Completely integrable Hamiltonian systems look promising for controllability since their first integrals are stable under an internal evolution, and one may hope to find a perturbation of a Hamiltonian which drives the first integrals at will. Action-angle coordinates are most convenient for this purpose. Written with respect to these coordinates, a Hamiltonian and first integrals depend only on the action ones. We introduce a suitable perturbation of a Hamiltonian by a term containing time-dependent parameters so that an evolution of action variables is nothing else than a displacement along a curve in a parameter space. Therefore, one can define this evolution in full by an appropriate choice of parameter functions. Similar holonomy controllability of finite level quantum systems is of special interes in connection with quantum computation. We provide geometric quantization of a time-dependent completely integrable Hamiltonian system in action-angle variables. Its Hamiltonian and first integrals have time-independent countable spectra. The holonomy control operator in this countable level quantum system is constructed.
Completely integrable Hamiltonian systems look promising for controllability since their first integrals are stable under an internal evolution, and one may hope to find a perturbation of a Hamiltonian which drives the first integrals at will. Action-angle coordinates are most convenient for this purpose. Written with respect to these coordinates, a Hamiltonian and first integrals depend only on the action ones. We introduce a suitable perturbation of a Hamiltonian by a term containing time-dependent parameters so that an evolution of action variables is nothing else than a displacement along a curve in a parameter space. Therefore, one can define this evolution in full by an appropriate choice of parameter functions. Similar holonomy controllability of finite level quantum systems is of special interes in connection with quantum computation. We provide geometric quantization of a time-dependent completely integrable Hamiltonian system in action-angle variables. Its Hamiltonian and first integrals have time-independent countable spectra. The holonomy control operator in this countable level quantum system is constructed.
We provide geometric quantization of a completely integrable Hamiltonian system in the action-angle variables around an invariant torus with respect to the angle polarization. The carrier space of this quantization is the pre-Hilbert space of smooth complex functions on the torus. A Hamiltonian of a completely integrable system in this carrier space has a countable spectrum. If it is degenerate, its eigenvalues are countably degenerate. We study nonadiabatic perturbations of this Hamiltonian by a term depending on classical time-dependent parameters. It is originated by a connection on the parameter space, and is linear in the temporal derivatives of parameters. One can choose it commuting with a degenerate Hamiltonian of a completely integrable system. Then the corresponding evolution operator acts in the eigenspaces of this Hamiltonian, and is an operator of parallel displacement along a curve in the parameter space.
The subject of the four manuscripts which make up this dissertation is the concept of integrability in quantum mechanical systems with two degrees of freedom, including the relation between quantum integrability and classical integrability in such systems. In the first manuscript, I consider a two-spin model both classically and quantum mechanically. In the six-dimensional parameter space of this model, classical integrability is satisfied on a five-dimensional hypersurface, and level crossings occur on four-dimensional manifolds that are completely embedded in the integrability hypersurface except for some lower-dimensional sub-manifolds. Under mild assumptions, the classical integrability condition can be reconstructed from a purely quantum mechanical study of level degeneracies in finite-dimensional invariant blocks of the Hamiltonian matrix. In the second manuscript, I analyze the functional dependence [special characters omitted] of the Hamiltonian operator on the action operators for an integrable quantum system, and I compare it with the corresponding functional relationship H (p1, q1; p2, q 2) = HC(J1, J2) in the classical limit of that system. It is demonstrated that the functional dependence is not the same quantum mechanically and (semi-)classically. The former is shown to converge toward the latter in some asymptotic regime associated with the classical limit, but the convergence may be non-uniform. This analysis is carried out for an integrable one- parameter two-spin model. Additional results presented for the (integrable) circular billiard model exhibit similar quantum corrections to the predictions of semiclassical quantization. In the third and the fourth manuscripts, I investigate the relations between symmetry, integrability, and the assignment of quantum numbers to eigenstates for two-spin and spin-boson models. I calculate quantum invariants in the form of expectation values for selected operators and monitor their dependence on the Hamiltonian parameters along loops within, without, and across the integrability region in parameter space. I find clear-cut signatures of integrability and nonintegrability in the observed traces of quantum invariants evaluated in invariant Hilbert subspaces. The results support the notion that quantum integrability depends on the existence of action operators as constituent elements of the Hamiltonian.
The question of controllability is investigated for a quantum control system in which the Hamiltonian operator components carry explicit time dependence which is not under the control of an external agent. We consider the general situation in which the state moves in an infinite-dimensional Hilbert space, a drift term is present, and the operators driving the state evolution may be unbounded. However, considerations are restricted by the assumption that there exists an analytic domain, dense in the state space, on which solutions of the controlled Schrödinger equation may be expressed globally in exponential form. The issue of controllability then naturally focuses on the ability to steer the quantum state on a finite-dimensional submanifold of the unit sphere in Hilbert space—and thus on analytic controllability. A relatively straightforward strategy allows the extension of Lie-algebraic conditions for strong analytic controllability derived earlier for the simpler, time-independent system in which the drift Hamiltonian and the interaction Hamiltonian have no intrinsic time dependence. Enlarging the state space by one dimension corresponding to the time variable, we construct an augmented control system that can be treated as time independent. Methods developed by Kunita can then be implemented to establish controllability conditions for the one-dimension-reduced system defined by the original time-dependent Schrödinger control problem. The applicability of the resulting theorem is illustrated with selected examples.
For a (classically) integrable quantum-mechanical system with two degrees of freedom, the functional dependence H=HQ(J1,J2) of the Hamiltonian operator on the action operators is analyzed and compared with the corresponding functional relationship H(p1,q1;p2,q2)=HC(J1,J2) in the classical limit of that system. The former converges toward the latter in some asymptotic regime associated with the classical limit, but the convergence is, in general, nonuniform. The existence of the function H=HQ(J1,J2) in the integrable regime of a parametric quantum system explains empirical results for the dimensionality of manifolds in parameter space on which at least two levels are degenerate. The analysis is carried out for an integrable one-parameter two-spin model. Additional results presented for the (integrable) circular billiard model illuminate the same conclusions from a different angle.
This work develops a symplectic framework for quantum computing to be applied to classical Hamiltonian systems, exploiting the intrinsic geometric compatibility between unitary quantum evolution and symplectic phase-space dynamics in a two-fold way. The first part is devoted in establishing an exact correspondence between quantum evolution and classical Hamiltonian flow on a Kahler manifold. This correspondence enables a geometric quantization scheme that identifies a family of classical Hamiltonian systems admitting exponentially compressed quantum representations-appropriate for quantum simulation. In the second part we demonstrate that Liouville-integrable Hamiltonian dynamics induce finite-dimensional unitary evolution through action-angle variables and Koopman-von Neumann encoding. This allows efficient quantum representation and parallel evolution of large phase-space ensembles, where entangled encodings provide exponential compression in ensemble size and enable quantum speed-ups in observable estimation via amplitude estimation techniques. For non-integrable systems, Lie canonical perturbation theory is incorporated to construct near-symplectic transformations that map dynamics to approximately integrable forms, preserving unitary evolution up to a controlled error. We derive the resulting quantum computational complexity of the proposed quantum-symplectic scheme, revealing both an exponential compression in memory requirements and a potential polynomial speed-up with respect to the system size. Finally, the transport evolution equation governing the quantum phase-space observables is obtained.
Abstract — The question of controllability is investigated for a quantum control system in which the Hamiltonian operator components carry explicit time dependence which is not under the control of an external agent. We consider the general situation in which the state moves in an infinite-dimensional Hilbert space, a drift term is present, and the operators driving the state evolution may be unbounded. However, considerations are restricted by the assumption that there exists an analytic domain, dense in the state space, on which solutions of the controlled Schrödinger equation may be expressed globally in exponential form. The issue of controllability then naturally focuses on the ability to steer the quantum state on a finitedimensional submanifold of the unit sphere in Hilbert space – and thus on analytic controllability. A relatively straightforward strategy allows the extension of Lie-algebraic conditions for strong analytic controllability derived earlier for the simpler, time-independent system in which the drift Hamiltonian and the interaction Hamiltonia have no intrinsic time dependence. Enlarging the state space by one dimension corresponding to the time variable, we construct an augmented control system that can be treated as time-independent. Methods developed by Kunita can then be implemented to establish controllability conditions for the one-dimension-reduced system defined by the original time-dependent Schrödinger control problem. The applicability of the resulting theorem is illustrated with selected examples. I.