Searching for an optimal control in the presence of saddles on the quantum-mechanical observable landscape

Abstract

The broad success of theoretical and experimental quantum optimal control is intimately connected to the topology of the underlying control landscape. For several common quantum control goals, including the maximization of an observable expectation value, the landscape has been shown to lack local optima if three assumptions are satisfied: (i) the quantum system is controllable, (ii) the Jacobian of the map from the control field to the evolution operator is full-rank, and (iii) the control field is not constrained. In the case of the observable objective, this favorable analysis shows that the associated landscape also contains saddles, i.e., critical points that are not local suboptimal extrema. In this paper, we investigate whether the presence of these saddles affects the trajectories of gradient-based searches for an optimal control. We show through simulations that both the detailed topology of the control landscape and the parameters of the system Hamiltonian influence whether the searches are attracted to a saddle. For some circumstances with a special initial state and target observable, optimizations may approach a saddle very closely, reducing the efficiency of the gradient algorithm. Encounters with such attractive saddles are found to be quite rare. Neither the presence of a large number of saddles on the control landscape nor a large number of system states increase the likelihood that a search will closely approach a saddle. Even for applications that encounter a saddle, well-designed gradient searches with carefully chosen algorithmic parameters will readily locate optimal controls.

Learning Unitaries by Gradient Descent

“…Recently, it has been shown that quantum alternating operator unitaries can perform universal quantum computation [3]. In the infinitesimal time setting, QAOA also encompasses the more general problem of the application of time varying quantum controls [4][5][6][7][8][9][10][11][12][13][14]. In this work, we study the quantum alternating operator formalism as a general framework of performing arbitrary unitary transformations.…”

“…In this work, we ask whether optimal learning sequences for performing the target unitary U can be obtained by using gradient descent optimization on the parameters t, τ of V( t, τ ). The matrices A, B are sampled from the Gaussian Unitary Ensemble (GUE) so that the algebra generated by A, B via commutation is with probability one complete in u(d), i.e., the system is controllable [4][5][6][7][8][9][10][11][12][13][14]. The parameters t, τ represent the times for which the generators of V( t, τ ) are applied.…”

“…In general, the control landscape for learning the unitary U is highly non-convex [4][5][6][7][8][9][10][11][12][13][14]. Gradient descent algorithms do not necessarily converge to a globally optimal solution in the parameters of a non-convex space [25], and they frequently converge instead to some undesired critical point of the loss function landscape.…”

Optimization Landscape of Quantum Control Systems

“…In this way, the influence of saddles was systematically investigated in Ref. [59]. An extremal control problem was identified that contains a highly attractive saddle submanifold.…”

The Optimization Landscape of Hybrid Quantum-Classical Algorithms: from Quantum Control to NISQ Applications

“…2(a). There are additional saddle points on the landscape that may slow down the optimization, but they will not halt even a greedy algorithm search for global optimal solutions [13]. When the control resources are insufficient, e.g., when the control pulses have very limited time duration and bandwidth, false traps will emerge to likely halt the optimization procedure in a suboptimal minimum (Fig.…”