Quantum-classical Hybrid Systems Research Articles

Hybrid quantum-classical systems: Quasi-free Markovian dynamics

In the case of a quantum-classical hybrid system with a finite number of degrees of freedom, the problem of characterizing the most general dynamical semigroup is solved, under the restriction of being quasi-free. This is a generalization of a Gaussian dynamics, and it is defined by the property of sending (hybrid) Weyl operators into Weyl operators in the Heisenberg description. The result is a quantum generalization of the Lévy–Khintchine formula; Gaussian and jump contributions are included. As a byproduct, the most general quasi-free quantum-dynamical semigroup is obtained; on the classical side the Liouville equation and the Kolmogorov–Fokker–Planck equation are included. As a classical subsystem can be observed, in principle, without perturbing it, information can be extracted from the quantum system, even in continuous time; indeed, the whole construction is related to the theory of quantum measurements in continuous time. While the dynamics is formulated to give the hybrid state at a generic time [Formula: see text], we show how to extract multi-time probabilities and how to connect them to the quantum notions of positive operator-valued measure and instrument. The structure of the generator of the dynamical semigroup is analyzed, in order to understand how to go on to non-quasi-free cases and to understand the possible classical-quantum interactions; in particular, all the interaction terms which allow to extract information from the quantum system necessarily vanish if no dissipation is present in the dynamics of the quantum component. A concrete example is given, showing how a classical component can input noise into a quantum one and how the classical system can extract information on the behavior of the quantum one.

Measurement of cryoelectronics heating using a local quantum dot thermometer in silicon

Silicon technology offers the enticing opportunity for monolithic integration of quantum and classical electronic circuits. However, the power consumption levels of classical electronics may compromise the local chip temperature and hence the fidelity of qubit operations. Here, we utilize a quantum-dot-based thermometer embedded in an industry-standard silicon field-effect transistor (FET), to assess the local temperature increase produced by an active FET placed in close proximity. We study the impact of both static and dynamic operation regimes. When the FET is operated statically, we find a power budget of 45 nW at 100 nm separation whereas at 216 μm the power budget raises to 150 μW. When operated dynamically, we observe negligible temperature increase for the switch frequencies tested up to 10 MHz. Our work describes a method to accurately map out the available power budget at a distance from a solid-state quantum processor and indicate under which conditions cryoelectronics circuits may allow the operation of hybrid quantum-classical systems.

Non-Abelian phase and geometric force in a quantum-classical hybrid system

Non-Abelian phase and geometric force in a quantum-classical hybrid system

Satellite Image Representations for Quantum Classifiers

Existing quantum hardware is limited in the number of bits and length of the series of operations. Nevertheless, by shifting parts of the computation on classical hardware, hybrid quantum-classical systems utilize quantum hardware for scaled-down machine learning approaches, which is quantum machine learning. Due to the theoretically possible computational speed-up of quantum computers compared to classical computers and the increasing volume and speed of data generated in earth observation, attempts are now being made to use quantum computers for satellite image processing. However, satellite imagery is too large and high dimensional, and transformations that reduce the dimensionality are necessary to fit the classical data in the limited input domain of quantum circuits. This paper presents and compares several dimensionality reduction techniques as part of hybrid quantum-classical systems to represent satellite images with up to 256×256×3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$256\ imes 256\ imes 3$$\\end{document} values with only 16 values. We evaluate the representations of two benchmark datasets with supervised classification by four different quantum circuit architectures. We demonstrate the potential use of quantum machine learning for satellite image classification and give a comprehensive overview of the impact of various satellite image representations on the performance of quantum classifiers. It shows that autoencoder models are best suited to create small-scale representations, outperforming commonly used methods such as principle component analysis.

Markovian master equations for quantum-classical hybrid systems

The problem of constructing a consistent quantum-classical hybrid dynamics is afforded in the case of a quantum component in a separable Hilbert space and a continuous, finite-dimensional classical component. In the Markovian case, the problem is formalized by the notion of hybrid dynamical semigroup. A classical component can be observed without perturbing the system and information on the quantum component can be extracted, thanks to the quantum-classical interaction. This point is formalized by showing how to introduce positive operator valued measures and operations compatible with the hybrid dynamical semigroup; in this way the notion of hybrid dynamics is connected to quantum measurements in continuous time. Then, the case of the most general quasi-free generator is presented and the various quantum-classical interaction terms are discussed. To bee quasi-free means to send, in the Heisenberg description, hybrid Weyl operators into multiples of Weyl operators; the results on the structure of quasi-free semigroups were proved in Reference [12]. Even in the pure quantum case, a quasi-free semigroup is not restricted to have only a Gaussian structure, but also jump-type terms are allowed. An important result is that, to have interactions producing a flow of information from the quantum component to the classical one, suitable dissipative terms must be present in the generator. Finally, some possibilities are discussed to go beyond the quasi-free case.

Effective implementation of text{L}{0}-regularised compressed sensing with chaotic-amplitude-controlled coherent Ising machines

Coherent Ising machine (CIM) is a network of optical parametric oscillators that can solve large-scale combinatorial optimisation problems by finding the ground state of an Ising Hamiltonian. As a practical application of CIM, Aonishi et al., proposed a quantum-classical hybrid system to solve optimisation problems of l_0-regularisation-based compressed sensing. In the hybrid system, the CIM was an open-loop system without an amplitude control feedback loop. In this case, the hybrid system is enhanced by using a closed-loop CIM to achieve chaotic behaviour around the target amplitude, which would enable escaping from local minima in the energy landscape. Both artificial and magnetic resonance image data were used for the testing of our proposed closed-loop system. Compared with the open-loop system, the results of this study demonstrate an improved degree of accuracy and a wider range of effectiveness.

Quantum-Classical Hybrid Systems and their Quasifree Transformations

We study continuous variable systems, in which quantum and classical degrees of freedom are combined and treated on the same footing. Thus all systems, including the inputs or outputs to a channel, may be quantum-classical hybrids. This allows a unified treatment of a large variety of quantum operations involving measurements or dependence on classical parameters. The basic variables are given by canonical operators with scalar commutators. Some variables may commute with all others and hence generate a classical subsystem. We systematically study the class of "quasifree" operations, which are characterized equivalently either by an intertwining condition for phase-space translations or by the requirement that, in the Heisenberg picture, Weyl operators are mapped to multiples of Weyl operators. This includes the well-known Gaussian operations, evolutions with quadratic Hamiltonians, and "linear Bosonic channels", but allows for much more general kinds of noise. For example, all states are quasifree. We sketch the analysis of quasifree preparation, measurement, repeated observation, cloning, teleportation, dense coding, the setup for the classical limit, and some aspects of irreversible dynamics, together with the precise salient tradeoffs of uncertainty, error, and disturbance. Although the spaces of observables and states are infinite dimensional for every non-trivial system that we consider, we treat the technicalities related to this in a uniform and conclusive way, providing a calculus that is both easy to use and fully rigorous.

Effective nonlinear Ehrenfest hybrid quantum-classical dynamics

The definition of a consistent evolution equation for statistical hybrid quantum-classical systems is still an open problem. In this paper, we analyze the case of Ehrenfest dynamics on systems defined by a probability density and identify the relations of the nonlinearity of the dynamics with the obstructions to define a consistent dynamics for the first quantum moment of the distribution. This first quantum moment represents the physical states as a family of classically-parametrized density matrices {hat{rho }}(xi ), for xi a classical point; and it is the most common representation of hybrid systems in the literature. Due to this obstruction, we consider higher order quantum moments, and argue that only a finite number of them are physically measurable. Because of this, we propose an effective solution for the hybrid dynamics problem based on approximating the distribution by those moments and representing the states by them.

Quantization of symplectic fibrations and canonical metrics

We relate Berezin–Toeplitz quantization of higher rank vector bundles to quantum-classical hybrid systems and quantization in stages of symplectic fibrations. We apply this picture to the analysis and geometry of vector bundles, including the spectral gap of the Berezin transform and the convergence rate of Donaldson’s iterations toward balanced metrics on stable vector bundles. We also establish refined estimates in the scalar case to compute the rate of Donaldson’s iterations toward balanced metrics on Kähler manifolds with constant scalar curvature.

Multi-Objective Evolutionary Architecture Search for Parameterized Quantum Circuits

Recent work on hybrid quantum-classical machine learning systems has demonstrated success in utilizing parameterized quantum circuits (PQCs) to solve the challenging reinforcement learning (RL) tasks, with provable learning advantages over classical systems, e.g., deep neural networks. While existing work demonstrates and exploits the strength of PQC-based models, the design choices of PQC architectures and the interactions between different quantum circuits on learning tasks are generally underexplored. In this work, we introduce a Multi-objective Evolutionary Architecture Search framework for parameterized quantum circuits (MEAS-PQC), which uses a multi-objective genetic algorithm with quantum-specific configurations to perform efficient searching of optimal PQC architectures. Experimental results show that our method can find architectures that have superior learning performance on three benchmark RL tasks, and are also optimized for additional objectives including reductions in quantum noise and model size. Further analysis of patterns and probability distributions of quantum operations helps identify performance-critical design choices of hybrid quantum-classical learning systems.

Evolution of hybrid quantum–classical wavefunctions

A gauge-invariant wave equation for the dynamics of hybrid quantum–classical systems is formulated by combining the variational setting of Lagrangian paths in continuum theories with Koopman wavefunctions in classical mechanics. We identify gauge transformations with unobservable phase factors in the classical phase-space and we introduce gauge invariance in the variational principle underlying a hybrid wave equation previously proposed by the authors. While the original construction ensures a positive-definite quantum density matrix, the present model also guarantees the same property for the classical Liouville density. After a suitable wavefunction factorization, gauge invariance is achieved by resorting to the classical Lagrangian paths made available by the Madelung transform of Koopman wavefunctions. Due to the appearance of a phase-space analogue of the Berry connection, the new hybrid wave equation is highly nonlinear and it is proposed here as a platform for further developments in quantum–classical dynamics. Indeed, the associated model is Hamiltonian and appears to be the first to ensure a series of consistency properties beyond positivity of quantum and classical densities. For example, the model possesses a quantum–classical Poincaré integral invariant and its special cases include both the mean-field model and the Ehrenfest model from chemical physics.

Hybrid quantum–classical chaotic NEMS

We present an exactly solvable model of a hybrid quantum–classical system of a Nitrogen-Vacancy (NV) center spin (quantum spin) coupled to a nanocantilever (classical) and analyze the enforcement of the regular or chaotic classical dynamics onto the quantum spin dynamics. The main problem we focus in this paper is whether the classical dynamical chaos may induce chaotic effects in the quantum spin dynamics or not. We explore several characteristic criteria of the quantum chaos, such as quantum Poincaré recurrences, generation of coherence and energy level distribution and observe interesting chaotic effects in the spin dynamics. Dynamical chaos imposed in the cantilever dynamics through the kicking pulses induces the stochastic dynamics on the quantum subsystem. We consider a quantum system of two and three levels and show that in a two-level case, type of stochasticity is not conforming all the characteristic features of the quantum chaos and is distinct from it. We also explore the effect of quantum feedback on dynamics of the cantilever and the entire system.

Measure synchronization in hybrid quantum–classical systems

Measure synchronization in hybrid quantum–classical systems is investigated in this paper. The dynamics of the classical subsystem is described by the Hamiltonian equations, while the dynamics of the quantum subsystem is governed by the Schrödinger equation. By increasing the coupling strength in between the quantum and classical subsystems, we reveal the existence of measure synchronization in coupled quantum–classical dynamics under energy conservation for the hybrid systems.

Evolution of Hybrid Quantum-Classical Wavefunctions

A gauge-invariant wave equation for the dynamics of hybrid quantum-classical systems is formulated by combining the variational setting of Lagrangian paths in continuum theories with Koopman wavefunctions in classical mechanics. We identify gauge transformations with unobservable phase factors in the classical phase-space and we introduce gauge invariance in the variational principle underlying a hybrid wave equation previously proposed by the authors. While the original construction ensures a positive-definite quantum density matrix, the present model also guarantees the same property for the classical Liouville density. After a suitable wavefunction factorization, gauge invariance is achieved by resorting to the classical Lagrangian paths made available by the Madelung transform of Koopman wavefunctions. Due to the appearance of a phase-space analogue of the Berry connection, the new hybrid wave equation is highly nonlinear and it is proposed here as a platform for further developments in quantum-classical dynamics. Indeed, the associated model is Hamiltonian and appears to be the first to ensure a series of consistency properties beyond positivity of quantum and classical densities. For example, the model possesses a quantum-classical Poincar\'e integral invariant and its special cases include both the mean-field model and the Ehrenfest model from chemical physics.

Survey of NISQ Era Hybrid Quantum-Classical Machine Learning Research

Quantum computing is a rapidly growing field that has received a significant amount of support in the past decade in industry and academia. Several physical quantum computers are now freely available to use through cloud services, with some implementations supporting upwards of hundreds of qubits. These advances mark the beginning of the Noisy Intermediate-Scale Quantum (NISQ) era of quantum computing, paving the way for hybrid quantum-classical systems. This work provides an introductory overview of gate-model quantum computing through the Visual IoT/Robotics Programming Language Environment and a survey of recent applications of NISQ era quantum computers to hybrid quantum-classical machine learning.

About the computation of finite temperature ensemble averages of hybrid quantum-classical systems with molecular dynamics

Molecular or condensed matter systems are often well approximated by hybrid quantum-classical models: the electrons retain their quantum character, whereas the ions are considered to be classical particles. We discuss various alternative approaches for the computation of equilibrium (canonical) ensemble averages for observables of these hybrid quantum-classical systems through the use of molecular dynamics (MD)-i.e. by performing dynamics in the presence of a thermostat and computing time-averages over the trajectories. Often, in classical or ab initio MD, the temperature of the electrons is ignored and they are assumed to remain at the instantaneous ground state given by each ionic configuration during the evolution. Here, however, we discuss the general case that considers both classical and quantum subsystems at finite temperature canonical equilibrium. Inspired by a recent formal derivation for the canonical ensemble for quantum classical hybrids, we discuss previous approaches found in the literature, and provide some new formulas.

Enhancement of quantum correlations and a geometric phase for a driven bipartite quantum system in a structured environment

We study the role of driving in an initial maximally entangled state evolving under the presence of a structured environment in a weak and strong regime. We focus on the enhancement and degradation of maximal concurrence when the system is driven on and out of resonance for a general evolution, as well as the effect of adding a transverse coupling among the particles of the model. We further investigate the role of driving in the acquisition of a geometric phase for the maximally entangled state. As the model studied herein can be used to model experimental situations such as hybrid quantum classical systems feasible with current technologies, this knowledge can aid the search for physical setups that best retain quantum properties under dissipative dynamics.

Spintronics Meets Nonadiabatic Molecular Dynamics: Geometric Spin Torque and Damping on Dynamical Classical Magnetic Texture due to an Electronic Open Quantum System.

We analyze a quantum-classical hybrid system of steadily precessing around the fixed axis slow classical localized magnetic moments (LMMs), forming a head-to-head domain wall, surrounded by fast electrons driven out of equilibrium by LMMs and residing within a metallic wire whose connection to macroscopic reservoirs makes electronic quantum system an open one. The model captures the essence of dynamical noncollinear magnetic textures encountered in spintronics, while making it possible to obtain the exact time-dependent nonequilibrium density matrix of electronic systems and split it into four contributions. The Fermi surface contribution generates dissipative (or dampinglike in spintronics terminology) spin torque on LMMs, as the counterpart of electronic friction in nonadiabatic molecular dynamics (MD). Among two Fermi sea contributions, one generates geometric torque dominating in the adiabatic regime, which remains as the only nonzero contribution in a closed system with disconnected reservoirs. Locally geometric torque can have nondissipative (or fieldlike in spintronics terminology) component, acting as the counterpart of geometric magnetism force in nonadiabatic MD, as well as a much smaller dampinglike component acting as "geometric friction." Such current-independent geometric torque is absent from widely used micromagnetics or atomistic spin dynamics modeling of magnetization dynamics based on the Landau-Lifshitz-Gilbert equation, while previous analyses of how to include our Fermi-surface dampinglike torque have severely underestimated its total magnitude.

Galilean covariance of quantum-classical hybrid systems of the Sudarshan type

We revisit quantum-classical hybrid systems of the Sudarshan type under the light of Galilean covariance. We show that these kind of hybrids cannot be given as a unitary representation of the Galilei group and at the same time conserve the total linear momentum unless the interaction term only depends on the relative canonical velocities.

Entropy and canonical ensemble of hybrid quantum classical systems

In this work we generalize and combine Gibbs and von Neumann approaches to build, for the first time, a rigorous definition of entropy for hybrid quantum-classical systems. The resulting function coincides with the two cases above when the suitable limits are considered. Then, we apply the MaxEnt principle for this hybrid entropy function and obtain the natural candidate for the hybrid canonical ensemble (HCE). We prove that the suitable classical and quantum limits of the HCE coincide with the usual classical and quantum canonical ensembles since the whole scheme admits both limits, thus showing that the MaxEnt principle is applicable and consistent for hybrid systems.