Active control of quantum systems enables diverse applications ranging from quantum computation to manipulation of molecular processes. Maximum speeds and related bounds have been identified from uncertainty principles and related inequalities, but such bounds utilize only coarse system information and loosen significantly in the presence of constraints and complex interaction dynamics. We show that an integral-equation-based formulation of conservation laws in quantum dynamics leads to a systematic framework for identifying fundamental limits to any quantum control scenario. We demonstrate the utility of our bounds in three scenarios-three-level driving, decoherence suppression, and maximum-fidelity gate implementations-and show that in each case our bounds are tight or nearly so. Global bounds complement local-optimization-based designs, illuminating performance levels that may be possible, as well as those that cannot be surpassed.
Active control of quantum systems enables diverse applications ranging from quantum computation to manipulation of molecular processes. Maximum speeds and related bounds have been identified from uncertainty principles and related inequalities, but such bounds utilize only coarse system information and loosen significantly in the presence of constraints and complex interaction dynamics. We show that an integral-equation-based formulation of conservation laws in quantum dynamics leads to a systematic framework for identifying fundamental limits to any quantum control scenario. We demonstrate the utility of our bounds in three scenarios-three-level driving, decoherence suppression, and maximum-fidelity gate implementations-and show that in each case our bounds are tight or nearly so. Global bounds complement local-optimization-based designs, illuminating performance levels that may be possible, as well as those that cannot be surpassed.
The precise control of devices on the quantum scale is crucial for all quantum computing applications. The limitations of quantum control are therefore of great interest to those developing quantum hardware or researching quantum information protocols. We show how moment-sum-of-squares techniques can be combined with quantum dynamics and filtering theory to construct a hierarchy of convex optimization problems that furnish hard, computable bounds on the best achievable performance in quantum control tasks. These bounds can serve as witnesses of fundamental limitations, certificates of optimality, or performance targets and thus complement existing controller design strategies. In particular, our work allows us to compute bounds for common quantum control tasks, such as maximizing the fidelity of the final quantum state relative to a target state given a finite time budget or finding the minimum time required to reach the target state with near-unit fidelity. Our work also naturally applies to feedback-controlled quantum systems subjected to continuous observations, e.g., homodyne detection or photon counting, which are of emerging interest for the next generation of quantum devices. We provide an open-source package, MarkovBounds.jl, that is built upon the optimization ecosystem in the Julia programming language and exposes a high-level interface to the proposed bounding framework.
Speed of state transitions in macroscopic systems is a crucial concept for foundations of nonequilibrium statistical mechanics as well as various applications in quantum technology represented by optimal quantum control.While extensive studies have made efforts to obtain rigorous constraints on dynamical processes since Mandelstam and Tamm, speed limits that provide tight bounds for macroscopic transitions have remained elusive.Here, by employing the local conservation law of probability, the fundamental principle in physics, we develop a general framework for deriving qualitatively tighter speed limits for macroscopic systems than many conventional ones.We show for the first time that the speed of the expectation value of an observable defined on an arbitrary graph, which can describe general many-body systems, is bounded by the "gradient" of the observable, in contrast with conventional speed limits depending on the entire range of the observable.This framework enables us to derive novel quantum speed limits for macroscopic unitary dynamics.Unlike previous bounds, the speed limit decreases when the expectation value of the transition Hamiltonian increases; this intuitively describes a new trade-off relation between time and the quantum phase difference.Our bound is dependent on instantaneous quantum states and thus can achieve the equality condition, which is conceptually distinct from the Lieb-Robinson bound.We also find that, beyond expectation values of macroscopic observables, the speed of macroscopic quantum coherence can be bounded from above by our general approach.The newly obtained bounds are verified in transport phenomena in particle systems and nonequilibrium dynamics in many-body spin systems.We also demonstrate that our strategy can be applied for finding new speed limits for macroscopic transitions in stochastic systems, including quantum ones, where the bounds are expressed by the entropy production rate.Our work elucidates novel speed limits on the basis of local conservation law, providing fundamental limits to various types of nonequilibrium quantum macroscopic phenomena.
Fundamental trade-off relations, such as quantum speed limit and quantum thermodynamic uncertainty relation, describe the performance limits of quantum systems by imposing that improvements in speed or precision necessitate a substantial thermodynamic cost. Quantum feedback control, which is a pivotal technique for manipulating quantum dynamics based on measurement outcomes, is widely employed to enhance system performance. Nevertheless, its impact on these fundamental bounds remains an open question. This work elucidates this influence by establishing a theoretical framework for quantum speed limit and quantum thermodynamic uncertainty relation under a paradigmatic Markovian feedback protocol. We derive general inequalities incorporating the effects of feedback control on speed and precision. Through numerical simulations on a simple two-level system and quantum error correction, a key application of quantum feedback control, we validate our derived bounds and demonstrate that feedback control can indeed improve both speed and precision beyond those achievable limits in uncontrolled systems. Next, to elucidate the mechanism behind these improvements and the qualitative difference from uncontrolled dynamics, we analyze the governing thermodynamic costs, which are the fundamental quantities that constrain speed and precision, within a simple model. Our analysis reveals that feedback can improve the time scaling order of these costs. This modification of the dynamical scaling is the origin of the qualitative performance gain, signifying that the feedback-induced improvements of performance are not merely quantitative but represent a fundamental shift. Consequently, our work offers a comprehensive understanding of how feedback control impacts the fundamental limits on the speed and precision of quantum systems, providing crucial insights for designing high-performance quantum technologies.
Abstract In quantum control, quantum speed limits provide fundamental lower bounds on the time that is needed to implement certain unitary transformations. Using Lie algebraic methods, we link these speed limits to symmetries of the control Hamiltonians and provide quantitative bounds that can be calculated without solving the controlled system dynamics. In particular we focus on two scenarios: On one hand, we provide bounds on the time that is needed for a control system to implement a given target unitary U and on the other hand we bound the time to implement the most difficult to reach evolution e − i θ H s of a target Hamiltonian H s . We apply our abstract bounds on physically relevant systems like coupled qubits, spin chains, globally controlled Rydberg atoms and NMR molecules and compare our results to the existing literature. We hope that our bounds can aid experimentalists to identify bottlenecks and design faster quantum control systems.
This paper presents a novel theoretical framework for dynamic control of quantum decoherence, specifically tailored for application within quantum communication and quantum computation defense mechanisms. The core of this work lies in the construction of a rigorous mathematical model of quantum decoherence, followed by the design of dynamic control algorithms utilizing established control theory principles. These algorithms are actively employed to suppress or manipulate decoherence, thereby bolstering the security of quantum information. We demonstrate a shift in perspective, framing decoherence control as a controllable systems problem, which provides a new approach to quantum defense. The theoretical foundations presented here offer a robust methodology for enhancing the resilience of quantum systems against environmental noise, a critical step towards practical, secure quantum technologies. Key areas explored include decoherence rate modeling, optimal control design, and adaptive control strategies. This research contributes significantly to the ongoing efforts in quantum information security and provides a practical roadmap for developing robust quantum defense protocols.
We investigate the problem of control of quantum systems. Various notions of controllability of quantum systems are discussed and Lie algebraic techniques are employed to derive necessary and sufficient conditions for the resulting degrees of controllability. The results are employed to study the degree of controllability for various model systems. The problem of optimal control is formulated as a problem of maximisation of the expectation value of an observable of the system subject to dynamical constraints and costs. A class of iterative algorithms is discussed, and shown to converge to a solution of the Euler-Lagrange equations, which is a necessary condition for optimality of a solution. A numerical realisation of the algorithm is presented and potential implementation of the resulting control fields in the laboratory is discussed. The algorithm is applied to find optimal control fields for a variety of applications including selective and simultaneous excitation of individual quantum dots in a globally addressed ensemble, creation of superposition states, and selective excitation of certain vibrational modes of a simple molecule such as Hydrogen Fluoride. In each case we systematically explore the parameter space of the algorithm to select solutions with desirable physical features from the multitude of possible choices. As there are many possible optimal control fields which give similar yields the question of whether some are more robust than others is investigated. A number of optimal fields are analysed quantitatively in terms of robustness with regard to temporal and spectral noise affecting the fields, uncertainty in the model parameters and the effect of experimental limitations such as restrictions on the bandwidth of the fields. It is shown that whilst the solutions tend to be sensitive to uncertainty in the energy levels chosen for the model, they appear to be surprisingly tolerant to even severe bandwidth limitations.