Geometric Control Methods for Quantum Computations (original) (raw)

Methods of geometric control theory for quantum computations

Journal of Mathematical Sciences, 2007

The applications of geometric control theory methods on Lie groups and homogeneous spaces to the theory of quantum computations are investigated. It is shown that these methods are very useful for the problem of constructing a universal set of gates for quantum computations. The well-known result that the set of all one-bit gates, together with no more than one two-bit gate, is universal is considered from the control theory viewpoint.

Geometric methods for construction of quantum gates

Journal of Mathematical Sciences, 2008

The applications of geometric control theory methods on Lie groups and homogeneous spaces to the theory of quantum computations are investigated. These methods are shown to be very useful for the problem of constructing a universal set of gates for quantum computations: the well-known result that the set of all one-bit gates together with almost any one two-bit gate is universal is considered from the control theory viewpoint. Differential geometric structures such as the principal bundle for the canonical vector bundle on a complex Grassmann manifold, the canonical connection form on this bundle, the canonical symplectic form on a complex Grassmann manifold, and the corresponding dynamical systems are investigated. The Grassmann manifold is considered as an orbit of the co-adjoint action, and the symplectic form is described as the restriction of the canonical Poisson structure on a Lie coalgebra. The holonomy of the connection on the principal bundle over the Grassmannian and its relation with the Berry phase is considered and investigated for the trajectories of Hamiltonian dynamical systems.

A Geometric Algebra Perspective on Quantum Computational Gates and Universality in Quantum Computing

Advances in Applied Clifford Algebras, 2011

We investigate the utility of geometric (Clifford) algebras (GA) methods in two specific applications to quantum information science. First, using the multiparticle spacetime algebra (MSTA, the geometric algebra of a relativistic configuration space), we present an explicit algebraic description of one and two-qubit quantum states together with a MSTA characterization of one and two-qubit quantum computational gates. Second, using the above mentioned characterization and the GA description of the Lie algebras SO (3) and SU (2) based on the rotor group Spin + (3, 0) formalism, we reexamine Boykin's proof of universality of quantum gates. We conclude that the MSTA approach does lead to a useful conceptual unification where the complex qubit space and the complex space of unitary operators acting on them become united, with both being made just by multivectors in real space. Finally, the GA approach to rotations based on the rotor group does bring conceptual and computational advantages compared to standard vectorial and matricial approaches.

Quantum control via geometry: An explicit example

Physical Review A, 2008

We explicitly compute the optimal cost for a class of example problems in geometric quantum control. These problems are defined by a Cartan decomposition of su(2 n ) into orthogonal subspaces

Geometry of quantum computation

2013

Preface Basic of Quantum Computation Quantum Computers Based on Exactly Solvable Models & Geometric Phases Quantum Processor Based on the Three-Level Quantum System Methods of Geometric Control Theory for Quantum Computations Analytic Methods in Quantum Computation References Index.

Quantum control and representation theory

Journal of Physics A: Mathematical and Theoretical, 2009

A new notion of controllability for quantum systems that takes advantage of the linear superposition of quantum states is introduced. We call such notion von Neumann controllabilty and it is shown that it is strictly weaker than the usual notion of pure state and operator controlability. We provide a simple and effective characterization of it by using tools from the theory of unitary representations of Lie groups. In this sense we are able to approach the problem of control of quantum states from a new perspective, that of the theory of unitary representations of Lie groups. A few examples of physical interest and the particular instances of compact and nilpotent dynamical Lie groups are discussed.

Geometric Control For Analysing the Quantum Speed Limit and the Physical Limitations of Computers

2015

This thesis studies the role of Finsler geometry in quantum time optimal control of systems with constrained control field power and other constraints. The systems considered are all finite dimensional systems with pure states. A Finsler metric is constructed such that its geodesics are the time optimal trajectories for the quantum time evolution operator on the special unitary group. This metric is shown to be right invariant. The geodesic equation, in the form of an Euler-Poincaré equation is found. It is also shown that the geodesic lengths of this same metric equal the optimal times for implementing any desired quantum gate. In a special case, where all are control fields are equally constrained, the desired geodesics are found in closed form. The results obtained are discussed in the general context of natural computation.

Group-Theoretical Aspects of Control of Quantum Systems

2001

The subject of control of quantum-mechanical systems has been a fruitful area of investigation lately. The growing interest of researchers in various aspect of control of quantum systems can be attributed at least partly to the fact that the ability to manipulate quantum systems in a controlled fashion will be an essential prerequisite for many exciting new applications from control of chemical reactions to quantum computing devices. One issue of interest, which we shall address in this paper, is the question which target states can be dynamically reached from a given initial state of the system by application of suitable control elds, given a certain type of interaction of the system with those elds. This paper is organized as follows. In section 2, the concept pure and mixed quantum states is introduced and examples of both are given. Section 3 deals with the evolution of quantum states and the partitioning of states into kinematical equivalence classes arising from the constraint...

Control and Representation of n-qubit Quantum Systems

Just as any state of a single qubit or 2-level system can be obtained from any other state by a rotation operator parametrized by three real Euler angles, we show how any state of an n-qubit or 2^n-level system can be obtained from any other by a compact unitary transformation with 2^(n+1)-1 real angles, 2^n of which are azimuthal-like and the rest polar-like. The results follow from a modeling of the Hilbert space of n-qubits by a minimal left ideal of an associative algebra. This representation is expected to be useful in the design of new compact control techniques or more efficient algorithms in quantum computing.