Superoscillation is a counterintuitive phenomenon for its mathematical feature of faster than Fourier, which has allowed novel optical imaging beyond the diffraction limit. In this article we introduce a superoscillating quantum control protocol realized by sequential selections within the framework of weak measurement, which steers the apparatus (target) by repeatedly applying optimal pre- and postselections to the system (controller). Our protocol accelerates the adiabatic transport of trapped ions and the adiabatic quantum search algorithm at a finite energy cost. We demonstrate the accuracy and robustness of the protocol in the presence of decoherence and fluctuating noise and elucidate the trade-off between fidelity and rounds of selections. Our findings open avenues for quantum-state manipulation and wave-packet control using superoscillation across a range of quantum platforms.
Superoscillation is a counterintuitive phenomenon for its mathematical feature of faster than Fourier, which has allowed novel optical imaging beyond the diffraction limit. In this article we introduce a superoscillating quantum control protocol realized by sequential selections within the framework of weak measurement, which steers the apparatus (target) by repeatedly applying optimal pre- and postselections to the system (controller). Our protocol accelerates the adiabatic transport of trapped ions and the adiabatic quantum search algorithm at a finite energy cost. We demonstrate the accuracy and robustness of the protocol in the presence of decoherence and fluctuating noise and elucidate the trade-off between fidelity and rounds of selections. Our findings open avenues for quantum-state manipulation and wave-packet control using superoscillation across a range of quantum platforms.
Superoscillations occur when a globally band-limited function locally oscillates faster than its highest Fourier component. We generalize this effect to arbitrary quantum-mechanical operators as a weak value, where the preselected state is a superposition of eigenstates of the operator with eigenvalues bounded to a range, and the postselection state is a local position. Superbehavior of this operator occurs whenever the operator's weak value exceeds its eigenvalue bound. We give illustrative examples of this effect for total angular momentum and energy. In the latter case, we demonstrate a sequence of harmonic oscillator potentials where a finite-energy state converges everywhere on the real line, using only bounded superpositions of states whose asymptotic energy vanishes---``energy out of nothing.'' This limit requires postselecting the particle in a region whose size diverges in the considered limit. We further show in this example that the superenergy is associated with superoscillations in time with a rate given by the local superenergy divided by the reduced Planck's constant. This example demonstrates the possibility of mimicking a high-energy state with coherent superpositions of nearly zero-energy states for as wide a spatial region as desired. We provide numerical evidence of these features to further bolster and elucidate our claims.
With the advent of quantum technologies, control issues are becoming increasingly important. In this paper, we address control in phase space under a global constraint provided by a minimal energylike cost function and a local (in Fourier space) constraint resulting from a robustness criterion. The inverse-engineering Lagrangian formalism developed here generalizes the formalism commonly used to describe the superoscillation phenomenon. It is applied to both nondissipative and dissipative quantum mechanics, and extended to stochastic thermodynamics. Interestingly, our approach also allows one to improve sensing capabilities by appropriate control of the system.
Adiabatic quantum algorithms must evolve slowly enough to suppress non-adiabatic transitions while remaining fast enough to be practical. In open systems, this trade-off is reshaped by decoherence. For Hamiltonians subject to dephasing Lindbladians, Avron et al. [Phys. Rev. A 82, 040304(R) (2010)] showed that a unique timetable exists that maximizes the fidelity with a target state. This optimal schedule is characterized by a constant tunneling rate along the adiabatic path. In this work, we revisit their analysis and apply it to the adiabatic Grover search framework, obtaining closed-form expressions for the optimal evolution schedule, the minimum runtime, and the resulting achievable fidelity. Moreover, by invoking an energy-time uncertainty argument, we identify a critical dephasing threshold beyond which further noise-assisted acceleration is prohibited, showing that the apparent advantage of strong dephasing is a Zeno-type stabilization mechanism rather than a conventional algorithmic speedup over unitary Grover search.
Novel concepts, perspectives and challenges in measuring and controlling an open quantum system via sequential schemes are shown. We discuss how similar protocols, relying both on repeated quantum measurements and dynamical decoupling control pulses, can allow to: (i) Confine and protect quantum dynamics from decoherence in accordance with the Zeno physics. (ii) Analytically predict the probability that a quantum system is transferred into a target quantum state by means of stochastic sequential measurements. (iii) Optimally reconstruct the spectral density of environmental noise sources by orthogonalizing in the frequency domain the filter functions driving the designed quantum-sensor. The achievement of these tasks will enhance our capability to observe and manipulate open quantum systems, thus bringing advances to quantum science and technologies.
The challenge to fully grasp and to harness the bizarre phenomena which take place in the quantum realm is crucially linked to the ability to engineer and manipulate quantum systems with high precision. The scientific progress in this direction is opening new routes for the investigation of fundamental quantum physics, and it is at the same time paving the way to the development of the first quantum technologies. \n\nThis thesis is dedicated to the search of strategies for high-fidelity and noise-tolerant quantum control.\nA versatile theoretical framework is set up for the design of optimized control protocols based on accelerated adiabatic processes. The benefits of the adiabatic dynamics, rooted in its intrinsic stability versus control imperfections, are released from the requirement of slow evolution: this is achieved thanks to the theoretical construction of accelerating control Hamiltonians which dynamically counteract deviations from the target dynamics. Our framework provides a methodical way to engineer these Hamiltonians by introducing fast modulations of the initially-available control parameters, and it results from the interplay between the theory of counterdiabatic fields and Floquet-Magnus average Hamiltonian theory. As compared to other shortcut-to-adiabaticity strategies, our control method keeps the system always close to the exact adiabatic path while not requiring the implementation of new Hamiltonian couplings. Moreover, the flexibility in the construction of the accelerating Hamiltonians offers a fertile ground for the hybridization with optimal control techniques.\n\nThe general setup is applied to exemplary control problems of relevance in quantum optics and quantum information processing. First, a protocol is designed for the generation of entanglement between effective two-level systems coupled through a transmission line, in the experimental context of circuit quantum electrodynamics. Second, an accelerated version of the widespread technique of stimulated Raman adiabatic passage is presented, which dramatically enlarges its range of applicability for implementations both in the optical and in the microwave regime.\n\nThe results strongly support the potential of the quantum control framework developed in this thesis. In particular, our methodology offers the opportunity to flank high fidelities and small error sensitivity with on-demand adaptability to experimental constraints and the flexibility for integrations with other control techniques. This makes it a valuable resource for applications in many branches of quantum science, with a natural outlet in quantum computation and simulation.
Realizing coherent quantum computation requires precise and robust manipulation of quantum systems through quantum control protocols. Most quantum control techniques rely on heuristic methods for designing the driving pulses that steer the system towards a target state. Such methods are often based on brute-force optimization and offer limited understanding of the solution landscape. In contrast, quantum algorithms offer a rich body of analytical methods with rigorous error guarantees for implementing unitary and non-unitary transformations, which suggests a promising direction for developing new approaches to quantum control. Among various such algorithms, quantum signal processing (QSP) has emerged as a powerful framework for quantum algorithm design, implementation, and optimization. However, its potential for quantum control remains largely unexplored. In this work, we establish QSP-Control, an analytical framework for quantum control of qubit-oscillator dynamics. We focus on dispersively coupled qubit-oscillator systems and employ the QSP formalism to mitigate unwanted nonlinear effects arising from cross-Kerr interactions. In addition, we develop constructions for precise manipulation of Fock states by designing Fock-state-selective operators, based on structural parallels between the Jaynes-Cummings interaction and QSP. These findings demonstrate how several practically relevant problems in quantum control can be mapped to forms amenable to QSP, offering both a systematic design framework and an interpretable perspective on quantum control.