Saturable Quantum Speed Limits for Imaginary-Time Evolution

Kohei Kobayashi

REVIEW 3 major objections 3 minor 29 references

Saturable Quantum Speed Limits for Imaginary-Time Evolution

T0 review · 3 major / 3 minor · reviewed 2026-08-05 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Imaginary-time evolution obeys a quantum speed limit set by the Fubini–Study angle and the time-averaged energy variance.

desk verdict The central geometric QSL for pure-state imaginary-time evolution is correct and clean, but both quantitative applications contain algebra errors that need fixing before the claims in the abstract can be trusted. read the letter →

arxiv 2508.10361 v1 pith:5MAH4IVP submitted 2025-08-14 quant-ph

Kohei Kobayashi This is my paper · ORCID

classification quant-ph

keywords quantumspeedlimitimaginary-timeevolutionFubini-Studyanglenon-unitarydynamicsGroversearchground-statepreparationenergyvariancegeodesicsaturation

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

This paper tries to establish that imaginary-time evolution—the non-unitary, norm-decreasing dynamics used in ground-state preparation—obeys its own quantum speed limit. The bound says the time to reach a final normalized state is at least the Fubini–Study angle between initial and final states divided by the time-averaged energy variance. It is not merely a lower bound: for trajectories that move along a geodesic in projective Hilbert space the bound becomes an equality, and the paper identifies two exactly saturating examples. This matters because it turns the runtime of idealized imaginary-time algorithms into a geometric quantity and explains, from the geometry alone, why the imaginary-time Grover search reaches the marked state in $O(\log N)$ time instead of the unitary $O(\sqrt{N})$.

What carries the argument

The load-bearing object is the angular cost $\Theta(t)=\arccos(|\langle\psi_0|\phi(t)\rangle|)$, which measures how distinguishable the normalized evolving state is from the initial state. It is paired with the standard Fubini–Study fact that this angle never exceeds the length of any dynamical path in projective Hilbert space. The identity doing the work is $\|\frac{d}{dt}|\phi\rangle\|=\Delta H(t)$, the instantaneous standard deviation of the Hamiltonian, which converts geometry into a time bound; saturation is controlled by the tangent-space alignment condition $P_\perp(-H+\langle H\rangle)|\phi\rangle \propto P_\perp|\psi_0\rangle$.

What would settle it

For a two-level system with $H=E|0\rangle\langle 0|$ and initial state $(\cos\theta,\sin\theta)$, numerically integrate Eq. (22) and compare $T$ with $\Theta(T)/\overline{\Delta H}$ for several $T$ and $\theta$; if any instance gives $T < \Theta/\overline{\Delta H}$ the bound is violated. The paper predicts equality at $\theta=\pi/4$; a direct experimental or numerical measurement of the normalized state's path length on the Bloch sphere would either confirm the geodesic interpretation or show where it fails.

Watch

Extended reading notes

Core claim

The paper's central claim is the inequality (10): for pure-state imaginary-time dynamics $d|\psi\rangle/dt = -H|\psi\rangle$ with normalized state $|\phi(t)\rangle$, any evolution taking the state from $|\psi_0\rangle$ to $|\phi(T)\rangle$ satisfies $T \ge \Theta(T)/\overline{\Delta H}$, where $\Theta(T)=\arccos(|\langle\psi_0|\phi(T)\rangle|)$ is the Fubini–Study angle and $\overline{\Delta H}$ is the time average of the instantaneous variance of $H$. The physical content is that the normalized state's speed in projective Hilbert space is exactly the energy spread $\Delta H(t)$, so the geodesic distance provides a universal lower bound. The bound is tight when the instantaneous tangent-spac

Load-bearing premise

The derivation assumes an ideal pure-state process in which the norm decay is simply divided out and no cost is charged for normalization or postselection; if real implementations must pay that cost, the bound constrains the idealized trajectory, not the full resource count.

Editorial extensions

If this is right

If the bound is correct, no pure-state imaginary-time protocol can reach a distinguishable final state faster than $\Theta(T)/\overline{\Delta H}$; the only way to speed it up is to raise the time-averaged energy spread of the driving Hamiltonian.
Saturation is generic for evolutions confined to a two-dimensional subspace with the tangent velocity aligned toward the initial state, giving a constructive scheduling rule for optimal imaginary-time evolution.
For the two-level Hamiltonian $H=E|0\rangle\langle 0|$ with initial angle $\theta=\pi/4$, the minimal annealing time is exactly $\Theta/\overline{\Delta H}$.
For imaginary-time Grover search, the same saturated bound reproduces $T\simeq (1/g)[\frac{1}{2}\ln N+\ln(1/\varepsilon)]$, so the logarithmic speedup is a direct consequence of the geodesic, constant-speed motion.
The bound is restricted to pure states; extending it to mixed states or open-system imaginary-time evolution is left open by the paper.

Reading between the lines

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

The logarithmic runtime in the Grover example is essentially forced by the gap $g=E_\perp-E_w$: the geodesic speed $g\sin\theta\cos\theta$ integrates to the angle $\theta(0)-\theta(T)$, so the QSL converts an exponential decay of $\tan\theta$ into a linear-in-$t$ decay of $\Theta$; this suggests gap scheduling, not state-specific details, controls imaginary-time search speed.
Because the bound counts only the ideal renormalized trajectory, it should be viewed as a lower bound on the postselected process; any probabilistic or variational implementation that spends resources on normalization, amplitude amplification, or estimating $\Delta H$ will have a larger true runtime, a point the paper notes but does not quantify.
The saturation condition is testable in analog or dissipative settings: prepare a two-level system in the equal superposition, drive with $H=E|0\rangle\langle 0|$, and check that $T\overline{\Delta H}=\Theta(T)$ holds at every intermediate time; deviations would show where the idealized geodesic picture breaks.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: Eq. (10) is the real content, and it is correct. For pure-state ITE with time-independent H, T ≥ Θ(T)/ΔH follows from the standard Fubini–Study length inequality once you look at the normalized state; its speed is exactly the energy variance. The saturation condition is also right, and the conclusion is honest about the pure-state limitation. What is genuinely new is modest: the explicit inequality and the saturation criterion in this ITE setting. The Grover example qualitatively reproduces the known O(log N) scaling, which is already in the literature. That is a useful, readable note, no more.

The soft spots are where the stress-test note lands. Appendix A evaluates arctan(cot θ) as θ, but the correct value is π/2 − θ. So Eq. (23) and the identity in Eq. (24) are wrong except at θ = π/4, the one case used for saturation. That means the closed-form expression for the minimal time is not generally valid as stated. In the Grover analysis, the conversion from target success probability to r is also off: for success 1−ε you need r(T) ≲ √ε, not r(T) ≤ ε, so the ε-dependent term in Eq. (45) is too large by a factor of 2. These are correctable, but they sit in the two quantitative applications, so they matter for the abstract's claim that the bound is "analytically evaluated."

One caveat worth keeping in view: the QSL is for the idealized differential equation and does not account for the postselection or probabilistic cost of implementing ITE. The paper acknowledges this, so it is not a hidden flaw, but readers should not interpret T as an algorithm runtime without that additional resource cost.

Verdict: the central inequality is sound and worth having; the example sections need a correction pass. I would send this to a serious referee, with the requirement that the errors in Appendix A and Eq. (45) be fixed. After that it works as a short letter.

Referee Report

3 major / 3 minor

Summary. The paper derives a geometric quantum speed limit (QSL) for imaginary-time evolution of pure states under d|ψ>/dt = -H(t)|ψ>. The main result is Eq. (10): T ≥ Θ(T)/∆H, where Θ(T) is the Fubini–Study angle between the initial state and the normalized evolving state, and ∆H is the time-averaged energy dispersion. A saturation condition is given in Section II.B, and two applications are presented: a two-level system (Section III.A) and an imaginary-time Grover search (Section III.B), claiming a closed-form minimal time and T = O(log N) scaling. The central inequality is correctly derived, but the quantitative evaluation of the examples contains algebraic errors, in particular in Appendix A and in the success-probability condition in the Grover example.

Significance. If corrected, the main result is a clean and potentially useful geometric QSL for ITE. The derivation is parameter-free, and the saturation analysis is constructive. The novelty relative to the standard Mandelstam–Tamm bound is modest—the bound is essentially the MT bound applied to the normalized state—but the ITE context and the explicit saturation examples give the paper value. The Grover application qualitatively reproduces the expected O(log N) runtime. However, the paper's quantitative claims are currently not reliable because of the algebra errors detailed below; these are fixable, so the underlying approach appears sound.

major comments (3)

[Appendix A, Eq. (23)] The evaluation of the integral is incorrect. For 0<θ<π/2, arctan(cot θ) = π/2 − θ, not θ. The correct result is ∫_0^T ∆H(t)dt = π/2 − θ − arctan(e^{-ET} cot θ). The printed Eq. (23), θ − arctan(e^{-ET} cot θ), is nonzero at T=0 (it equals 2θ − π/2) and does not equal Θ(T) except when θ=π/4. Thus the closed-form 'minimal time' expression in Section III.A is not established for general θ. Interestingly, the corrected integral shows that the two-level example actually saturates the QSL for all θ, not only at θ=π/4; the manuscript's formula obscures this.
[Eq. (24) and surrounding text] The trigonometric identity in Eq. (24) is a valid angle-addition identity, but it is not equal to the overlap ⟨ψ0|ϕ(T)⟩ of Eq. (21) for general θ. For example, as T→∞, the right-hand side of Eq. (24) tends to cos θ, whereas the true overlap tends to sin θ. The jump from Eq. (24) to Eq. (25) is therefore only justified at θ=π/4. This is a direct consequence of the mistaken branch in Eq. (23). The text should be rewritten to use the correct antiderivative and to state explicitly that the saturation is generic for this two-level model.
[Eq. (45), Grover example] The success-probability condition is wrong. Achieving success probability 1−ε means cos²θ(T) ≥ 1−ε, so the failure probability is sin²θ(T) = r²/(1+r²) ≤ ε, which gives r(T) ≤ √(ε/(1−ε)) ≈ √ε, not r(T) ≤ ε. The correct runtime bound is T ≥ (1/g)[(1/2)ln(N−1) + (1/2)ln((1−ε)/ε)] ≃ (1/g)[(1/2)ln N + (1/2)ln(1/ε)]. The stated expression is too large by ln(1/ε)/g and is not a valid lower bound for the stated success probability: a time with success probability exactly 1−ε can violate r(T) ≤ ε. The O(log N) scaling survives, but the claimed quantitative reproduction of the runtime is incorrect.

minor comments (3)

[Eq. (28)] The normalized state for θ=π/4 is written as (|0⟩ + e^{-Et}|1⟩)/√(1+e^{-2Et}), but the correct expression is (e^{-Et}|0⟩ + |1⟩)/√(1+e^{-2Et}). The coefficients are swapped; the sentence claiming the trajectory goes toward |1⟩ is inconsistent with the printed formula.
[After Eq. (16)] The phrase 'approaching the target state corresponds to dΘ/dt < 0' is confusing. In the two-level and Grover examples the target ground state is not |ψ0⟩, and Θ(t) increases during the evolution. Please clarify the sign convention.
[General presentation] There are several typos: 'achived' should be 'achieved'; reference [20] contains 'Mnnich'; the intro has a stray '?' in '[5 ? –7]'; Eq. (26) contains a missing equation reference '(??)'. Equation (24) is also typeset in a way that loses the fraction, making it hard to read.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Eq. (10) is a self-contained geometric inequality; the examples are exact evaluations, not fits.

full rationale

The central derivation (Eqs. (1)-(10)) is not circular. The paper defines the normalized state |phi(t)> in Eq. (2), computes its instantaneous speed as the energy dispersion Delta H(t) in Eq. (9), and applies the standard Fubini-Study inequality that the geodesic angle cannot exceed the path length (Eqs. (5)-(6)). This is a rigorous mathematical chain with no fitted parameters, no data, and no calibration; Theta(T) is the angle to the actual solution, so the inequality is a genuine geometric bound, not an identity imposed by definition. The saturation condition (Eqs. (12)-(13)) is derived from Cauchy-Schwarz, and the examples are exact solutions of the Schrodinger equation, so the reported tightness is a demonstration, not a restatement of inputs. The only self-citation, Ref. [16], is background and not load-bearing. The two-level example contains an algebraic error in Appendix A (the identity arctan(cot theta) = theta should have an extra pi/2 - theta term), and the Grover condition in Eq. (45) uses r(T) <= eps instead of r(T) <= sqrt(eps/(1-eps)); these are correctness issues in the application formulas, but they do not make the derivation circular.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

No new physical entities, mediators, forces, or dimensions are introduced. The QSL is purely an inequality on the normalized state of an existing model. The only special choice listed is the initial angle used to engineer saturation in the two-level example.

free parameters (1)

initial angle theta in two-level saturation example = pi/4 (chosen)
The claimed saturation identity in Section III A only holds for theta = pi/4; for other values Eq. (24) is false. This special choice is made to demonstrate tightness and is not a fitted parameter in the main bound.

assumptions (4)

standard math Fubini-Study geodesic distance is no greater than the length of any path between two pure states (Eq. (6))
Invoked to go from trajectory length to Theta(T) <= L; standard projective-space geometry.
standard math Cauchy-Schwarz inequality applied to tangent vectors P_perp(-H+<H>)|phi> and P_perp|psi0> (Eq. (16))
Used to derive the saturation condition and the bound magnitude.
domain assumption The imaginary-time Schrodinger equation d|psi>/dt = -H|psi> is the correct model for ITE
Adopted at the start of Section II; all runtime bounds are relative to this continuous-time idealization.
domain assumption Initial state is pure and the evolving state is renormalized at every instant (Eq. (2))
The bound applies only to pure states; the paper notes mixed-state generalization is open.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Saturable Quantum Speed Limits for Imaginary-Time Evolution." pith.science (2026). https://pith.science/paper/5MAH4IVP

@misc{pith2026250810361,
  author       = {Pith},
  title        = {Pith review of: Saturable Quantum Speed Limits for Imaginary-Time Evolution},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5MAH4IVP}},
  note         = {Machine review of arXiv:2508.10361}
}

abstract

We derive a Geometric quantum speed limit (QSL) for imaginary-time evolution, where the dynamics is governed by a non-unitary Schr\"{o}dinger equation. By introducing a cost function based on the angular distance between the normalized evolving state and the initial state, we obtain a lower bound on the evolution time expressed as the ratio between this angle and the time-averaged energy dispersion. Our bound is analytical, general, and applicable to arbitrary time-independent Hamiltonians. We analytically evaluate this bound for two physically motivated cases. First, we apply it to a two-level system and derive an expression for the minimal time. Second, we analyze the imaginary-time version of Grover search problem and rigorously reproduce the well-known logarithmic scaling $T=\mathcal{ O}(\log N)$ within our QSL framework.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

29 extracted references · 28 canonical work pages

[1]

Deffner and E

S. Deffner and E. Lutz, Quantum Speed Limit for Non- Markovian Dynamics, Phys. Rev. Lett. 111, 010402 (2013)

work page 2013
[2]

Caneva, M

T. Caneva, M. Murphy, T. Calarco, R. Fazio, S. Mon- tangero, V. Giovannetti, and G. E. Santoro, Optimal Control at the Quantum Speed Limit, Phys. Rev. Lett. 103, 240501 (2009)

work page 2009
[3]

G. C. Hegerfeldt, Driving at the Quantum Speed Limit: Optimal Control of a Two-Level System, Phys. Rev. Lett. 111, 260501 (2013)

work page 2013
[4]

Brouzos, A

I. Brouzos, A. I. Streltsov, A. Negretti, R. S. Said, T. Caneva, S. Montangero, T. and Calarco, Quantum speed limit and optimal control of many-boson dynamics, Phys. Rev. A 92, 062110 (2015)

work page 2015
[5]

M. M. Taddei, B. M. Escher, L. Davidovich, and R. L. de Matos Filho, Quantum speed limit for physical processes, Phys. Rev. Lett. 110, 050402 (2013)

work page 2013
[6]

Gessner and A

M. Gessner and A. Smerzi, Statistical speed of quan- tum states: Generalized quantum Fisher information and Schatten speed, Phys. Rev. A 97, 022109 (2018)

work page 2018
[7]

Zwierz, C

M. Zwierz, C. A. Perez-Delgado, and P. Kok, General op- timality of the Heisenberg limit for quantum metrology, Phys. Rev. Lett. 105 180402 (2011)

work page 2011
[8]

Lloyd, Ultimate physical limits to computation, Na- ture, 406, 1047–1054 (2000)

S. Lloyd, Ultimate physical limits to computation, Na- ture, 406, 1047–1054 (2000)

work page 2000

Show all 29 references

[9]

Suzuki and K

K. Suzuki and K. Takahashi, Performance evaluation of adiabatic quantum computation via quantum speed limits and possible applications to many-body systems, Phys. Rev. Res. 2, 032016 (2019)

2019
[10]

Mandelstam and I

L. Mandelstam and I. Tamm, The uncertainty relation between energy and time in non-relativistic quantum me- chanics, Izv. Akad. Nauk SSSR 9, 122 (1945) [J. Phys. (U.S.S.R.) 9 249 (1945)]

1945
[11]

Margolus and L

N. Margolus and L. B. Levitin, The maximum speed of dynamical evolution, Physica D 120, 188 (1998)

1998
[12]

del Campo, I

A. del Campo, I. L. Egusquiza, M. B. Plenio, and S. F. Huelga, Quantum speed limits in open system dynamics, Phys. Rev. Lett. 110, 050403 (2013)

2013
[13]

Y. J. Zhang, W. Han, Y. J. Xia, J. P. Cao, and H. Fan, Quantum speed limit for arbitrary initial states, Sci. Rep. 4, 4890 (2014)

2014
[14]

D. P. Pires, M. Cianciaruso, L. C. C’eleri, G. Adesso, and D. O. Soares-Pinto, Generalized geometric quantum speed limits, Phys. Rev. X 6, 021031 (2016)

2016
[15]

Deffner, Geometric quantum speed limits: a case for Wigner phase space, New J

S. Deffner, Geometric quantum speed limits: a case for Wigner phase space, New J. Phys. 19, 103018 (2017)

2017
[16]

Kobayashi and N

K. Kobayashi and N. Yamamoto, Quantum speed limit for robust state characterization and engineering, Phys. Rev. A 102, 042606 (2020)

2020
[17]

L. P. Garc ´ ıa-Pintos and A. del Campo, Quantum speed limits under continuous quantum measurements, New J. Phys. 21, 033012 (2019)

2019
[18]

Hou L, Shao B and Wang C Quantum speed limit under the influence of measurement-based feedback control, Int. J. Theor. Phys. 62, 47 (2023)

2023
[19]

McArdle, T

S. McArdle, T. Jones, E. Suguru, L. Ying, S. Benjamin, and Y. Xiao, Variational ansatz-based quantum simula- tion of imaginary time evolution, npj Quantum Inf. 5, 1–6 (2019)

2019
[20]

Motta, C

M. Motta, C. Sun, A. T. K. Tan, M. J. O’Rourke, E. Ye, 7 A. J. Mnnich, F. G. S. L. Brandao, and K.-L. Chan, De- termining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution, Nat. Phys. 16, 205–210 (2020)

2020
[21]

Yeter-Aydeniz, R

K. Yeter-Aydeniz, R. C. Pooser, and G. Siopsis, Practi- cal quantum computation of chemical and nuclear energy levels using quantum imaginary time evolution and Lanc- zos algorithms, arXiv:1912.06226v1 (2019)

1912 arXiv
[22]

Kosugi, Y

T. Kosugi, Y. Nishiya, H. Nishi, and Y. Matsushita, Probabilistic imaginary-time evolution by using forward and backward real-time evolution with a single an- cilla: first-quantized eigensolver of quantum chemistry for ground states, Phys. Rev. Res. 4, 033121 (2022)

2022
[23]

Xie, S.-J

H.-N. Xie, S.-J. Wei, F. Yang, Z.-A. Wang, C.-T. Chen, H. Fan, and G.-L. Long, A probabilistic imaginary time evolution algorithm based on non-unitary quantum cir- cuit, arXiv:2210.05293 (2022)

2022 arXiv
[24]

Nishi, T

H. Nishi, T. Kosugi, Y. Nishiya, and Y. Matsushita, Quadratic acceleration of multi-step probabilistic algo- rithms for state preparation Phys. Rev. Res. 6, L022041 (2024)

2024
[25]

Nishi, K

H. Nishi, K. Hamada, Y. Nishiya, T. Kosugi, and Y. Mat- sushita, Optimal scheduling in probabilistic imaginary- time evolution on a quantum computer, Phys. Rev. Res. 5, 043048 (2023)

2023
[26]

H. H. S. Chan, D. M. Ramo, and N. Fitzpatrick, Simu- lating non-unitary dynamics using quantum signal pro- cessing with unitary block encoding, arXiv:2303.06161 (2023)

2023 arXiv
[27]

T. L. Silva, M. M. Taddei, S. Carrazza, and L. Aolita, Fragmented imaginary-time evolution for early-stage quantum signal processors, Sci. Rep. 13, 18258 (2023)

2023
[28]

Okada, M

S. Okada, M. Ohzeki, and K. Tanaka, Difference be- tween quantum annealing by imaginary-time and real- time Schr¨ odinger equation of Grover’s search, J. Phys. Soc. Jpn. 88, 024803 (2019)

2019
[29]

Uhlmann and B

A. Uhlmann and B. Crell, Geometry of state spaces, in Entanglement and Decoherence, edited by A. Buch- leitner, C. Viviescas, and M. Tiersch, Lecture Notes in Physics Vol. 768 (Springer, Berlin, Heidelberg, 2009), pp. 1-60

2009

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

Deffner and E

S. Deffner and E. Lutz, Quantum Speed Limit for Non- Markovian Dynamics, Phys. Rev. Lett. 111, 010402 (2013)

work page 2013

[2] [2]

Caneva, M

T. Caneva, M. Murphy, T. Calarco, R. Fazio, S. Mon- tangero, V. Giovannetti, and G. E. Santoro, Optimal Control at the Quantum Speed Limit, Phys. Rev. Lett. 103, 240501 (2009)

work page 2009

[3] [3]

G. C. Hegerfeldt, Driving at the Quantum Speed Limit: Optimal Control of a Two-Level System, Phys. Rev. Lett. 111, 260501 (2013)

work page 2013

[4] [4]

Brouzos, A

I. Brouzos, A. I. Streltsov, A. Negretti, R. S. Said, T. Caneva, S. Montangero, T. and Calarco, Quantum speed limit and optimal control of many-boson dynamics, Phys. Rev. A 92, 062110 (2015)

work page 2015

[5] [5]

M. M. Taddei, B. M. Escher, L. Davidovich, and R. L. de Matos Filho, Quantum speed limit for physical processes, Phys. Rev. Lett. 110, 050402 (2013)

work page 2013

[6] [6]

Gessner and A

M. Gessner and A. Smerzi, Statistical speed of quan- tum states: Generalized quantum Fisher information and Schatten speed, Phys. Rev. A 97, 022109 (2018)

work page 2018

[7] [7]

Zwierz, C

M. Zwierz, C. A. Perez-Delgado, and P. Kok, General op- timality of the Heisenberg limit for quantum metrology, Phys. Rev. Lett. 105 180402 (2011)

work page 2011

[8] [8]

Lloyd, Ultimate physical limits to computation, Na- ture, 406, 1047–1054 (2000)

S. Lloyd, Ultimate physical limits to computation, Na- ture, 406, 1047–1054 (2000)

work page 2000