Controlling open quantum systems: tools, achievements, and limitations

Optimal Control in Large Open Quantum Systems: The Case of Transmon Readout and Reset

We present a framework that combines the adjoint-state method together with reverse-time backpropagation to solve prohibitively large open-system quantum control problems. Our approach enables the optimization of arbitrary cost functions with fully general controls applied on large open quantum systems described by a Lindblad master equation. It is scalable, computationally efficient, and has a low-memory footprint. We apply this framework to optimize two inherently dissipative operations in superconducting qubits which lag behind in terms of fidelity and duration compared to other unitary operations: the dispersive readout and all-microwave reset of a transmon qubit. Our results show that while standard pulses for dispersive readout are nearly optimal, adding a transmon drive during the protocol can yield 2× improvements in fidelity and duration. We further demonstrate a 2× improvement in reset fidelity and duration through pulse shaping, indicating significant potential for enhancement in reset protocols. Our approach can readily be applied to optimize quantum controls in a vast range of applications such as reservoir engineering, autonomous quantum error correction, and leakage-reduction units.

Combining the generalized quantum master equation approach with quasiclassical mapping Hamiltonian methods to simulate the dynamics of electronic coherences

The generalized quantum master equation (GQME) approach provides a powerful general-purpose framework for simulating the inherently quantum mechanical dynamics of a subset of electronic reduced density matrix elements of interest in complex molecular systems. Previous studies have found that combining the GQME approach with quasiclassical mapping Hamiltonian (QC/MH) methods can dramatically improve the accuracy of electronic populations obtained via those methods. In this paper, we perform a complimentary study of the advantages offered by the GQME approach for simulating the dynamics of electronic coherences, which play a central role in optical spectroscopy, quantum information science, and quantum technology. To this end, we focus on cases where the electronic coherences predicted for the spin-boson benchmark model by direct application of various QC/MH methods are inaccurate. We find that similar to the case of electronic populations, combining the QC/MH methods with the GQME approach can dramatically improve the accuracy of the electronic coherences obtained via those methods. We also provide a comprehensive analysis of how the performance of GQMEs depends on the choice of projection operator and electronic basis and show that the accuracy and feasibility of the GQME approach can benefit from casting the GQME in terms of the eigen-basis of the observable of interest.

Unifying methods for optimal control in non-Markovian quantum systems via process tensors

The large dimensionality of environments is the limiting factor in applying optimal control to open quantum systems beyond the Markovian approximation. Various methods exist to simulate non-Markovian systems, which effectively reduce the environment to a number of active degrees of freedom. Here, we show that several of these methods can be expressed in terms of a process tensor in the form of a matrix-product-operator, which serves as a unifying framework to show how they can be used in optimal control and to compare their performance. The matrix-product-operator form provides a general scheme for computing gradients using back propagation and allows the efficiency of the different methods to be compared via the bond dimensions of their respective process tensors.

Bexcitonics: Quasiparticle approach to open quantum dynamics

We develop a quasiparticle approach to capture the dynamics of open quantum systems coupled to bosonic thermal baths of arbitrary complexity based on the Hierarchical Equations of Motion (HEOM). This is done by generalizing the HEOM dynamics and mapping it into that of the system in interaction with a few bosonic fictitious quasiparticles that we call bexcitons. Bexcitons arise from a decomposition of the bath correlation function into discrete features. Specifically, bexciton creation and annihilation couple the auxiliary density matrices in the HEOM. The approach provides a systematic strategy to construct exact quantum master equations that include the system-bath coupling to all orders even for non-Markovian environments. Specifically, by introducing different metrics and representations for the bexcitons it is possible to straightforwardly generate different variants of the HEOM, demonstrating that all these variants share a common underlying quasiparticle picture. Bexcitonic properties, while unphysical, offer a coarse-grained view of the correlated system-bath dynamics and its numerical convergence. For instance, we use it to analyze the instability of the HEOM when the bath is composed of underdamped oscillators and show that it leads to the creation of highly excited bexcitons. The bexcitonic picture can also be used to develop more efficient approaches to propagate the HEOM. As an example, we use the particle-like nature of the bexcitons to introduce mode-combination of bexcitons in both number and coordinate representation that uses the multi-configuration time-dependent Hartree to efficiently propagate the HEOM dynamics.

Optimizing Performance of Quantum Operations with Non-Markovian Decoherence: The Tortoise or the Hare?

The interaction between a quantum system and its environment limits our ability to control it and perform quantum operations on it. We present an efficient method to find optimal controls for quantum systems coupled to non-Markovian environments, by using the process tensor to compute the gradient of an objective function. We consider state transfer for a driven two-level system coupled to a bosonic environment, and characterize performance in terms of speed and fidelity. This allows us to determine the best achievable fidelity as a function of process duration. We show there can be a trade-off between speed and fidelity, and that slower processes can have higher fidelity by exploiting non-Markovian effects.

Control of the von Neumann Entropy for an Open Two-Qubit System Using Coherent and Incoherent Drives

This article is devoted to developing an approach for manipulating the von Neumann entropy S(ρ(t)) of an open two-qubit system with coherent control and incoherent control inducing time-dependent decoherence rates. The following goals are considered: (a) minimizing or maximizing the final entropy S(ρ(T)); (b) steering S(ρ(T)) to a given target value; (c) steering S(ρ(T)) to a target value and satisfying the pointwise state constraint S(ρ(t))≤S¯ for a given S¯; (d) keeping S(ρ(t)) constant at a given time interval. Under the Markovian dynamics determined by a Gorini-Kossakowski-Sudarshan-Lindblad type master equation, which contains coherent and incoherent controls, one- and two-step gradient projection methods and genetic algorithm have been adapted, taking into account the specifics of the objective functionals. The corresponding numerical results are provided and discussed.

Parity-Time Symmetric Holographic Principle

Originating from the Hamiltonian of a single qubit system, the phenomenon of the avoided level crossing is ubiquitous in multiple branches of physics, including the Landau-Zener transition in atomic, molecular, and optical physics, the band structure of condensed matter physics and the dispersion relation of relativistic quantum physics. We revisit this fundamental phenomenon in the simple example of a spinless relativistic quantum particle traveling in (1+1)-dimensional space-time and establish its relation to a spin-1/2 system evolving under a PT-symmetric Hamiltonian. This relation allows us to simulate 1-dimensional eigenvalue problems with a single qubit. Generalizing this relation to the eigenenergy problem of a bulk system with N spatial dimensions reveals that its eigenvalue problem can be mapped onto the time evolution of the edge state with (N-1) spatial dimensions governed by a non-Hermitian Hamiltonian. In other words, the bulk eigenenergy state is encoded in the edge state as a hologram, which can be decoded by the propagation of the edge state in the temporal dimension. We argue that the evolution will be PT-symmetric as long as the bulk system admits parity symmetry. Our work finds the application of PT-symmetric and non-Hermitian physics in quantum simulation and provides insights into the fundamental symmetries.

Convex Optimization for Nonequilibrium Steady States on a Hybrid Quantum Processor

Finding the transient and steady state properties of open quantum systems is a central problem in various fields of quantum technologies. Here, we present a quantum-assisted algorithm to determine the steady states of open system dynamics. By reformulating the problem of finding the fixed point of Lindblad dynamics as a feasibility semidefinite program, we bypass several well-known issues with variational quantum approaches to solving for steady states. We demonstrate that our hybrid approach allows us to estimate the steady states of higher dimensional open quantum systems and discuss how our method can find multiple steady states for systems with symmetries.

The Surprising Ease of Finding Optimal Solutions for Controlling Nonlinear Phenomena in Quantum and Classical Complex Systems

This Perspective addresses the often observed surprising ease of achieving optimal control of nonlinear phenomena in quantum and classical complex systems. The circumstances involved are wide-ranging, with scenarios including manipulation of atomic scale processes, maximization of chemical and material properties or synthesis yields, Nature's optimization of species' populations by natural selection, and directed evolution. Natural evolution will mainly be discussed in terms of laboratory experiments with microorganisms, and the field is also distinct from the other domains where a scientist specifies the goal(s) and oversees the control process. We use the word "control" in reference to all of the available variables, regardless of the circumstance. The empirical observations on the ease of achieving at least good, if not excellent, control in diverse domains of science raise the question of why this occurs despite the generally inherent complexity of the systems in each scenario. The key to addressing the question lies in examining the associated control landscape, which is defined as the optimization objective as a function of the control variables that can be as diverse as the phenomena under consideration. Controls may range from laser pulses, chemical reagents, chemical processing conditions, out to nucleic acids in the genome and more. This Perspective presents a conjecture, based on present findings, that the systematics of readily finding good outcomes from controlled phenomena may be unified through consideration of control landscapes with the same common set of three underlying assumptions─the existence of an optimal solution, the ability for local movement on the landscape, and the availability of sufficient control resources─whose validity needs assessment in each scenario. In practice, many cases permit using myopic gradient-like algorithms while other circumstances utilize algorithms having some elements of stochasticity or introduced noise, depending on whether the landscape is locally smooth or rough. The overarching observation is that only relatively short searches are required despite the common high dimensionality of the available controls in typical scenarios.

Information theoretical limits for quantum optimal control solutions: error scaling of noisy control channels

Accurate manipulations of an open quantum system require a deep knowledge of its controllability properties and the information content of the implemented control fields. By using tools of information and quantum optimal control theory, we provide analytical bounds (information-time bounds) to characterize our capability to control the system when subject to arbitrary sources of noise. Moreover, since the presence of an external noise field induces open quantum system dynamics, we also show that the results provided by the information-time bounds are in very good agreement with the Kofman-Kurizki universal formula describing decoherence processes. Finally, we numerically test the scaling of the control accuracy as a function of the noise parameters, by means of the dressed chopped random basis (dCRAB) algorithm for quantum optimal control.

Anti-Zeno purification of spin baths by quantum probe measurements

The quantum Zeno and anti-Zeno paradigms have thus far addressed the evolution control of a quantum system coupled to an immutable bath via non-selective measurements performed at appropriate intervals. We fundamentally modify these paradigms by introducing, theoretically and experimentally, the concept of controlling the bath state via selective measurements of the system (a qubit). We show that at intervals corresponding to the anti-Zeno regime of the system-bath exchange, a sequence of measurements has strongly correlated outcomes. These correlations can dramatically enhance the bath-state purity and yield a low-entropy steady state of the bath. The purified bath state persists long after the measurements are completed. Such purification enables the exploitation of spin baths as long-lived quantum memories or as quantum-enhanced sensors. The experiment involved a repeatedly probed defect center dephased by a nuclear spin bath in a diamond at low-temperature.

Resonant and non-resonant optimizations by multi-constraint quantum control theory in molecular rotational states

It is a promising research for optimization of quantum gate in the field of quantum computation. We investigate the feasibility of implementing the single-qubit gate (Hadamard) in molecular rotational system. By applying the Multi-constraint quantum optimal control method, the excepted final states can be achieved based on the molecular rotational states both in resonant and non-resonant cases with the control pulses. The permanent electric dipole moment is ignored in non-resonance. Besides, the zero-pulse area constraint and the constant fluence constraint are employed to optimize shapes of control pulses. Finally, we show that the Hadamard gate can be realized with the high fidelity (0.9999) and also examine the dependence of the fidelity on pulse fluence as well as the control pulse.

Controlling the uncontrollable: Quantum control of open-system dynamics

Control of open quantum systems is essential for the realization of contemporary quantum science and technology. We demonstrate such control using a thermodynamically consistent framework, taking into account the fact that the drive can modify the system's interaction with the environment. Such an effect is incorporated within the dynamical equation, leading to control-dependent dissipation. This relation serves as the key element for open-system control. The control paradigm is displayed by analyzing entropy-changing state-to-state transformations, such as heating and cooling. The difficult task of controlling quantum gates is achieved for nonunitary reset maps with complete memory loss. In addition, we identify a mechanism for controlling unitary gates by actively removing entropy from the system to the environment. We demonstrate a universal set of single- and double-qubit unitary gates under dissipation.

Highly Adiabatic Time-Optimal Quantum Driving at Low Energy Cost

Time-efficient control schemes for manipulating quantum systems are of great importance in quantum technologies, where environmental forces rapidly degrade the quality of pure states over time. In this Letter, we formulate an approach to time-optimal control that circumvents the boundary-value problem that plagues the quantum brachistochrone equation at the expense of relaxing the form of the control Hamiltonian. In this setting, a coupled system of equations, one for the control Hamiltonian and another one for the duration of the protocol, realizes an ansatz-free approach to quantum control theory. We show how driven systems, in the form of a Landau-Zener type Hamiltonian, can be efficiently maneuvered to speed up a given state transformation in a highly adiabatic manner and with a low energy cost.

Optimization two-qubit quantum gate by two optical control methods in molecular pendular states

Implementation of quantum gates are important for quantum computations in physical system made of polar molecules. We investigate the feasibility of implementing gates based on pendular states of the molecular system by two different quantum optical control methods. Firstly, the Multi-Target optimal control theory and the Multi-Constraint optimal control theory are described for optimizing control fields and accomplish the optimization of quantum gates. Numerical results show that the controlled NOT gate (CNOT) can be realized under the control of above methods with high fidelities (0.975 and 0.999) respectively. In addition, in order to examine the dependence of the fidelity on energy difference in the same molecular system, the SWAP gate in the molecular system is also optimized with high fidelity (0.999) by the Multi-Constraint optimal control theory with the zero-area and constant-fluence constraints.

Optimal Control of Families of Quantum Gates

Quantum optimal control (QOC) enables the realization of accurate operations, such as quantum gates, and supports the development of quantum technologies. To date, many QOC frameworks have been developed, but those remain only naturally suited to optimize a single targeted operation at a time. We extend this concept to optimal control with a continuous family of targets, and demonstrate that an optimization based on neural networks can find families of time-dependent Hamiltonians realizing desired classes of quantum gates in minimal time.

One decade of quantum optimal control in the chopped random basis

The chopped random basis (CRAB) ansatz for quantum optimal control has been proven to be a versatile tool to enable quantum technology applications such as quantum computing, quantum simulation, quantum sensing, and quantum communication. Its capability to encompass experimental constraints-while maintaining an access to the usually trap-free control landscape-and to switch from open-loop to closed-loop optimization (including with remote access-or RedCRAB) is contributing to the development of quantum technology on many different physical platforms. In this review article we present the development, the theoretical basis and the toolbox for this optimization algorithm, as well as an overview of the broad range of different theoretical and experimental applications that exploit this powerful technique.

Laser control of a dark vibrational state of acetylene in the gas phase—Fourier transform pulse shaping constraints and effects of decoherence

We propose a methodology to tackle the laser control of a non-stationary dark ro-vibrational state of acetylene (C2H2), given realistic experimental limitations in the 7.7 μm (1300 cm−1) region. Simulations are performed using the Lindblad master equation, where the so-called Lindblad parameters are used to describe the effect of the environment in the dilute gas phase. A phenomenological representation of the parameters is used, and they are extracted from high-resolution spectroscopy line broadening data. An effective Hamiltonian is used for the description of the system down to the rotational level close to experimental accuracy. The quality of both the Hamiltonian and Lindblad parameters is assessed by a comparison of a calculated infrared spectrum with the available experimental data. A single shaped laser pulse is used to perform the control, where elements of optics and pulse shaping using masks are introduced with emphasis on experimental limitations. The optimization procedure, based on gradients, explicitly takes into account the experimental constraints. Control performances are reported for shaping masks of increasing complexity. Although modest performances are obtained, mainly due to the strong pulse shaping constraints, we gain insights into the control mechanism. This work is the first step toward the conception of a realistic experiment that will allow for population characterization and manipulation of a non-stationary vibrational “dark” state. Effects of the collisions on the laser control in the dilute gas phase, leading to decoherence in the molecular system, are clearly shown.

Shortcuts to adiabaticity for open systems in circuit quantum electrodynamics

Shortcuts to adiabaticity are powerful quantum control methods, allowing quick evolution into target states of otherwise slow adiabatic dynamics. Such methods have widespread applications in quantum technologies, and various shortcuts to adiabaticity protocols have been demonstrated in closed systems. However, realizing shortcuts to adiabaticity for open quantum systems has presented a challenge due to the complex controls in existing proposals. Here, we present the experimental demonstration of shortcuts to adiabaticity for open quantum systems, using a superconducting circuit quantum electrodynamics system. By applying a counterdiabatic driving pulse, we reduce the adiabatic evolution time of a single lossy mode from 800 ns to 100 ns. In addition, we propose and implement an optimal control protocol to achieve fast and qubit-unconditional equilibrium of multiple lossy modes. Our results pave the way for precise time-domain control of open quantum systems and have potential applications in designing fast open-system protocols of physical and interdisciplinary interest, such as accelerating bioengineering and chemical reaction dynamics.

High-Efficiency Arbitrary Quantum Operation on a High-Dimensional Quantum System

The ability to manipulate quantum systems lies at the heart of the development of quantum technology. The ultimate goal of quantum control is to realize arbitrary quantum operations (AQUOs) for all possible open quantum system dynamics. However, the demanding extra physical resources impose great obstacles. Here, we experimentally demonstrate a universal approach of AQUO on a photonic qudit with the minimum physical resource of a two-level ancilla and a log_{2}d-scale circuit depth for a d-dimensional system. The AQUO is then applied in a quantum trajectory simulation for quantum subspace stabilization and quantum Zeno dynamics, as well as incoherent manipulation and generalized measurements of the qudit. Therefore, the demonstrated AQUO for complete quantum control would play an indispensable role in quantum information science.

Experimental Simulation of Open Quantum System Dynamics via Trotterization

Digital quantum simulators provide a diversified tool for solving the evolution of quantum systems with complicated Hamiltonians and hold great potential for a wide range of applications. Although much attention is paid to the unitary evolution of closed quantum systems, dissipation and noise are vital in understanding the dynamics of practical quantum systems. In this work, we experimentally demonstrate a digital simulation of an open quantum system in a controllable Markovian environment with the assistance of a single ancillary qubit. By Trotterizing the quantum Liouvillians, the continuous evolution of an open quantum system is effectively realized, and its application in error mitigation is demonstrated by adjusting the simulated noise intensities. High-order Trotter for open quantum dynamics is also experimentally investigated and shows higher accuracy. Our results represent a significant step toward hardware-efficient simulation of open quantum systems and error mitigation in quantum algorithms in noisy intermediate-scale quantum systems.

Efficient Exploration of Hamiltonian Parameter Space for Optimal Control of Non-Markovian Open Quantum Systems

We present a general method to efficiently design optimal control sequences for non-Markovian open quantum systems, and illustrate it by optimizing the shape of a laser pulse to prepare a quantum dot in a specific state. The optimization of control procedures for quantum systems with strong coupling to structured environments-where time-local descriptions fail-is a computationally challenging task. We modify the numerically exact time evolving matrix product operator (TEMPO) method, such that it allows the repeated computation of the time evolution of the reduced system density matrix for various sets of control parameters at very low computational cost. This method is potentially useful for studying numerous optimal control problems, in particular in solid state quantum devices where the coupling to vibrational modes is typically strong.

Selective Enhancement of Spectroscopic Features by Quantum Optimal Control

Tailored light can be used to steer atomic motions into selected quantum pathways. In optimal control theory (OCT), the target is usually expressed in terms of the molecular wave function, a quantity that is not directly observable in experiment. We present simulations using OCT that optimize the spectroscopic signal itself. By shaping the optical pump, the x-ray stimulated Raman signal, which occurs solely during the passage through conical intersections, is temporally controlled and amplified by up to 2 orders of magnitude. This enhancement can be crucial in order to bring small coherence-based signatures above the detectable threshold. Our approach is applicable to any signal that depends on the expectation value of a positive definite operator.

Lindblad parameters from high resolution spectroscopy to describe collision-induced rovibrational decoherence in the gas phase-Application to acetylene

Within the framework of the Lindblad master equation, we propose a general methodology to describe the effects of the environment on a system in the dilute gas phase. The phenomenological parameters characterizing the transitions between rovibrational states of the system induced by collisions can be extracted from experimental transition kinetic constants, relying on energy gap fitting laws. As the availability of these kinds of experimental data can be limited, this work relied on experimental line broadening coefficients, however still using energy gap fitting laws. The 3 μm infrared spectral range of acetylene was chosen to illustrate the proposed approach. The method shows fair agreement with available experimental data while being computationally inexpensive. The results are discussed in the context of state laser quantum control.

Geometric Control of Collective Spontaneous Emission

Dipole spin-wave states of atomic ensembles with wave vector k(ω) mismatched from the dispersion relation of light are difficult to access by far-field excitation but may support rich phenomena beyond the traditional phase-matched scenario in quantum optics. We propose and demonstrate an optical technique to efficiently access these states. In particular, subnanosecond laser pulses shaped by a home-developed wideband modulation method are applied to shift the spin wave in k space with state-dependent geometric phase patterning, in an error-resilient fashion and on timescales much faster than spontaneous emission. We verify this control through the redirection, switch off, and recall of collectively enhanced emission from a ^{87}Rb gas with ∼75% single-step efficiency. Our work represents a first step toward efficient control of electric dipole spin waves for studying many-body dissipative dynamics of excited gases, as well as for numerous quantum optical applications.

Persistent dark states in anisotropic central spin models

Long-lived dark states, in which an experimentally accessible qubit is not in thermal equilibrium with a surrounding spin bath, are pervasive in solid-state systems. We explain the ubiquity of dark states in a large class of inhomogeneous central spin models using the proximity to integrable lines with exact dark eigenstates. At numerically accessible sizes, dark states persist as eigenstates at large deviations from integrability, and the qubit retains memory of its initial polarization at long times. Although the eigenstates of the system are chaotic, exhibiting exponential sensitivity to small perturbations, they do not satisfy the eigenstate thermalization hypothesis. Rather, we predict long relaxation times that increase exponentially with system size. We propose that this intermediate chaotic but non-ergodic regime characterizes mesoscopic quantum dot and diamond defect systems, as we see no numerical tendency towards conventional thermalization with a finite relaxation time.

Using Matrix-Product States for Open Quantum Many-Body Systems: Efficient Algorithms for Markovian and Non-Markovian Time-Evolution

This paper presents an efficient algorithm for the time evolution of open quantum many-body systems using matrix-product states (MPS) proposing a convenient structure of the MPS-architecture, which exploits the initial state of system and reservoir. By doing so, numerically expensive re-ordering protocols are circumvented. It is applicable to systems with a Markovian type of interaction, where only the present state of the reservoir needs to be taken into account. Its adaption to a non-Markovian type of interaction between the many-body system and the reservoir is demonstrated, where the information backflow from the reservoir needs to be included in the computation. Also, the derivation of the basis in the quantum stochastic Schrödinger picture is shown. As a paradigmatic model, the Heisenberg spin chain with nearest-neighbor interaction is used. It is demonstrated that the algorithm allows for the access of large systems sizes. As an example for a non-Markovian type of interaction, the generation of highly unusual steady states in the many-body system with coherent feedback control is demonstrated for a chain length of N=30.

Exploiting chemistry and molecular systems for quantum information science

The power of chemistry to prepare new molecules and materials has driven the quest for new approaches to solve problems having global societal impact, such as in renewable energy, healthcare and information science. In the latter case, the intrinsic quantum nature of the electronic, nuclear and spin degrees of freedom in molecules offers intriguing new possibilities to advance the emerging field of quantum information science. In this Perspective, which resulted from discussions by the co-authors at a US Department of Energy workshop held in November 2018, we discuss how chemical systems and reactions can impact quantum computing, communication and sensing. Hierarchical molecular design and synthesis, from small molecules to supramolecular assemblies, combined with new spectroscopic probes of quantum coherence and theoretical modelling of complex systems, offer a broad range of possibilities to realize practical quantum information science applications.

An Optimal Control Framework for the Automated Design of Personalized Cancer Treatments

One of the key challenges in current cancer research is the development of computational strategies to support clinicians in the identification of successful personalized treatments. Control theory might be an effective approach to this end, as proven by the long-established application to therapy design and testing. In this respect, we here introduce the Control Theory for Therapy Design (CT4TD) framework, which employs optimal control theory on patient-specific pharmacokinetics (PK) and pharmacodynamics (PD) models, to deliver optimized therapeutic strategies. The definition of personalized PK/PD models allows to explicitly consider the physiological heterogeneity of individuals and to adapt the therapy accordingly, as opposed to standard clinical practices. CT4TD can be used in two distinct scenarios. At the time of the diagnosis, CT4TD allows to set optimized personalized administration strategies, aimed at reaching selected target drug concentrations, while minimizing the costs in terms of toxicity and adverse effects. Moreover, if longitudinal data on patients under treatment are available, our approach allows to adjust the ongoing therapy, by relying on simplified models of cancer population dynamics, with the goal of minimizing or controlling the tumor burden. CT4TD is highly scalable, as it employs the efficient dCRAB/RedCRAB optimization algorithm, and the results are robust, as proven by extensive tests on synthetic data. Furthermore, the theoretical framework is general, and it might be applied to any therapy for which a PK/PD model can be estimated, and for any kind of administration and cost. As a proof of principle, we present the application of CT4TD to Imatinib administration in Chronic Myeloid leukemia, in which we adopt a simplified model of cancer population dynamics. In particular, we show that the optimized therapeutic strategies are diversified among patients, and display improvements with respect to the current standard regime.

Generation and manipulation of Schrödinger cat states in Rydberg atom arrays

Quantum entanglement involving coherent superpositions of macroscopically distinct states is among the most striking features of quantum theory, but its realization is challenging because such states are extremely fragile. Using a programmable quantum simulator based on neutral atom arrays with interactions mediated by Rydberg states, we demonstrate the creation of "Schrödinger cat" states of the Greenberger-Horne-Zeilinger (GHZ) type with up to 20 qubits. Our approach is based on engineering the energy spectrum and using optimal control of the many-body system. We further demonstrate entanglement manipulation by using GHZ states to distribute entanglement to distant sites in the array, establishing important ingredients for quantum information processing and quantum metrology.

Uncomputability and complexity of quantum control

In laboratory and numerical experiments, physical quantities are known with a finite precision and described by rational numbers. Based on this, we deduce that quantum control problems both for open and closed systems are in general not algorithmically solvable, i.e., there is no algorithm that can decide whether dynamics of an arbitrary quantum system can be manipulated by accessible external interactions (coherent or dissipative) such that a chosen target reaches a desired value. This conclusion holds even for the relaxed requirement of the target only approximately attaining the desired value. These findings do not preclude an algorithmic solvability for a particular class of quantum control problems. Moreover, any quantum control problem can be made algorithmically solvable if the set of accessible interactions (i.e., controls) is rich enough. To arrive at these results, we develop a technique based on establishing the equivalence between quantum control problems and Diophantine equations, which are polynomial equations with integer coefficients and integer unknowns. In addition to proving uncomputability, this technique allows to construct quantum control problems belonging to different complexity classes. In particular, an example of the control problem involving a two-mode coherent field is shown to be NP-hard, contradicting a widely held believe that two-body problems are easy.

Shortcut to Equilibration of an Open Quantum System

We present a procedure to accelerate the relaxation of an open quantum system towards its equilibrium state. The control protocol, termed the shortcut to equilibration, is obtained by reverse-engineering the nonadiabatic master equation. This is a nonunitary control task aimed at rapidly changing the entropy of the system. Such a protocol serves as a shortcut to an abrupt change in the Hamiltonian, i.e., a quench. As an example, we study the thermalization of a particle in a harmonic well. We observe that for short protocols the accuracy improves by 3 orders of magnitude.

Quantum thermodynamics and open-systems modeling

A comprehensive approach to modeling open quantum systems consistent with thermodynamics is presented. The theory of open quantum systems is employed to define system bath partitions. The Markovian master equation defines an isothermal partition between the system and bath. Two methods to derive the quantum master equation are described: the weak coupling limit and the repeated collision model. The role of the eigenoperators of the free system dynamics is highlighted, in particular, for driven systems. The thermodynamical relations are pointed out. Models that lead to loss of coherence, i.e., dephasing are described. The implication of the laws of thermodynamics to simulating transport and spectroscopy is described. The indications for self-averaging in large quantum systems and thus its importance in modeling are described. Basic modeling by the surrogate Hamiltonian is described, as well as thermal boundary conditions using the repeated collision model and their use in the stochastic surrogate Hamiltonian. The problem of modeling with explicitly time dependent driving is analyzed. Finally, the use of the stochastic surrogate Hamiltonian for modeling ultrafast spectroscopy and quantum control is reviewed.

Pathways to New Applications for Quantum Control

In 1998, the first successful quantum control experiment with application to a molecular framework was conducted with a shaped laser pulse, optimizing the branching ratio between different organometallic reaction channels. This work induced a vast activity in quantum control during the next 10 years, and different optimization aims were achieved in the gas phase, liquid phase, and even in biologically relevant molecules like light-harvesting complexes. Accompanying and preceding this development were important advances in theoretical quantum control simulations. They predicted several control scenarios and explained how and why quantum control experiments work. After many successful proofs of concept in molecular science, the big challenge is to expand its huge conceptual potential of directly being able to steer nuclear and/or electronic motion to more applied implementations. In this Account, based on several recent advances, we give a personal evaluation of where the field of molecular quantum control is at the moment and especially where we think promising applications can be in the near future. One of these paths leads to synthetic chemistry. The synthesis of novel pharmaceutical compounds or natural products often involves many synthetic steps, each one devouring resources and lowering the product yield. Shaped laser pulses can possibly act as photonic reagents and shorten the synthetic route toward the desired product. Chemical synthesis usually takes place in solution, and by including explicit solvent molecules in our quantum control simulations, we were able to identify their highly inhomogeneous influence on chemical reactions and how this affects potential quantum control. More important, we demonstrated for a synthetically relevant example that these complications can be overcome in theory, and laser pulses can be optimized to initiate the desired carbon-carbon bond formation. Putting this into context with the recently emerging concept of flow chemistry, which brings several practical advantages to the application of laser pulses, we want to encourage experimental groups to exploit this concept. Another path was opened by several additions to the commonly used laser pulse optimization algorithm (optimal control theory, OCT), several of which were developed in our group. The OCT algorithm as such is a purely mathematical optimization procedure, with no direct relation to experimental requirements. This means that usually the electric fields obtained out of OCT optimizations do not resemble laser pulses that can be achieved experimentally. However, the previously mentioned additions are aimed at closing the gap toward the experiment. In a recent quantum control study of our group, these algorithmic developments came to fruition. We were able to suggest a shaped laser pulse which can induce a long-living wave packet in the excited state of uracil. This might pave the way for novel experiments dedicated to investigating the formation of biological photodamage in DNA and RNA. The pulse we suggest is surprisingly simple because of the extended OCT algorithm and fulfills all criteria to be experimentally accessible.

Horizons of cybernetical physics

The subject and main areas of a new research field-cybernetical physics-are discussed. A brief history of cybernetical physics is outlined. The main areas of activity in cybernetical physics are briefly surveyed, such as control of oscillatory and chaotic behaviour, control of resonance and synchronization, control in thermodynamics, control of distributed systems and networks, quantum control.This article is part of the themed issue 'Horizons of cybernetical physics'.