Open-system quantum dynamics described by non-Hermitian effective Hamiltonians have become a subject of considerable interest. Studies of non-Hermitian physics have revealed general principles, including relationships between the topology of the complex eigenvalue space and the breakdown of adiabatic control strategies. We study here the control of a single non-Hermitian qubit, similar to recently realized experimental systems in which the non-Hermiticity arises from an open spontaneous emission channel. We review the topological features of the resulting non-Hermitian Hamiltonian and then present two distinct results. First, we illustrate how to realize any continuous and differentiable pure-state trajectory in the dynamics of a qubit that are conditioned on no emission. This result implicitly provides a work around for the breakdown of standard adiabatic following in such non-Hermitian systems. Second, we use Pontryagin's maximum principle to derive optimal trajectories connecting boundary states on the Bloch sphere, using a cost function which balances the desired dynamics against the controller energy used to realize them. We demonstrate that the latter approach can effectively find trajectories which maintain high state purity, even in the case of inefficient detection.
Open-system quantum dynamics described by non-Hermitian effective Hamiltonians have become a subject of considerable interest. Studies of non-Hermitian physics have revealed general principles, including relationships between the topology of the complex eigenvalue space and the breakdown of adiabatic control strategies. We study here the control of a single non-Hermitian qubit, similar to recently realized experimental systems in which the non-Hermiticity arises from an open spontaneous emission channel. We review the topological features of the resulting non-Hermitian Hamiltonian and then present two distinct results. First, we illustrate how to realize any continuous and differentiable pure-state trajectory in the dynamics of a qubit that are conditioned on no emission. This result implicitly provides a work around for the breakdown of standard adiabatic following in such non-Hermitian systems. Second, we use Pontryagin's maximum principle to derive optimal trajectories connecting boundary states on the Bloch sphere, using a cost function which balances the desired dynamics against the controller energy used to realize them. We demonstrate that the latter approach can effectively find trajectories which maintain high state purity, even in the case of inefficient detection.
As quantum systems are understood better from a fundamental physics perspective, we are gaining more insight and ideas for their many applications. Among them are quantum computation, sensing, and materials. Some difficulties that have been encountered are short lifetimes and coupling to unwanted modes while attempting to control these systems, resulting in error propagation. Previously, physicists have pursued Hermitian descriptions of systems to preserve unitary evolution and real-valued measurements. However, non-Hermitian systems not only provide a realistic description of physical systems within larger environments, but also a rich topological landscape from their complex energy spectrum. In this thesis, we will explore how to control a non-Hermitian qubit to utilize its complex spectrum and the role of its Hamiltonian in energetics and time dynamics. Because superconducting qubits offer flexibility in parameter tuning, we can tune the values of the non-Hermitian Hamiltonian to explore this complex spectrum across regions of both $\PT$-symmetry and $\PT$-symmetry breaking. This complex energy spectrum forms a Riemann surface that has a branch cut and branch points (exceptional points). Encircling these exceptional points adiabatically allows for state transport that can non-trivially convert between eigenstates. Previous experimental studies of this process revealed a chirality: eigenstate conversion occurs when following the state with higher gain, but breaks down when following the state with higher loss. We explore how the interplay between gain and loss in modes of the system results in the breakdown of this adiabaticity and how we can use a method based on shortcuts to adiabaticity to circumvent this. Finally, we leverage this tunability to further investigate how the components of the non-Hermitian Hamiltonian relate to classical thermodynamic notions of internal energy and work in time-dependent protocols via the Jarzynski equality.
This article provides a review of recent developments in the formulation and execution of optimal control strategies for the dynamics of quantum systems. A brief introduction to the concept of optimal control, the dynamics of of open quantum systems, and quantum information processing is followed by a presentation of recent developments regarding the two main tasks in this context: state-specific and state-independent optimal control. For the former, we present an extension of conventional theory (Pontryagin's principle) to quantum systems which undergo a non-Markovian time-evolution. Owing to its importance for the realization of quantum information processing, the main body of the review, however, is devoted to state-independent optimal control. Here, we address three different approaches: an approach which treats dissipative effects from the environment in lowest-order perturbation theory, a general method based on the time--evolution superoperator concept, as well as one based on the Kraus representation of the time-evolution superoperator. Applications which illustrate these new methods focus on single and double qubits (quantum gates) whereby the environment is modeled either within the Lindblad equation or a bath of bosons (spin-boson model). While these approaches are widely applicable, we shall focus our attention to solid-state based physical realizations, such as semiconductor- and superconductor-based systems. While an attempt is made to reference relevant and representative work throughout the community, the exposition will focus mainly on work which has emerged from our own group.
We address a control-constrained optimal control problem pertaining to the transformation of quantum states. Our objective is to steer a quantum system from an initial state to a desired target state while adhering to the principles of the Liouville-von Neumann equation. To achieve this, we introduce a cost functional that balances the dual goals of fidelity maximization and energy consumption minimization. We derive optimality conditions in the form of the Pontryagin Maximum Principle (PMP) for the matrix-valued dynamics associated with this problem. Subsequently, we present a time discretized computational scheme designed to solve the optimal control problem. This computational scheme is rooted in an indirect method grounded in the PMP, showcasing its versatility and efficacy. To illustrate the practicality and applicability of our methodology, we employ it to address the case of a spin ½ particle subjected to interaction with a magnetic field. Our findings shed light on the potential of this approach to tackle complex quantum control scenarios and contribute to the broader field of quantum state transformations.
The ability to control matter at the quantum level is a central requirement for the continued advance of quantum technologies. Recent experimental progress has enabled continuous monitoring of quantum systems in time, giving rise to dynamics conditioned on measurement outcomes. Such measurement-conditioned evolution is naturally described by non-Hermitian Hamiltonians, whose complex eigenvalues encode gain and loss of population associated with the measurement record. Non-Hermitian quantum dynamics exhibit geometric features without Hermitian counterparts, including non-orthogonal eigenspaces, exceptional points, and complex geometric phases. This thesis investigates geometric structures underlying non-Hermitian quantum dynamics from two complementary perspectives: the geometry of quantum states under measurement-conditioned evolution, and the geometry of Hamiltonians and parameter spaces relevant for control in experimentally accessible platforms. First, measurement-conditioned non-Hermitian dynamics are formulated within quantum information geometry. The normalized evolution preserves the rank of the density operator, confining the dynamics to fixed-rank manifolds with a natural geometric structure. This framework enables optimization of the success probability of prescribed non-Hermitian trajectories and, via a purification-bundle formulation, leads to a non-Hermitian quantum speed limit expressed in terms of the Bures geometry. Explicit shortest geodesics and their associated non-Hermitian generators are derived, including results on the existence of time-independent generators in finite dimensions. The thesis then addresses control of non-Hermitian dynamics from a complementary geometric viewpoint focused on Hamiltonians and parameter spaces. A complex mixing angle is introduced for the detuned parity--time-symmetric dimer, providing a transparent mapping between parameter space and eigenspaces and enabling implementable counter-diabatic control on a superconducting transmon qubit. This approach is validated experimentally through accurate eigenstate tracking and exceptional-point encircling, and is subsequently generalized to arbitrary two-level systems, bipartite lattice models, and multilevel transmons. Finally, the geometric origin of complex Berry phases is identified, with predictions indicating their observability on the transmon platform and their role in purity stabilization and noise-assisted chiral transfer. Overall, the thesis establishes geometry as a unifying and operational language for the analysis and control of non-Hermitian quantum systems, addressing both foundational theory and experimentally relevant protocols.
This work explores the relationship between optimal control theory and adiabatic passage techniques in quantum systems. The study is based on a geometric analysis of the Hamiltonian dynamics constructed from Pontryagin's maximum principle. In a three-level quantum system, we show that the stimulated Raman adiabatic passage technique can be associated to a peculiar Hamiltonian singularity. One deduces that the adiabatic pulse is solution of the optimal control problem only for a specific cost functional. This analysis is extended to the case of a four-level quantum system.
Optimal Control Theory is a powerful mathematical tool, which has known a rapid development since the 1950s, mainly for engineering applications. More recently, it has become a widely used method to improve process performance in quantum technologies by means of highly efficient control of quantum dynamics. This tutorial aims at providing an introduction to key concepts of optimal control theory which is accessible to physicists and engineers working in quantum control or in related fields. The different mathematical results are introduced intuitively, before being rigorously stated. This tutorial describes modern aspects of optimal control theory, with a particular focus on the Pontryagin Maximum Principle, which is the main tool for determining open-loop control laws without experimental feedback. The different steps to solve an optimal control problem are discussed, before moving on to more advanced topics such as the existence of optimal solutions or the definition of the different types of extremals, namely normal, abnormal, and singular. The tutorial covers various quantum control issues and describes their mathematical formulation suitable for optimal control. The connection between the Pontryagin Maximum Principle and gradient-based optimization algorithms used for high-dimensional quantum systems is described. The optimal solution of different low-dimensional quantum systems is presented in detail, illustrating how the mathematical tools are applied in a practical way.