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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- Findingown data§ Optimal Control at the Quantum Speed Limit · p. 4
For the spin-chain model, the energy spread ∆E2(t) is almost constant except near the final time, where large oscillations occur corresponding to deceleration of the spin wave.
“The energy spread is almost constant save near the final time, where large oscillations are present, corresponding to the deceleration of the spinwave.”
Quote in the paperNot a substantial claim - Findingown data§ Optimal Control at the Quantum Speed Limit · p. 4
For the spin-chain model, the average energy spreading ∆E2(t) is larger for smaller transfer time T, allowing for a higher average excitation velocity.
“The average energy spreading ∆ E 2 ( t ) is larger for smaller time transfer T allowing for a higher average excitation velocity.”
Quote in the paperSubstantial claim - Findingown data§ Optimal Control at the Quantum Speed Limit · p. 2
For the Landau-Zener model, when T is at or above T_QSL, the infidelity decreases exponentially with the number of Krotov iterations.
“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.”
Quote in the paperSubstantial claim - Findingcitation§ Optimal Control at the Quantum Speed Limit · p. 2
For the Landau-Zener model, the initial guess pulse Γ_gs(t) being almost constantly zero for most of the central part of the evolution is preserved by the recursive optimization of optimal control.
“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]”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [2] V. F. Krotov, Global Methods in Optimal Control Theory (Marcel Dekker, New York, 1996).
Not checked in the free run.
- Findingown data§ Optimal Control at the Quantum Speed Limit · p. 3
For the spin-chain model, the infidelity converges to zero only for transfer times T above a critical time T_QSL.
“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 .”
Quote in the paperSubstantial claim
- Findingnone given§ Optimal Control at the Quantum Speed Limit · p. 1
The connection between optimal control and the quantum speed limit emerges at the time duration T < T_QSL for which optimal control fails to converge.
“The connection between OC and the QSL emerges at the time duration T < T QSL for which OC fails to converge.”
Quote in the paperNot a substantial claim - Findingown data§ Optimal Control at the Quantum Speed Limit · p. 4
The optimal control method achieved an improvement of the spin-chain transfer time with respect to Ref. [18] of up to two orders of magnitude.
“Indeed, we achieved an improvement of the transfer time with respect to Ref. [18] of up to two orders of magnitude.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [18] V. Balachandran and J. Gong, Phys. Rev. A 77 , 012303 (2008). source
Not checked in the free run.
- Findingown data§ Optimal Control at the Quantum Speed Limit · p. 2
For the Landau-Zener model, the QSL time T_QSL estimated from the second derivative of the infidelity agrees with the theoretical time-independent estimate from the Bhattacharyya bound, with no fitting parameters.
“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.”
Quote in the paperSubstantial claim- 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].
- Findingown data§ Optimal Control at the Quantum Speed Limit · p. 3
For the spin-chain model, the sign of the second derivative of the infidelity cannot be used as a reliable signature of T_QSL.
“Consequently, for the present model the sign of the second derivative cannot be used as a reliable signature of T QSL .”
Quote in the paperSubstantial claim - Findingcitation§ Optimal Control at the Quantum Speed Limit · p. 4
The optimal control method achieved a transfer time faster than that obtained for Ising coupling in Ref. [19].
“It is also worth mentioning that we achieved a transfer time faster than that obtained (for Ising coupling) in Ref. [19].”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [19] N. Khaneja and S. J. Glaser, Phys. Rev. A 66 , 060301(R) (2002). source
Not checked in the free run.
- Findingown data§ Optimal Control at the Quantum Speed Limit · p. 3
For the spin-chain model, apart from small N where boundary effects are more pronounced, the estimated T*_QSL has a linear dependence on chain length N.
“Apart from the case of small N , where boundary effects are more pronounced, the resulting T ∗ QSL appears to have a linear dependence on the chain length N - see Fig. 3 (right).”
Quote in the paperSubstantial claim - Findingown data§ Optimal Control at the Quantum Speed Limit · p. 1
The method is able to produce infidelities arbitrarily close to zero only above a certain threshold T_QSL.
“We then analyze the performance of the process as a function of T and show that the method is able to produce infidelities arbitrarily close to zero only above a certain threshold T QSL , which we compare with the dynamical bounds that affect the system.”
Quote in the paperSubstantial claim - Interpretationown data§ Optimal Control at the Quantum Speed Limit · p. 2
For the Landau-Zener model, the efficiency of optimal control is ultimately set by the dynamical bound of Eq. (2).
“The excellent agreement shows that the OC efficiency is ultimately set by the dynamical bound of Eq. (2).”
Quote in the paperSubstantial claim- 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].
- Interpretationreasoning§ Optimal Control at the Quantum Speed Limit · p. 2
For the Landau-Zener model, the QSL time T_QSL can be estimated by a time-independent formula using H0 = H[Γ=0] as the Hamiltonian.
“suggesting that for this simple model an estimation of a finite resource QSL bound T QSL for T can be deduced by a time-independent formula, assuming H 0 = H [Γ = 0] as the Hamiltonian.”
Quote in the paperSubstantial claim- 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].
- Interpretationown data§ Optimal Control at the Quantum Speed Limit · p. 1
The effectiveness of the Krotov algorithm for quantum optimal control is limited by the fundamental dynamical bounds on the maximum speed at which a quantum system can evolve (the quantum speed limit).
“We explore its ultimate limit in paradigmatic cases, and demonstrate that it coincides with the maximum speed limit allowed by quantum evolution.”
- Also stated in: § Optimal Control at the Quantum Speed Limit · p. 1 · plain
“In this work we will show that the effectiveness of the Krotov algorithm for quantum OC is related to fundamental bounds that affect the maximum speed at which a quantum system can evolve in its Hilbert space.”
- Also stated in: § Optimal Control at the Quantum Speed Limit · p. 1 · plain
“We found a good agreement between these (in principle) independent quantities, meaning that the effectiveness of our control pulses is only limited by the dynamical bounds of the system.”
Quote in the paperSubstantial claim - Also stated in: § Optimal Control at the Quantum Speed Limit · p. 1 · plain
- Interpretationown data§ Optimal Control at the Quantum Speed Limit · p. 4
For the spin-chain model, the agreement in scaling of the numerical and theoretical QSL estimates shows that even with additional constraints, optimal control reaches the ultimate dynamical bounds set by quantum mechanics.
“As before, the agreement of scaling of the two results shows that even in the presence of additional constraints the OC reaches the ultimate dynamical bounds set by quantum mechanics.”
Quote in the paperSubstantial claim - Interpretationown data§ Optimal Control at the Quantum Speed Limit · p. 4
There are fundamental constraints governing the efficiency of the Krotov quantum optimal control algorithm, dictated by the maximum speed at which a quantum state can evolve in time.
“In summary, we have demonstrated that there are fundamental constraints governing the efficiency of Krotov quantum OC algorithm dictated by the maximum speed at which a quantum state can evolve in time.”
Quote in the paperSubstantial claim - Premisecitation§ Optimal Control at the Quantum Speed Limit · p. 4
For the spin-chain model, the bound on the minimum transfer time is T_QSL/(N-1) ≥ max{π/(2∆E1); π/(2∆E2)}, which must be satisfied by any H(t) that brings the initial state to the target state in time T.
“The bound on the minimum transfer time is then given by T QSL / ( N -1) ≥ max { π 2∆ E 1 ; π 2∆ E 2 } , which needs to be satisfied by any H ( t ) that brings | ψ (0) 〉 to the target state | ψ G 〉 in a time T .”
Quote in the paperSubstantial claim0 of 2 sources checked - Interpretationnone given§ Optimal Control at the Quantum Speed Limit · p. 4
The improvement of the numerical T*_QSL over the theoretical QSL estimate by a factor of η ≈ 3 is attributed to the difficulty in formulating the quantum speed limit for the many-body problem.
“which we attribute to the difficulty in formulating the quantum speed limit for our many-body problem.”
Quote in the paperSubstantial claim
- May combine two claims. 4
May combine two claims.
Findingown data§ Optimal Control at the Quantum Speed Limit · p. 4For the spin-chain model, both the numerical T*_QSL and the analytical QSL estimate are linearly dependent on N, but the numerical results show an improvement over the theoretical prediction by a factor of η ≈ 3.
“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”
Quote in the paperSubstantial claimMay combine two claims.
Findingcitation§ Optimal Control at the Quantum Speed Limit · p. 4The optimal control method uses only single-site local pulses, whereas the method in Ref. [19] used global pulses operating jointly on the whole chain.
“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.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [19] N. Khaneja and S. J. Glaser, Phys. Rev. A 66 , 060301(R) (2002). source
Not checked in the free run.
May combine two claims.
Study design and conductnone given§ Optimal Control at the Quantum Speed Limit · p. 3For the spin-chain model, a numerical estimate T*_QSL can be obtained as the smallest time T that achieves the target infidelity threshold I* in a fixed number of iterations n, and the result does not depend significantly on I* and n.
“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 ).”
Quote in the paperSubstantial claimMay combine two claims.
Interpretationown data§ Optimal Control at the Quantum Speed Limit · p. 4Optimal control finds a solution x_opt(t) that initially accelerates the spin excitation, then transfers it with constant velocity up to the end of the chain, where deceleration occurs.
“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.”
Quote in the paperNot a substantial claim
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?
The maximum bound gives the shortest time scale for convergence of quantum optimal control algorithms.
“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]
Caneva et al showed that the minimal elapsed time in the Krotov algorithm corresponds exactly to the quantum speed limit time.
“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 Landau-Zener model optimized with the Krotov algorithm, there is a minimum elapsed time given by Bhattacharyya's bound below which the algorithm fails to converge, while above it the fidelity approaches one.
“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]
Caneva et al explicitly showed the connection between the quantum speed limit and optimal control.
“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]
The connection between the quantum speed limit and optimal control first became evident in the work of Caneva et al.
“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]
If the initial and final states are the respective ground states of the Landau-Zener Hamiltonian at the start and end of the protocol, then the minimal time equals the quantum speed limit time.
“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]
The quantum speed limit time from Caneva et al's equation assumes that the system's Hamiltonian is time-independent.
“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]
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.
“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]
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.
“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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Quantum Speed Limits across the Quantum-to-Classical Transition
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Introduction to quantum optimal control for quantum sensing with nitrogen-vacancy centers in diamond
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Self-citationQuantum Speed Limit is Not Quantum
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Communication at the quantum speed limit along a spin chain
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Self-citationInformation Theoretical Analysis of Quantum Optimal Control
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“(4) equals zero and the coefficient 2 in Xt should be omitted.”
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Scalable quantum computation via local control of only two qubits
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Self-citationSpeeding up and slowing down the relaxation of a qubit by optimal control
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Self-citationClassical-driving-assisted quantum speed-up
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Tight, robust, and feasible quantum speed limits for open dynamics
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“Such bounds are typically applied to estimate the speed of computational gates [5, 6], the precision in quantum metrology [7–11], the performance in quantum optimal control [4, 12–19], and the charging power in quantum thermodynamics [20, 21].”
BackgroundSpeed limits for quantum gates in multiqubit systems
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Error budgeting for a controlled-phase gate with strontium-88 Rydberg atoms
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Self-citationOptimal control methods for quantum batteries
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Unifying Quantum and Classical Speed Limits on Observables
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Quantum Speed Limits for Leakage and Decoherence
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Optimizing entangling quantum gates for physical systems
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Speeding up critical system dynamics through optimized evolution
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Charting the circuit QED design landscape using optimal control theory
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Optimal control of a qubit in an optical cavity
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Charting the circuit QED design landscape using optimal control theory
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Automatic spin-chain learning to explore the quantum speed limit
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Zermelo navigation and a speed limit to quantum information processing
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Zeno Dynamics, Indistinguishability of State, and Entanglement
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Demonstration of Quantum Brachistochrones between Distant States of an Atom
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Self-citationFast and robust quantum state transfer via a topological chain
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Distinct Bound of the Quantum Speed Limit via the Gauge Invariant Distance
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Universal Bound on Energy Cost of Bit Reset in Finite Time
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Energetic cost of quantum control protocols
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Solution to the Quantum Zermelo Navigation Problem
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Self-citationProbing Quantum Speed Limits with Ultracold Gases
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Time-optimal navigation through quantum wind
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Quantum speed limit and optimal control of many-boson dynamics
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Self-citationSpeed Limits for Macroscopic Transitions
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Vibrational state inversion of a Bose-Einstein condensate: optimal control and state tomographyPreprint
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Quantum Field Thermal Machines
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- 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 › Computer Science › Artificial Intelligence › 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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