Traps in the control landscapes of controllable multilevel systems — Open problems

Statement

Consider a closed $N$-level system driven by one real control $f$,

$$i\,\dot U_t = \big(H_0 + f(t)\,V\big)\,U_t, \qquad U_0 = I, \qquad f \in L^2([0,T];\mathbb R),$$

with $H_0$ and $V$ Hermitian $N\times N$ matrices, and assume that it is completely controllable (the Lie algebra generated by $iH_0$ and $iV$ is $\mathfrak{su}(N)$ or $\mathfrak u(N)$). Typical objectives are

an expectation value, $J_O(f) = \mathrm{Tr}\big[O\,U_T\,\rho_0\,U_T^\dagger\big]$, for an observable $O$ and an initial state $\rho_0$;
a transition probability, $J_{i\to f}(f) = |\langle f|U_T|i\rangle|^2$;
a gate fidelity, $J_W(f) = \tfrac{1}{N^2}\big|\mathrm{Tr}\big(W^\dagger U_T\big)\big|^2$, for a target gate $W$.

A control $f$ is a trap if it is a local maximum of $J$ but $J(f) < \sup J$ [1].

Question. Is there, for every controllable pair $(H_0, V)$ with $N \ge 3$, a time $T^*$ such that for all $T \ge T^*$ these objectives have no traps on $L^2([0,T])$? If not, which systems and objectives have traps?

Why it matters

Gradient methods such as GRAPE and Krotov are local. If the landscape has no traps, a local search from almost any starting pulse reaches a global optimum; a trap with a large basin of attraction can stop it. The answer explains why local optimal control works so often in practice, and tells when global methods are needed (see Control landscapes).

Known results

The kinematic picture is trap-free. As functions on the unitary group, objectives such as $J_O$ have only global maxima, global minima and saddles [2, 3]. The dynamical landscape $J(f)$ inherits this structure at controls where the map $f \mapsto U_T$ is locally surjective (regular controls). Traps can therefore only occur at singular controls, where this map loses rank [4, 1].

Two-level systems are settled. For $N = 2$ the absence of traps has been proved: first for the Landau–Zener system [5], then for general qubit control problems with long enough controls [6], and then in the ultrafast regime — for state transfer between a wide range of initial and final states [7] and for gate generation, including the phase-shift gates that earlier analyses had left out [8, 9].

Higher-order traps for $N \ge 3$. In a three-level $\Lambda$ system, the zero control is a critical point that is not a global maximum and near which the Taylor expansion of the objective, up to some order, shows no direction of increase. Such points slow down local searches and are called higher-order traps [10, 11]. Their practical significance was disputed in a Comment and Reply [12, 13]. Traps of arbitrarily high order exist in some strongly degenerate controllable systems [14]. In three-level systems with one forbidden transition, degenerate Bohr frequencies raise the order (a seventh-order trap in a symmetric ladder system), and when the system is made uncontrollable the zero control becomes a true trap [1]. A higher-order trap is not necessarily a local maximum; all known examples of higher-order traps in controllable systems are constant controls [1].

True traps at short times. de Fouquieres and Schirmer found critical points of the $L^2$ landscape that are traps, for state-transfer and gate problems [15]. According to [1], these examples have $N \ge 4$ and control times that are not very large, and the traps disappear at larger times.

Generic systems. Russell, Rabitz and Wu argued, with a geometric transversality argument, that the landscape is trap-free for all Hamiltonians outside a set of measure zero [16]. A Comment pointed out errors in the proof [17], and the authors replied [18]; the generic statement is therefore contested.

The gate objective matters. On $SU(N)$, the trace fidelity $\mathrm{Re}\,\mathrm{Tr}(W^\dagger U)$ has kinematic traps [15]; for every $N \ge 5$ there are kinematic local extrema that are not global [19]. These kinematic traps disappear when the objective ignores the global phase — an optimization over the projective unitary group, as with $J_W$ above [15].

Status

Partial progress. The question is answered for $N = 2$: there are no traps. For $N \ge 3$, there is no proof that controllable systems are trap-free at long times, and no true trap of such a system at long times is known; the known obstructions are higher-order traps at constant controls. The proof for generic systems is disputed.

Where it is posed

Alexander N. Pechen and David J. Tannor Are there Traps in Quantum Control Landscapes? Phys. Rev. Lett. 106, 120402 (2011) doi

The title question. The Letter shows that the dynamical landscape can contain special critical points with trapping behaviour, contrary to earlier claims that it is always trap-free.

Alexander Pechen and Nikolay Il'in Trap-free manipulation in the Landau-Zener system Phys. Rev. A 86, 052117 (2012) doi · arXiv:1304.1357

Abstract and introduction. Deciding, for a given system, whether traps exist or not is called the central open problem of the analysis of traps.

Boris Volkov, Anastasia Myachkova and Alexander Pechen Phenomenon of a stronger trapping behavior in Λ -type quantum systems with symmetry Phys. Rev. A 111, 022617 (2025) doi

Introduction. A rigorous proof of the existence or absence of traps in the dynamical landscape is described as still not fully solved; absence is proved only for $N = 2$.

References

Boris Volkov, Anastasia Myachkova and Alexander Pechen Phenomenon of a stronger trapping behavior in Λ -type quantum systems with symmetry Phys. Rev. A 111, 022617 (2025) doi

Herschel A. Rabitz, Michael M. Hsieh and Carey M. Rosenthal Quantum Optimally Controlled Transition Landscapes Science 303, 1998-2001 (2004) doi

Tak-San Ho and Herschel Rabitz Why do effective quantum controls appear easy to find? Journal of Photochemistry and Photobiology A: Chemistry 180, 226-240 (2006) doi

Re-Bing Wu, Ruixing Long, Jason Dominy, Tak-San Ho and Herschel Rabitz Singularities of quantum control landscapes Phys. Rev. A 86, 013405 (2012) doi · arXiv:0907.2354

Alexander Pechen and Nikolay Il'in Trap-free manipulation in the Landau-Zener system Phys. Rev. A 86, 052117 (2012) doi · arXiv:1304.1357

A. N. Pechen and N. B. Il’in Coherent control of a qubit is trap-free Proc. Steklov Inst. Math. 285, 233-240 (2014) doi · arXiv:1407.5175

Alexander Pechen and Nikolay Il’in Control landscape for ultrafast manipulation by a qubit J. Phys. A: Math. Theor. 50, 075301 (2017) doi · arXiv:1909.09216

Boris O Volkov, Oleg V Morzhin and Alexander N Pechen Quantum control landscape for ultrafast generation of single-qubit phase shift quantum gates J. Phys. A: Math. Theor. 54, 215303 (2021) doi · arXiv:2104.12699

Boris Olegovich Volkov and Alexander Nikolaevich Pechen On the detailed structure of quantum control landscape for fast single qubit phase-shift gate generation Izv. Math. 87, 906-919 (2023) doi

Alexander N. Pechen and David J. Tannor Are there Traps in Quantum Control Landscapes? Phys. Rev. Lett. 106, 120402 (2011) doi

Alexander N. Pechen and David J. Tannor Quantum Control Landscape for a Λ‐atom in the Vicinity of Second‐Order Traps Israel Journal of Chemistry 52, 467-472 (2012) doi · arXiv:1508.04169

Herschel Rabitz, Tak-San Ho, Ruixing Long, Rebing Wu and Constantin Brif Comment on “Are There Traps in Quantum Control Landscapes?” Phys. Rev. Lett. 108, 198901 (2012) doi · arXiv:1105.2367

Alexander N. Pechen and David J. Tannor Pechen and Tannor Reply: Phys. Rev. Lett. 108, 198902 (2012) doi

Boris Olegovich Volkov and Alexander Nikolaevich Pechen Higher-order traps for some strongly degenerate quantum control systems Russian Math. Surveys 78, 390-392 (2023) doi · arXiv:2304.03355

PIERRE DE FOUQUIERES and SOPHIE G. SCHIRMER A CLOSER LOOK AT QUANTUM CONTROL LANDSCAPES AND THEIR IMPLICATION FOR CONTROL OPTIMIZATION Infin. Dimens. Anal. Quantum. Probab. Relat. Top. 16, 1350021 (2013) doi

Benjamin Russell, Herschel Rabitz and Re-Bing Wu Control landscapes are almost always trap free: a geometric assessment J. Phys. A: Math. Theor. 50, 205302 (2017) doi

Dmitry V Zhdanov Comment on ‘Control landscapes are almost always trap free: a geometric assessment’ J. Phys. A: Math. Theor. 51, 508001 (2018) doi · arXiv:1811.01416

Benjamin Russell, Rebing Wu and Herschel Rabitz Reply to comment on ‘control landscapes are almost always trap free: a geometric assessment’ J. Phys. A: Math. Theor. 51, 508002 (2018) doi

Petre Birtea, Ioan Caşu and Dan Comănescu Constraint optimization and SU ( N ) quantum control landscapes J. Phys. A: Math. Theor. 55, 115301 (2022) doi