed numerical coefficients.

1.1F2F3F6A1

At $Δ = 0$ the lift term in [F2] vanishes. Let a block-diagonal operator be a sum of operators each supported within a single $J$ block. The Hamiltonian commutator vanishes on this subspace because the Hamiltonian is scalar on each block. Each $A_{δ} X A_{δ}^{†}$ maps a block-supported $X$ to the next lower block, while $A_{δ}^{†} A_{δ}$ preserves each block. Hence the dissipator preserves block-diagonal operators, and the generator restricted to them is $γ D$ , with $D$ independent of $γ$ . Every encoded logical operator starts in one block.

2.1F1F3F4F5F7step 1.1

In a fixed basis of the finite block-diagonal operator space, represent $D$ by a finite complex matrix $D$ . For every $k \geq 0$ , $(γ D)^{k} = γ^{k} D^{k}$ , so [F5] gives $e^{t (γ D)} = \sum_{k = 0}^{\infty} t^{k} (γ D)^{k} / k! = \sum_{k = 0}^{\infty} (γ t)^{k} D^{k} / k! = e^{(γ t) D}$ , with both series convergent in operator norm. Invariance means that every power of the full generator on this subspace equals the corresponding power of its restriction; applying [F5] to the full finite matrix therefore shows that its exponential restricts to $e^{t (γ D)}$ . Hence $N_{γ, t} (V X V^{†}) = N_{1, γ t} (V X V^{†})$ for every logical operator $X$ . Thus the two restricted channels agree, including their value on the code projector. For either restricted channel write $C$ for this common map and $Q = C (I_{H_{L}})$ . By [F7], for $B = Q^{- 1 / 2} Y Q^{- 1 / 2}$ the $(μ, ν)$ entry of its logical support-normalized recovery is $\sum_{a} ⟨ A_{a} μ ∣ B ∣ A_{a} ν ⟩ = Tr [B C (∣ ν ⟩ ⟨ μ ∣)]$ , by expanding the finite matrix entries. Thus that recovery map is determined by $C$ and $Q$ , independently of the Kraus family. The completion is irrelevant on $C (X)$ by [F7]. The recovered logical channels therefore coincide, giving $F_{γ} (t) = F_{1} (γ t)$ in the convention of [F1].

3.1F1step 2.1∎

Consequently the strict crossing set at rate $γ$ is ${s / γ : s \geq 0, F_{1} (s) < F_{0}}$ . Since multiplication by $1 / γ$ is an order-preserving bijection of $[0, \infty)$ , its infimum is $τ_{1} (F_{0}) / γ$ ; if the set is empty, both checking times are $+ \infty$ . The displayed numerical coefficients and baseline formulas are the article's reported results and do not follow from this rescaling alone. Thus the inverse-rate law is conditional on the model, while the reported constants retain their stated numerical or formula provenance.

Source notes

The article's $p_{d e c a y}$ denotes the modeled decay probability, $1 - e^{- γ t}$ , in its minimal-code baseline; $F_{t r}$ is the transpose-recovered entanglement fidelity. The two fitted coefficients are approximate and no uncertainty is inferred. No numerical replication, measurement, or repeated-cycle threshold theorem is claimed.

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Sources

Monolithic Quantum Error Correction in the Presence of Distinguishability · PDF p. 2, text immediately after Eq. (4) and the checking-time and transpose-fidelity discussion through Eq. (6); p. 16, final paragraph; p. 17, Fig. S5.2(a)-(b) and caption