Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics A Mathematical and Theoretical

Reply to comment on ‘control landscapes are almost always trap free: a geometric assessment’

Benjamin Russell, Re-Bing Wu, HERSCHEL A. RABITZ

Published Oct 31, 2018• 12 citations
Publisher

Abstract

Abstract This paper is a rebuttal of the comment by Zhdanov (2018 J. Phys. A: Math. Theor .) with regard to our paper (Russell et al 2017 J. Phys. A: Math. Theor . 50 205302). Parts of the comment either misrepresent the original claims in Russell et al (2017 J. Phys. A: Math. Theor . 50 205302) or consider issues that go beyond the scope of Russell et al (2017 J. Phys. A: Math. Theor . 50 205302). We address all of these points in the rebuttal. A remaining item about the prospect of an accumulation of null sets was already considered in a prior work (Russell et al 2018 J. Phys. A: Math. Theor . 51 335103). Importantly, the comment fails to consider the vast and continually growing set of successful quantum control tests under well-defined conditions, which serves as foundational evidence for the results in Russell et al (2017 J. Phys. A: Math. Theor . 50 205302). In summary, the remarks in the comment either misconstrue the claims of Russell et al (2017 J. Phys. A: Math. Theor . 50 205302) or raise points that are irrelevant to the goal of the original work. Thus, we conclude that the primary claim of the original paper stands: almost all closed finite dimensional quantum systems are trap free given adequate control resources.

DOI
10.1088/1751-8121/aaecf2

Related papers

Comment on ‘Control landscapes are almost always trap free: a geometric assessment’

Dmitry V Zhdanov

Abstract We analyze a recent claim that almost all closed, finite dimensional quantum systems have trap-free (i.e. free from local optima) landscapes (Russell et al 2017 J. Phys. A: Math. Theor . 50 , 205302). We point out several errors in the proof which compromise the authors’ conclusion.

arXiv:1811.01416•Journal of Physics A Mathematical and Theoretical•Oct 31, 2018•16 citations

Control landscapes are almost always trap free: a geometric assessment

Benjamin Russell, HERSCHEL A. RABITZ, Re-Bing Wu

Abstract A proof is presented that almost all closed, finite dimensional quantum systems have trap free (i.e. free from local optima) landscapes for a large and physically general class of circumstances, which includes qubit evolutions in quantum computing. This result offers an explanation for why gradient-based methods succeed so frequently in quantum control. The role of singular controls is analyzed using geometric tools in the case of the control of the propagator, and thus in the case of observables as well. Singular controls have been implicated as a source of landscape traps. The conditions under which singular controls can introduce traps, and thus interrupt the progress of a control optimization, are discussed and a geometrical characterization of the issue is presented. It is shown that a control being singular is not sufficient to cause control optimization progress to halt, and sufficient conditions for a trap free landscape are presented. It is further shown that the local surjectivity (full rank) assumption of landscape analysis can be refined to the condition that the end-point map is transverse to each of the level sets of the fidelity function. This mild condition is shown to be sufficient for a quantum system’s landscape to be trap free. The control landscape is shown to be trap free for all but a null set of Hamiltonians using a geometric technique based on the parametric transversality theorem. Numerical evidence confirming this analysis is also presented. This new result is the analogue of the work of Altifini, wherein it was shown that controllability holds for all but a null set of quantum systems in the dipole approximation. These collective results indicate that the availability of adequate control resources remains the most physically relevant issue for achieving high fidelity control performance while also avoiding landscape traps.

Journal of Physics A Mathematical and Theoretical•Apr 24, 2017•37 citations

Quantum Control Landscapes Are Almost Always Trap Free

Benjamin Russell, HERSCHEL A. RABITZ, Re-Bing Wu

A proof that almost all quantum systems have trap free (that is, free from local optima) landscapes is presented for a large and physically general class of quantum system. This result offers an explanation for why gradient methods succeed so frequently in quantum control in both theory and practice. The role of singular controls is analyzed using geometric tools in the case of the control of the propagator of closed finite dimension systems. This type of control field has been implicated as a source of landscape traps. The conditions under which singular controls can introduce traps, and thus interrupt the progress of a control optimization, are discussed and a geometrical characterization of the issue is presented. It is shown that a control being singular is not sufficient to cause a control optimization progress to halt and sufficient conditions for a trap free landscape are presented. It is further shown that the local surjectivity axiom of landscape analysis can be refined to the condition that the end-point map is transverse to each of the level sets of the fidelity function. This novel condition is shown to be sufficient for a quantum system's landscape to be trap free. The control landscape for a quantum system is shown to be trap free for all but a null set of Hamiltonians using a novel geometric technique based on the parametric transversality theorem. Numerical evidence confirming this is also presented. This result is the analogue of the work of Altifini, wherein it is shown that controllability holds for all but a null set of quantum systems in the dipole approximation. The presented results indicate that by-and-large limited control resources are the most physically relevant source of landscape traps.

arXiv:1608.06198•Aug 22, 2016•10 citations

Comment on “Are There Traps in Quantum Control Landscapes?”

HERSCHEL A. RABITZ, Tak‐San Ho, Ruixing Long +2 more

Many quantum control problems are formulated as a search for an optimal field that maximizes a physical objective. This search is performed over a landscape defined as the objective as a function of the control field. A recent Letter [A. N. Pechen and D. J. Tannor, Phys. Rev. Lett. 106, 120402 (2011)] asserts that the existence of special landscape critical points (CPs) with trapping characteristics is "contrary to recent claims in the literature" and "can have profound implications for both theoretical and experimental quantum control studies." We show here that these assertions are inaccurate and misleading.

arXiv:1105.2367•Physical Review Letters•May 8, 2012•29 citations

Common foundations of optimal control across the sciences: evidence of a free lunch

Benjamin Russell, HERSCHEL A. RABITZ

A common goal in the sciences is optimization of an objective function by selecting control variables such that a desired outcome is achieved. This scenario can be expressed in terms of a control landscape of an objective considered as a function of the control variables. At the most basic level, it is known that the vast majority of quantum control landscapes possess no traps, whose presence would hinder reaching the objective. This paper reviews and extends the quantum control landscape assessment, presenting evidence that the same highly favourable landscape features exist in many other domains of science. The implications of this broader evidence are discussed. Specifically, control landscape examples from quantum mechanics, chemistry and evolutionary biology are presented. Despite the obvious differences, commonalities between these areas are highlighted within a unified mathematical framework. This mathematical framework is driven by the wide-ranging experimental evidence on the ease of finding optimal controls (in terms of the required algorithmic search effort beyond the laboratory set-up overhead). The full scope and implications of this observed common control behaviour pose an open question for assessment in further work.This article is part of the themed issue 'Horizons of cybernetical physics'.

arXiv:1611.09406•Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences•Jan 23, 2017•18 citations

Singularities of quantum control landscapes

Re-Bing Wu, Ruixing Long, Jason Dominy +2 more

Quantum control landscape theory was formulated to assess the ease of finding optimal control fields in simulations and in the laboratory. The landscape is the observable as a function of the controls, and a primary goal of the theory is the analysis of landscape features. In what is referred to as the kinematic picture of the landscape, prior work showed that the landscapes are generally free of traps that could halt the search for an optimal control at a suboptimal observable value. The present paper considers the dynamical picture of the landscape, seeking the existence of singular controls, especially of a nonkinematic nature along with an assessment of whether they correspond to traps. We analyze the necessary and sufficient conditions for singular controls to be kinematic or nonkinematic critical solutions and the likelihood of their being encountered while maximizing an observable. An algorithm is introduced to seek singular controls on the landscape in simulations along with an associated Hessian landscape analysis. Simulations are performed for a large number of model finite-level quantum systems, showing that all the numerically identified kinematic and nonkinematic singular critical controls are not traps, in support of the prior empirical observations on the ease of finding high-quality optimal control fields.

arXiv:0907.2354•Physical Review A•Jul 9, 2012•63 citations
Abstract This paper is a rebuttal of the comment by Zhdanov (2018 J. Phys. A: Math. Theor .) with regard to our paper (Russell et al 2017 J. Phys. A: Math. Theor . 50 205302). Parts of the comment either misrepresent the original claims in Russell et al (2017 J. Phys. A: Math. Theor . 50 205302) or consider issues that go beyond the scope of Russell et al (2017 J. Phys. A: Math. Theor . 50 205302). We address all of these points in the rebuttal. A remaining item about the prospect of an accumulation of null sets was already considered in a prior work (Russell et al 2018 J. Phys. A: Math. Theor . 51 335103). Importantly, the comment fails to consider the vast and continually growing set of successful quantum control tests under well-defined conditions, which serves as foundational evidence for the results in Russell et al (2017 J. Phys. A: Math. Theor . 50 205302). In summary, the remarks in the comment either misconstrue the claims of Russell et al (2017 J. Phys. A: Math. Theor . 50 205302) or raise points that are irrelevant to the goal of the original work. Thus, we conclude that the primary claim of the original paper stands: almost all closed finite dimensional quantum systems are trap free given adequate control resources.
Abstract A proof is presented that almost all closed, finite dimensional quantum systems have trap free (i.e. free from local optima) landscapes for a large and physically general class of circumstances, which includes qubit evolutions in quantum computing. This result offers an explanation for why gradient-based methods succeed so frequently in quantum control. The role of singular controls is analyzed using geometric tools in the case of the control of the propagator, and thus in the case of observables as well. Singular controls have been implicated as a source of landscape traps. The conditions under which singular controls can introduce traps, and thus interrupt the progress of a control optimization, are discussed and a geometrical characterization of the issue is presented. It is shown that a control being singular is not sufficient to cause control optimization progress to halt, and sufficient conditions for a trap free landscape are presented. It is further shown that the local surjectivity (full rank) assumption of landscape analysis can be refined to the condition that the end-point map is transverse to each of the level sets of the fidelity function. This mild condition is shown to be sufficient for a quantum system’s landscape to be trap free. The control landscape is shown to be trap free for all but a null set of Hamiltonians using a geometric technique based on the parametric transversality theorem. Numerical evidence confirming this analysis is also presented. This new result is the analogue of the work of Altifini, wherein it was shown that controllability holds for all but a null set of quantum systems in the dipole approximation. These collective results indicate that the availability of adequate control resources remains the most physically relevant issue for achieving high fidelity control performance while also avoiding landscape traps.
A proof that almost all quantum systems have trap free (that is, free from local optima) landscapes is presented for a large and physically general class of quantum system. This result offers an explanation for why gradient methods succeed so frequently in quantum control in both theory and practice. The role of singular controls is analyzed using geometric tools in the case of the control of the propagator of closed finite dimension systems. This type of control field has been implicated as a source of landscape traps. The conditions under which singular controls can introduce traps, and thus interrupt the progress of a control optimization, are discussed and a geometrical characterization of the issue is presented. It is shown that a control being singular is not sufficient to cause a control optimization progress to halt and sufficient conditions for a trap free landscape are presented. It is further shown that the local surjectivity axiom of landscape analysis can be refined to the condition that the end-point map is transverse to each of the level sets of the fidelity function. This novel condition is shown to be sufficient for a quantum system's landscape to be trap free. The control landscape for a quantum system is shown to be trap free for all but a null set of Hamiltonians using a novel geometric technique based on the parametric transversality theorem. Numerical evidence confirming this is also presented. This result is the analogue of the work of Altifini, wherein it is shown that controllability holds for all but a null set of quantum systems in the dipole approximation. The presented results indicate that by-and-large limited control resources are the most physically relevant source of landscape traps.
Many quantum control problems are formulated as a search for an optimal field that maximizes a physical objective. This search is performed over a landscape defined as the objective as a function of the control field. A recent Letter [A. N. Pechen and D. J. Tannor, Phys. Rev. Lett. 106, 120402 (2011)] asserts that the existence of special landscape critical points (CPs) with trapping characteristics is "contrary to recent claims in the literature" and "can have profound implications for both theoretical and experimental quantum control studies." We show here that these assertions are inaccurate and misleading.
A common goal in the sciences is optimization of an objective function by selecting control variables such that a desired outcome is achieved. This scenario can be expressed in terms of a control landscape of an objective considered as a function of the control variables. At the most basic level, it is known that the vast majority of quantum control landscapes possess no traps, whose presence would hinder reaching the objective. This paper reviews and extends the quantum control landscape assessment, presenting evidence that the same highly favourable landscape features exist in many other domains of science. The implications of this broader evidence are discussed. Specifically, control landscape examples from quantum mechanics, chemistry and evolutionary biology are presented. Despite the obvious differences, commonalities between these areas are highlighted within a unified mathematical framework. This mathematical framework is driven by the wide-ranging experimental evidence on the ease of finding optimal controls (in terms of the required algorithmic search effort beyond the laboratory set-up overhead). The full scope and implications of this observed common control behaviour pose an open question for assessment in further work.This article is part of the themed issue 'Horizons of cybernetical physics'.
Quantum control landscape theory was formulated to assess the ease of finding optimal control fields in simulations and in the laboratory. The landscape is the observable as a function of the controls, and a primary goal of the theory is the analysis of landscape features. In what is referred to as the kinematic picture of the landscape, prior work showed that the landscapes are generally free of traps that could halt the search for an optimal control at a suboptimal observable value. The present paper considers the dynamical picture of the landscape, seeking the existence of singular controls, especially of a nonkinematic nature along with an assessment of whether they correspond to traps. We analyze the necessary and sufficient conditions for singular controls to be kinematic or nonkinematic critical solutions and the likelihood of their being encountered while maximizing an observable. An algorithm is introduced to seek singular controls on the landscape in simulations along with an associated Hessian landscape analysis. Simulations are performed for a large number of model finite-level quantum systems, showing that all the numerically identified kinematic and nonkinematic singular critical controls are not traps, in support of the prior empirical observations on the ease of finding high-quality optimal control fields.