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Optimal Control at the Quantum Speed Limit
Open accessAbstract
Optimal control theory is a promising candidate for a drastic improvement of the performance of quantum information tasks. We explore its ultimate limit in paradigmatic cases, and demonstrate that it coincides with the maximum speed limit allowed by quantum evolution.
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TL;DR
The paper shows that Krotov quantum optimal control can reach the quantum speed limit: infidelity converges to zero only when the fixed total time \(T\) exceeds a critical \(T_{\mathrm{QSL}}\), and this threshold agrees with dynamical bounds in Landau-Zener and spin-chain state-transfer models. It suggests optimal control can operationally characterize quantum speed limits in complex systems.
Key findings
- In the Landau-Zener model, Krotov infidelity does not converge to zero for \(T < T_{\mathrm{QSL}} \approx 1.5688\) (units of \(\hbar/\omega\)), while for \(T\) above this value it decreases exponentially; the second-derivative criterion crosses zero near \(T \approx 1.58\), matching the time-independent Bhattacharyya estimate with no fitting parameters.
- For Heisenberg spin-chain state transfer, infidelity converges to zero only above a critical time; the numerical \(T^*_{\mathrm{QSL}}\) scales linearly with chain length \(N\) (intercept \(b \sim 1.53\)), and for \(N=101\), target infidelity \(I^* = 5 \times 10^{-5}\) at \(n=10^5\) iterations gives \(T=56.66 \approx T^*_{\mathrm{QSL}}\) versus \(T=68.97\).
- Spin-chain optimal control improves the transfer time over the adiabatic approach of Ref. [18] by up to two orders of magnitude and is faster than Ref. [19] for Ising coupling, using only single-site local pulses whereas Ref. [19] used global pulses.
- The numerical spin-chain \(T^*_{\mathrm{QSL}}\) exceeds the analytical QSL estimate by a factor \(\eta \approx 3\), attributed to difficulty in formulating the QSL for the many-body problem; both estimators scale linearly with \(N\).
Methods
- Uses the Krotov recursive optimal-control algorithm to minimize infidelity \(I = 1 - |\langle \psi(T)|\psi_G\rangle|^2\) for fixed duration \(T\), starting from initial guess control pulses and analyzing convergence as a function of iteration number \(n\) and \(T\).
- Applies the approach to two models: a Landau-Zener Hamiltonian with control \(\Gamma(t)\), preparing the instantaneous ground state and targeting the ground state with \(\Gamma(T)=-\Gamma(0)\); and a one-dimensional Heisenberg spin chain with parabolic magnetic potential controls \(C(t)\) and \(d(t)\), transferring \(|0\rangle\) to \(|N-1\rangle\) in the single-spin-up sector.
- Compares numerical \(T_{\mathrm{QSL}}\) estimates with dynamical bounds: the Bhattacharyya bound for the Landau-Zener case, and time-averaged energy-spread bounds \(\Delta E_1\), \(\Delta E_2\) for the spin chain, with \(T_{\mathrm{QSL}}/(N-1) \ge \max\{\pi/(2\Delta E_1); \pi/(2\Delta E_2)\}\).
Limitations
- The paper states that for the spin-chain model the sign of the second derivative of infidelity cannot be used as a reliable signature of \(T_{\mathrm{QSL}}\), so the numerical estimate uses the smallest \(T\) achieving a target infidelity threshold \(I^*\) in a fixed number of iterations; the result does not depend significantly on \(I^*\) and \(n\).
- The paper attributes the factor \(\eta \approx 3\) discrepancy between numerical and analytical spin-chain QSL estimates to the difficulty in formulating the quantum speed limit for the many-body problem.
- Evident from the text: the findings are demonstrated in two paradigmatic models with Krotov optimal control and fixed \(T\), not as a general proof for arbitrary systems or control constraints; the Landau-Zener analytical estimate also assumes the dynamics can be effectively described by a time-independent Hamiltonian for most of the evolution.
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“The energy spread is almost constant save near the final time, where large oscillations are present, corresponding to the deceleration of the spinwave.”
Strength:plain· not yet calibratedCausal:correlational· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesA result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The average energy spreading ∆ E 2 ( t ) is larger for smaller time transfer T allowing for a higher average excitation velocity.”
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In the paper
“On the contrary, by progressively increasing T towards and above T QSL , the curvature changes sign and the infidelity in the large iteration limit decreases exponentially , as confirmed by the fit in Fig. 1.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIn the paper
“Remarkably, the peculiar feature of the initial guess Γ gs ( t ) of being almost constantly zero for most of the central part of the evolution is preserved by the recursive optimization of OC [2]”
What the sources say
- Not checked yetcites [2] V. F. Krotov, Global Methods in Optimal Control Theory (Marcel Dekker, New York, 1996).
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In the paper
“In particular, as shown in Fig. 3 (left) the infidelity appears to converge to zero only for values of T that are above a certain critical time T QSL .”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesThe paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The connection between OC and the QSL emerges at the time duration T < T QSL for which OC fails to converge.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes? Based: unclearIn the paper
“Indeed, we achieved an improvement of the transfer time with respect to Ref. [18] of up to two orders of magnitude.”
What the sources say
- Not checked yetcites [18] V. Balachandran and J. Gong, Phys. Rev. A 77 , 012303 (2008). source
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In the paper
“Finally, in Fig. 2 we show a comparison between the estimate T QSL through the second derivative of the infidelity and the theoretical time-independent estimate (2) for various Γ(0) /ω ratios. We stress that here we have no fitting parameters.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- The authors state this as an assumption: For the Landau-Zener model, the dynamics for most of the evolution can be approximated by the time-independent Hamiltonian H0 = H[Γ=0].
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In the paper
“Consequently, for the present model the sign of the second derivative cannot be used as a reliable signature of T QSL .”
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIn the paper
“It is also worth mentioning that we achieved a transfer time faster than that obtained (for Ising coupling) in Ref. [19].”
What the sources say
- Not checked yetcites [19] N. Khaneja and S. J. Glaser, Phys. Rev. A 66 , 060301(R) (2002). source
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In the paper
“The results are reported in Fig.4 (right) where the two quantities are compared: both estimators are linearly dependent on N , but the numerical results show an improvement over the theoretical prediction by a factor of η ≈ 3”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
In the paper
“Interestingly enough, however, our method only uses single-site local pulses while in Ref. [19] this was achieved using global pulses that operate jointly on the whole chain.”
What the sources say
- Not checked yetcites [19] N. Khaneja and S. J. Glaser, Phys. Rev. A 66 , 060301(R) (2002). source
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In the paper
“Nonetheless, a numerical estimate T ∗ QSL for such quantity can been obtained by considering the smallest time T which allows us to achieve the target infidelity threshold I ∗ in a fixed number of algorithm iterations n (the result does not depend significantly on the value of I ∗ and n ).”
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In the paper
“The picture that arises is that OC finds a solution x opt ( t ) that initially accelerates the spin excitation, and then transfers it with constant velocity up to the end of the chain, where deceleration occurs.”
Strength:plain· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesIt may state two separate claims as one.
The authors state this as an assumption.
For the Landau-Zener model, the dynamics for most of the evolution can be approximated by the time-independent Hamiltonian H0 = H[Γ=0].
“assuming H 0 = H [Γ = 0] as the Hamiltonian.”
“for most of the evolution time the dynamics can be effectively described by a time-independent Hamiltonian”
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
In the citing paper
“the shortest time-scale for quantum optimal control algorithms to converge [28]”
From Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control (2017) · cites this paper as [28]
In the citing paper
“Therefore, Caneva et al showed that using the Krotov algorithm the minimal elapsed time one can consider corresponds exactly to τ QSL .”
From Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control (2017) · cites this paper as [28]
In the citing paper
“Remarkably they found that there was a minimum value of τ given by Battacharyya's bound [51] (i.e. the Mandelstam-Tamm bound for arbitrary angles), below which the algorithm failed to converge, while for values above this they consistently found F → 1.”
From Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control (2017) · cites this paper as [28]
In the citing paper
“In a remarkable work, Caneva et al [28] showed explicitly the connection between the quantum speed limit and optimal control.”
From Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control (2017) · cites this paper as [28]
In the citing paper
“The relation and importance of this first became evident in the seminal work of Caneva et al [28] in the context of optimal control.”
From Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control (2017) · cites this paper as [28]
In the citing paper
“if we restrict |ψ 0 and |ψ T to be the respective ground states of the Landau-Zener Hamiltonian, equation ( 57), at the start and end of the protocol, Γ(0) = -γ and Γ(τ ) = γ , then τ min = τ QSL , precisely in line with [28,109].”
From Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control (2017) · cites this paper as [28]
In the citing paper
“the quantum speed limit time that emerged as the fundamental bound from Caneva et al [28], from equation (58), actually assumes that the system's Hamiltonian is time-independent.”
From Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control (2017) · cites this paper as [28]
In the citing paper
“Interestingly, Čencov's theorem states that the Fisher information metric is the only Riemannian metric on the set of probability distributions that is contractive under stochastic maps [47]”
From Generalized Geometric Quantum Speed Limits (2016) · cites this paper as [47]
In the citing paper
“They showed that the Krotov algorithm of quantum optimal control theory fails to converge when the duration of the process is chosen below the quantum speed limit time.”
From Energy–time uncertainty relation for driven quantum systems (2013) · cites this paper as [28]
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Metadata
- Authors
- Caneva, T., Murphy, M., Calarco, T., Fazio, R., Montangero, S., Giovannetti, V., & Santoro, G. E.
- Affiliations
- Scuola Internazionale Superiore di Studi Avanzati (IT)
- Universität Ulm (DE)
- Scuola Normale Superiore (IT)
- The Abdus Salam International Centre for Theoretical Physics (ICTP) (IT)
- Venue
- Physical Review Letters
- Publisher
- American Physical Society (APS)
- Published
- 2009-12-07
- Volume
- 103
- Issue
- 24
- Pages
- 240501-240501
- ISSN
- 0031-9007 (ISSN-L)
- 1079-7114
- OpenAlex
- W2160409148
- Semantic Scholar
- f1a2f8bcbbcd647e99b6a5a50dc08c24ff04e172
- Cited by
- 509
- References
- 25
- Citations by year
- 2026: 22 · 2025: 41 · 2024: 47 · 2023: 41 · 2022: 46 · 2021: 32 · 2020: 36 · 2019: 32 · 2018: 26 · 2017: 29 · 2016: 35 · 2015: 34 · 2014: 29 · 2013: 18 · 2012: 15
- FWCI
- 14.80
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review Letters (publishedVersion)
- arXiv (Cornell University) (publishedVersion)
- PubMed (publishedVersion)
- SDL (submittedVersion)
- Archivio istituzionale della ricerca (Alma Mater Studiorum Università di Bologna) (submittedVersion)
- Scuola Normale Superiore di Pisa (submittedVersion, cc-by)
- Padua Research Archive (University of Padova) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Information and Cryptography
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
- Keywords
- optimal control, quantum speed limit, quantum control, quantum information processing, quantum dynamics
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