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REVIEW 3 major objections 4 minor 43 references

A unifying framework for quantum algorithms for time-dependent non-unitary dynamics

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read This paper shows that time-dependent non-unitary linear dynamics can be autonomized with a clock variable into a time-independent linear system, and that combining it with Schrödingerization yields precision dependence log^{5/4}(1/ε)…

desk verdict The autonomization framework is clean and likely correct, but the advertised log^{5/4}(1/ε) precision dependence is not established: Lemma 2.1 omits the B(s)-derivatives that control the clock-direction regularity, so the main complexity claim is unsupported. read the letter →

arxiv 2608.07133 v1 pith:FPCQABBW submitted 2026-08-07 quant-ph

Xiaojing Dong , Yizhe Peng , Yue Yu This is my paper · ORCID
classification quant-ph MSC 81P6865M7065L05
keywords autonomizationclockvariabletime-dependentlineardifferentialequationsnon-unitarydynamicsquantumODEsolversSchrödingerizationblock-encodingFourierspectralmethod
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Quantum algorithms for linear differential equations have mostly targeted time-independent coefficients; time-dependent non-unitary dynamics forces a time-ordered exponential and resists direct adaptation of existing solvers. This paper builds a bridge: introduce a clock variable $s$, lift the non-autonomous system to a transport-type equation in $(t,s)$, Fourier-discretize $s$, and recover the target solution at the grid point $s=T$. The result is an explicit time-independent linear system together with a block-encoding, so any quantum ODE solver for autonomous problems can be applied as a black box. With the Schrödingerization solver, the matrix-query count depends on the error as $\log^{5/4}(1/\varepsilon)$, improving the $\log^2(1/\varepsilon)$ scaling of the best existing approaches under the assumption that $A(t)$ and $b(t)$ are analytic in time. Two numerical experiments on a time-modulated two-cavity system validate the recovery.

What carries the argument

The carrying object is the clock-variable autonomization identity together with its Fourier spectral discretization. The paper adds a clock register $s$, freezes the coefficient matrix as $B(s)$, and solves a convection equation whose method-of-characteristics solution reproduces the original propagator; a smooth initial profile $G(s)$ with $G(0)=1$ that decays super-exponentially (one of $G_m$, $G_w$, $G_f$) makes the problem effectively periodic and controls the largest Fourier mode as $\mu_{s,\max} = O(\log^\beta(1/\delta_s))$ with $\beta=3,1,1/2$ respectively. This produces the explicit time-independent matrix $\bar{A}$, its block-encoding, and the recovery map $x_h(T) = R_x w(T)$ projected at the midpoint $s=T$. The same matrix is then shifted by $\gamma = 1/(2T)$ to restore dissipativity for Schrödingerization, or fed directly into the BCOW linear-system construction.

What would settle it

Take an exactly solvable non-autonomous system whose coefficients are only $C^k$ rather than analytic, run the autonomization construction for a decreasing sequence of targets $\delta_s$, and measure the largest Fourier mode $\mu_{s,\max}$ needed to keep $\|G - P_h G\|_{L^\infty} \le \delta_s$; if $\mu_{s,\max}$ grows like $\delta_s^{-1/(k-1/2)}$ rather than $O(\log^\beta(1/\delta_s))$, that polynomial blow-up would falsify the $\log^{5/4}(1/\varepsilon)$ complexity for that regularity class.

Watch

Extended reading notes

Core claim

The central claim is that the time-ordered exponential is not an obstacle for time-dependent non-unitary systems. For the enlarged homogeneous system $du/dt = B(t)u$, define $z$ by $\partial_t z + \partial_s z = B(s)z$, $z(0,s)=G(s)u(0)$, with $G(0)=1$; then $z(t,s=t)=G(0)u(t)$, exactly. Fourier-spectral discretization in $s$ converts this into a static matrix $\bar{A} = -iP_s \otimes I + \sum_l |l\rangle\langle l| \otimes B(s_l)$, which the paper block-encodes from HAM-T oracles for $A(s)$ and $F(s)$ plus a Fourier-differentiation gadget. The paper proves two instantiations: with Schrödingerization, $O(g_A \alpha_A T \log^{\max\{1+\beta/2, 3\beta/2\}}(g_A/\varepsilon))$ queries to the matrix oracles, which becomes $\log^{5/4}(1/\varepsilon)$ for the super-exponentially decaying profile $G_f$ ($\beta=1/2$); with a Taylor-expansion BCOW solver, $O(C_A^2 g_A \alpha_A T \log^{3(\beta+1)/2}(\alpha_A T C_A^2 g_A/\varepsilon))$ queries, and the analysis removes the diagonalizability assumption by bounding complexity through the growth factor $C(A) = \sup_t \|e^{At}\|$.

Load-bearing premise

The load-bearing assumption is that $A(t)$ and $b(t)$ are real analytic in time, because only then does the Fourier spectral error in the clock variable decay fast enough to keep the number of grid points logarithmic in the required accuracy; with merely finitely smooth coefficients the mesh size would grow polynomially and the $\log^{5/4}(1/\varepsilon)$ dependence is not established.

Editorial extensions

If this is right

  • Any existing quantum ODE solver for time-independent non-unitary systems can be reused for time-dependent problems by consuming the autonomization matrix $\bar{A}$ and its block-encoding; time dependence is fully pushed into the HAM-T oracles.
  • For the analytic clock profile $G_f$, the Schrödingerization combination uses $O(g_A \alpha_A T \log^{5/4}(g_A/\varepsilon))$ queries to the matrix oracles, improving the $\varepsilon$-dependence of earlier constructions.
  • The BCOW combination inherits the same framework and its new analysis extends the algorithm to non-diagonalizable matrices, with cost controlled by the growth factor $C(A)$ instead of an eigenvector condition number.
  • The recovery postselects on the single grid point $s=T$ and then the $x$-block; with amplitude amplification the success probability is $\Omega(1)$, so the algorithm outputs a state $\varepsilon$-close to $|x(T)\rangle$ with a success flag.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A testable extension the paper leaves implicit: for coefficients that are only $C^k$ in time, the Fourier mesh would need to scale polynomially in $1/\delta_s$, so the $\log^{5/4}$ precision dependence is tied to analyticity; one could quantify the crossover by mollifying a $C^k$ coefficient and balancing regularization error against discretization error.
  • The modular structure suggests a menu of clock profiles: the $\beta=1/2$ profile is optimal for the Schrödingerization query count, while a compactly supported profile ($\beta=3$) gives simpler boundary handling and might be preferable in implementations where exact support matters.
  • The midpoint recovery condition ($s=T$ on an even grid) is a small numerical-engineering constraint that may generalize to other spectral discretizations, effectively giving a quantum method of characteristics for non-autonomous equations.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper proposes an 'autonomization' framework for quantum simulation of time-dependent non-unitary linear ODEs. The non-autonomous system (1.1) is lifted via the clock-variable construction of Cao–Jin–Liu to an autonomous transport-type PDE (2.2), whose Fourier spectral discretization yields the explicit time-independent matrix Ā in (2.15). The framework is modular: Ā can be fed into any quantum ODE solver for time-independent systems. The authors combine Ā with the Schrödingerization method and with the BCOW Taylor-expansion solver, deriving query complexities in Theorems 3.1 and 4.2. The headline claim is that, under real-analyticity and dissipativity of A(t), the precision dependence of the Schrödingerization-based combination scales as log^{5/4}(1/ε), improving on the log^2(1/ε) scaling of existing approaches. Numerical experiments on a two-cavity system validate recovery of the target trajectory for both combinations.

Significance. The modular autonomization idea is attractive: it decouples the treatment of time dependence from the choice of the underlying time-independent solver, and the block-encoding circuits in Section 2.3 are explicit. The non-diagonalizable BCOW analysis in Section 4.2 is a useful extension of the original Berry–Childs–Ostrander–Wang result. If the complexity claims were fully established, the log^{5/4}(1/ε) precision dependence would be a noteworthy improvement over the competing log^2(1/ε) scaling. However, the two load-bearing analytical steps — the spectral error bound in Lemma 2.1 and the treatment of the clock-interval length in Theorem 3.1 — are not justified under the stated assumptions. The advertised precision dependence is therefore not currently established, although the framework and numerics remain plausible and potentially salvageable with strengthened assumptions and a corrected proof.

major comments (3)
  1. [Lemma 2.1, Section 2.2] The proof of Lemma 2.1 is not provided; it is asserted by 'following the similar arguments in [30]'. This does not transfer to the present setting. The exact solution (2.5) is z(T,s) = Texp(∫_0^T B(s-T+τ)dτ)G(s-T)u_0, so ∂_s^r z(T,·) involves derivatives of B(s) of orders up to r, not just derivatives of G. The Fourier projection error (2.18) controls the error only through the regularity of the function being projected, which is z(T,·), not G alone. Consequently the bound ∥z(T,s_l) - z_h(T,s_l)∥ ≲ (Δs)^{r-1/2}∥G^{(r)}∥_{L^2(I_s)}∥u(0)∥ in Lemma 2.1 is not established. The assumption after (2.1) that A,b are real analytic on R does not supply uniform control of the relevant derivatives on the interval I_s, whose length grows with δ_s^{-1} for the profile G_f. A concrete counterexample to the claimed bound is A(t)=i e^t I with b=0, for which z(T,s) = exp(i(e^s - e^{s-T}))G(s-T)u_0; the r-th s-derivative grows like e^{rs} on I_s, so the spectral error is not governed by ∥G^{(r)}∥ alone. The proof must either estimate the full r-th derivative of z(T,·), including the B-dependent terms, or impose explicit regularity and boundedness assumptions on B(s) over I_s.
  2. [Assumptions after (2.1) and Theorem 3.1] The advertised log^{5/4}(1/ε) precision dependence is not derived under the stated assumptions. The interval I_s = [-R_s, R_s+2T] has R_s = 2 log^{1/2}(1/δ_s) for the headline profile G_f in (2.9), so the grid points s_l range over an interval whose length grows with ε^{-1}. The quantity α_A in (2.22) is defined as max_{0≤l≤N_s-1}∥A(s_l)∥ and enters the final bound O(g_A α_A T log^{max{1+β/2,3β/2}}(g_A/ε)) in Theorem 3.1. Real analyticity on R does not prevent α_A from growing on this expanding interval. For the dissipative, real-analytic example A(t)=i e^t I, one has α_A = e^{R_s+2T} = e^{O(log^{1/2}(1/δ_s))}, which converts the stated complexity into e^{O(log^{1/2}(1/ε))} instead of log^{5/4}(1/ε). The paper needs either a uniform boundedness assumption on A(s) and b(s) (and suitable derivatives) over I_s with a constant independent of ε, or a more careful statement that separates the ε-dependence of α_A from the precision dependence. As written, the abstract's 'precision dependence can scale as log^{5/4}(1/ε)' is not supported.
  3. [Equation (2.6), Section 2.1] The definition of G_m is inconsistent with the stated property G_m(0)=1. With η(s)=e^{1/(s^2-1)} for |s|<1, one has η(0)=e^{-1}, hence G_m(0)=e^{e^{-1}}≠1. If the intended mollifier is η(s)=1/(s^2-1), then G_m(0)=e^{-1}≠1. The manuscript uses G(0)=1 in the recovery map (2.17), in the success-probability computation in Theorem 3.1 step (2), and in Theorem 4.2. This is a concrete error in a central definition. The profile should be normalized by dividing by G_m(0), or η should be shifted so that η(0)=0; the derivative estimates in Theorem 2.2 are unaffected up to constants, but the text and recovery formulas must be corrected.
minor comments (4)
  1. [Remark 3.2 and Introduction] The claimed improvement over [26] is stated without giving the comparison complexity of [26]. To make the 'improving upon log^2(1/ε)' assertion verifiable, the authors should state the query complexity of [26] in the same notation and identify precisely the source of the saving.
  2. [Lemma 2.1 proof] In the proof of Lemma 2.1, the phrase '∀l≥0' is unclear because the grid index l ranges over 0≤l≤N_s-1 and the pointwise estimate is needed at the recovery point s=T=s_{N_s/2}; this should be stated directly.
  3. [Section 2.2, Eq. (2.18)] The pointwise Fourier projection estimate from [34] applies to periodic functions; for the profiles G_w and G_f, which are only approximately supported, the boundary values of G and its derivatives at s=±R_s are small but not exactly zero. The additional contribution of the boundary discontinuity to the estimate (2.18) should be addressed explicitly rather than left to the cited literature.
  4. [Acknowledgements] The acknowledgements mention 'NL acknowledges funding from ...', but the author list contains no 'NL'. This appears to be a leftover from a different version and should be corrected (or the name spelled out) before publication.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is an analytic reduction with explicit error estimates, and prior-work citations are used as external building blocks rather than as definitions of the target result.

full rationale

The paper's central chain is a mathematical derivation, not a fit. The enlarged system (2.1) is constructed so that F(t)r(t)=b(t) by the definitions of F and r, and Theorem 2.1 proves z(T,T)=G(0)u(T) by a direct characteristic calculation; the recovery map is then defined to extract u(T), so there is no fitted parameter being renamed as a prediction. The profile choices G_m, G_w, and G_f are explicit functions with stated estimates; the parameter a and the shift gamma are chosen to satisfy error and dissipativity bounds, not to reproduce the claimed complexity. The self-citations to [1] and [30] supply the clock-variable construction, the smooth-profile derivative estimates, and the Schroedingerization complexity lemma. These are prior published results whose stated assumptions do not include the present autonomization claim or the log^{5/4}(1/epsilon) precision target, so under the reviewing rules they count as independent support rather than circularity. The paper's possible correctness gap, namely that Lemma 2.1 bounds the Fourier error only through G^(r) and omits derivatives of B(s), and the fact that alpha_A may carry epsilon-dependence for real-analytic but unbounded coefficients, is a mathematical soundness concern, not a circularity: the claimed bound does not reduce by construction to an input or to a self-citation. The numerical experiments are benchmarked against a reference solution and do not feed fitted values back into the theorem. No step of the derivation is equivalent to its own inputs by definition, and no fitted parameter is relabeled as a prediction.

Assumptions & free parameters 2 free parameters · 7 assumptions · 0 invented entities

The central claim rests on the dissipativity and analyticity of the original ODE coefficients, the availability of exact HAM-T oracles, and the use of published Fourier and Schrödingerization estimates. No new physical entities are introduced; the clock variable is a mathematical device. The shift γ and profile parameter a are hand-chosen design constants rather than fitted data parameters.

free parameters (2)
  • shift γ = 1/(2T)
    Chosen so that the shifted generator \bar{A}_γ is dissipative (ω(\bar{A}) ≤ 1/(2T)); this value enters the amplitude amplification and the complexity bound. It is a hand-chosen design parameter, not fit to data.
  • profile scale a for G_w = 2 log^{1/2}(1/δ_s)
    Chosen so that the error-function profile decays to within δ_s outside [-1/2,1/2]; this controls the truncation error and the derivative bound in Theorem 2.2. A design parameter, not fit to data.
assumptions (7)
  • domain assumption A(t)+A†(t) ⪯ O for t ∈ [0,T]
    Assumed in Section 2 (page 5). Needed to ensure the shifted Schrödingerization system is dissipative; without it, the normalized output may grow exponentially and the complexity bound in Theorem 3.1 fails. The BCOW route relaxes this via the growth factor C(A).
  • domain assumption A(t) and b(t) are real analytic on R
    Assumed after Eq. (2.1). Used to make the Fourier spectral error decay fast enough that µ_{s,max}=O(log^β(1/δ_s)); this is the source of the log^{5/4} scaling. The paper notes finite regularity is left to future work.
  • domain assumption Exact HAM-T block-encodings of A(s_l) and F(s_l) are available for all grid points
    Assumed in Section 2.3 (Eqs. (2.21)-(2.22)). The algorithm and complexity are stated in the oracle model; the cost of implementing these oracles is not included.
  • ad hoc to paper G(s) can be chosen with G(0)=1 and (near-)compact support to impose periodic boundary conditions
    The recovery map relies on z(T,T)=G(0)u(T) and on the boundary conditions; the three profiles (2.6),(2.8),(2.9) are constructed for this purpose. The definition of G_m in (2.6) has an apparent typo.
  • standard math Standard pointwise Fourier projection error estimate (2.18)
    Used in Lemma 2.1 to bound the discretization error of the clock variable; taken from [34].
  • standard math Smooth-initialization and profile estimates from [30]
    Theorems 2.2 and Lemma 3.1 borrow from [30]'s analysis of Schrödingerization; these are published results treated as external lemmas.
  • standard math QSVT-based Hamiltonian simulation of time-independent H is available
    Used in Section 3.2 to simulate e^{-iH T} with block-encoding; standard result in the literature.

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Cite this review

Pith. "Pith review of A unifying framework for quantum algorithms for time-dependent non-unitary dynamics." pith.science (2026). https://pith.science/paper/FPCQABBW

@misc{pith2026260807133,
  author       = {Pith},
  title        = {Pith review of: A unifying framework for quantum algorithms for time-dependent non-unitary dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FPCQABBW}},
  note         = {Machine review of arXiv:2608.07133}
}
abstract

Quantum algorithms for simulating linear differential equations have attracted growing interest, driven by applications ranging from Hamiltonian dynamics to general non-unitary dynamics. While time-independent cases are well studied, time-dependent non-unitary dynamics remains considerably less explored, and it is unclear how to systematically adapt existing solvers for time-independent systems to such problems. In this work, we address this gap by introducing an autonomization framework based on the clock-variable formulation, a technique originally developed for time-dependent Hamiltonian systems in~\cite{CJL23TimeSchr}. By lifting the original non-autonomous system to an autonomous transport-type equation on an extended space and applying the Fourier spectral discretization in the clock variable, we obtain an explicit time-independent linear system, together with a suitable initial state and a recovery map for the target solution. Crucially, this formulation decouples the treatment of time dependence from the choice of the quantum ODE solver, thereby enabling the direct application of existing solvers designed for time-independent systems to the resulting autonomous problem. We combine this framework with Schr\"odingerization and a Taylor-expansion-based quantum ODE solver. In the Schr\"odingerization-based combination, our complexity analysis shows that the precision dependence can scale as $\log^{5/4}(1/\varepsilon)$, improving upon the $\log^2(1/\varepsilon)$ scaling found in existing approaches. Numerical experiments validate the autonomization formulation and confirm the successful recovery of the target solution.

Figures

Figures reproduced from arXiv: 2608.07133 by the authors.

Figure 1
Figure 1. Commutative diagram illustrating two routes from time-dependent non-unitary dynamics [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Our autonomization route: the time-dependent non-unitary system is first converted into a [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Construction of G(s) via error function. (3) G decays to zero at a super-exponential rate. One may also set a = 1 in (2.8), and the resulting function is denoted by Gf(s): G(s) = Gf(s) = ϕ1(s + 1 2 ) − ϕ1(s − 1 2 ) ϕ1( 1 2 ) − ϕ1(− 1 2 ) . (2.9) In this case, G(s) → 0 as |s| → +∞ at a super-exponential rate, as shown in Fig. 3b. It is evident that all these profiles satisfy G(0) = 1. Theorem 2.2. Let 0 < δs ≤ e −1 ,… view at source ↗
Figures from the paper (16 more)
Figure 4
Figure 4. Figure 4: HAM-T block-encoding oracles HAM-TA and HAM-TF . 11 [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: Block-encoding of DB = P l |l⟩⟨l| ⊗ B(sl), denoted by UB. Let Vθ be the corresponding controlled rotation, specified on the output of Oθ by Vθ|0⟩a|θl⟩w = Ry(2θl)|0⟩a|θl⟩w. The total ancilla register used in this block-encoding has w + 1 qubits. The unitary UD obtained …
Figure 6
Figure 6. Figure 6: Block-encoding of −iDs/µs,max, denoted by UD. block-encoding by the Fourier transform gives UP = (I ⊗(w+1) ⊗ Fs ⊗ I ⊗(n+1))(UD ⊗ I ⊗(n+1))(I ⊗(w+1) ⊗ F † s ⊗ I ⊗(n+1)), (2.27) which is a (µs,max, w + 1, 0)-block-encoding of DP = −iPs ⊗I ⊗(n+1). The circuit is shown in …
Figure 7
Figure 7. Figure 7: Block-encoding of DP = −iPs ⊗ I ⊗(n+1), denoted by UP . 13 [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 8
Figure 8. Figure 8: Block-encoding of the autonomization matrix [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]
Figure 9
Figure 9. Figure 9: Quantum circuit for the Schr¨odingerization procedure. [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]
Figure 10
Figure 10. Figure 10: Numerical and reference intracavity energies for autonomization–Schr¨odingerization. [PITH_FULL_IMAGE:figures/full_fig_p026_10.png]
Figure 11
Figure 11. Figure 11: Numerical and reference intracavity energies for autonomization–BCOW. [PITH_FULL_IMAGE:figures/full_fig_p027_11.png]
Figure 12
Figure 12. Figure 12: Block-encoding of Jk. 39 [PITH_FULL_IMAGE:figures/full_fig_p039_12.png]
Figure 13
Figure 13. Figure 13: Block-encoding of M2. Applying LCU to W, we obtain an exact (2, mA + 3, 0)-block-encoding of W, denoted by UW . The circuit is shown in [PITH_FULL_IMAGE:figures/full_fig_p040_13.png]
Figure 14
Figure 14. Figure 14: Block-encoding of W, denoted by UW . Step 2: Block-encoding of W˜ . The last diagonal block is decomposed as W˜ = M˜ 2 ⊗ I ⊗n − J˜ p ⊗ I ⊗n , (A.20) 40 [PITH_FULL_IMAGE:figures/full_fig_p040_14.png]
Figure 15
Figure 15. Figure 15: Block-encoding of W˜ , denoted by UW˜ . Step 3: Coherent routing for L1. Combining UW and UW˜ gives the block-encoding of L1. This step is not an LCU sum, since W and W˜ act on orthogonal block sectors. Therefore, the sector register coherently routes the input to the…
Figure 16
Figure 16. Figure 16: Block-encoding of L1, denoted by UL1 . Block-encoding of L2. We now construct the block-encoding of L2. For a rectangular matrix A ∈ CM×N with M, N ≤ 2 q , we use the standard square embedding Ae ∈ C 2 q×2 q , whose top-left block is A and whose remaining entries are …
Figure 17
Figure 17. Figure 17: Block-encoding of AF,e and AF ,e e . Step (b): Block-encoding of the sector-time shift. Define Sm := " Rm O AR,e e O # , AR,e e := |0⟩⟨m − 1|. Let ADD denote the cyclic shift on the joint sector-time register |s⟩|r⟩m, namely ADD|sm + r⟩ = |(sm + r + 1) mod 2m⟩. Equiva…
Figure 18
Figure 18. Figure 18: Block-encoding of L2, denoted by UL2 . Block-encoding of Cm,k,p(Ah). We now combine the block-encodings of L1 and L2. The con￾struction is first presented for the normalized case, and is then extended to the general case h > 0 and αA ≥ ∥A∥2. • Normalized case. In the …
Figure 19
Figure 19. Figure 19: LCU block-encoding of Cm,k,p(A), denoted by UC. • Extension to general h and αA. It remains to remove the normalized assumption. Sup￾pose that an exact (αA, mA, 0)-block-encoding of A is available, while the target matrix is Cm,k,p(Ah). If h = 1/αA, then the given blo…

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