The gate array version of quantum computation uses logical gates adopting convenient forms for computational algorithms based on the algorithms classical computation. Two-level quantum systems are the basic elements connecting the binary nature of classical computation with the settlement of quantum processing. Despite this, their design depends on specific quantum systems and the physical interactions involved, thus complicating the dynamics analysis. Predictable and controllable manipulation should be addressed in order to control the quantum states in terms of the physical control parameters. Resources are restricted to limitations imposed by the physical settlement. This work presents a formalism to decompose the quantum information dynamics in S U ( 2 2 d ) for 2 d -partite two-level systems into 2 2 d - 1 S U ( 2 ) quantum subsystems. It generates an easier and more direct physical implementation of quantum processing developments for qubits. Easy and traditional operations proposed by quantum computation are recovered for larger and more complex systems. Alternating the parameters of local and non-local interactions, the procedure states a universal exchange semantics on the basis of generalized Bell states. Although the main procedure could still be settled on other interaction architectures by the proper selection of the basis as natural grammar, the procedure can be understood as a momentary splitting of the 2 d information channels into 2 2 d - 1 pairs of 2 level quantum information subsystems. Additionally, it is a settlement of the quantum information manipulation that is free of the restrictions imposed by the underlying physical system. Thus, the motivation of decomposition is to set control procedures easily in order to generate large entangled states and to design specialized dedicated quantum gates. They are potential applications that properly bypass the general induced superposition generated by physical dynamics.
The gate array version of quantum computation uses logical gates adopting convenient forms for computational algorithms based on the algorithms classical computation. Two-level quantum systems are the basic elements connecting the binary nature of classical computation with the settlement of quantum processing. Despite this, their design depends on specific quantum systems and the physical interactions involved, thus complicating the dynamics analysis. Predictable and controllable manipulation should be addressed in order to control the quantum states in terms of the physical control parameters. Resources are restricted to limitations imposed by the physical settlement. This work presents a formalism to decompose the quantum information dynamics in S U ( 2 2 d ) for 2 d -partite two-level systems into 2 2 d - 1 S U ( 2 ) quantum subsystems. It generates an easier and more direct physical implementation of quantum processing developments for qubits. Easy and traditional operations proposed by quantum computation are recovered for larger and more complex systems. Alternating the parameters of local and non-local interactions, the procedure states a universal exchange semantics on the basis of generalized Bell states. Although the main procedure could still be settled on other interaction architectures by the proper selection of the basis as natural grammar, the procedure can be understood as a momentary splitting of the 2 d information channels into 2 2 d - 1 pairs of 2 level quantum information subsystems. Additionally, it is a settlement of the quantum information manipulation that is free of the restrictions imposed by the underlying physical system. Thus, the motivation of decomposition is to set control procedures easily in order to generate large entangled states and to design specialized dedicated quantum gates. They are potential applications that properly bypass the general induced superposition generated by physical dynamics.
Bell gems naturally split dynamics information from SU(22d)→U(1)22d−1−1×SU(2)22d−1
Quantum Computation and Quantum Information are continuously growing research areas which are based on nature and resources of quantum mechanics, as superposition and entanglement. In its gate array version, the use of convenient and appropriate gates is essential. But while those proposed gates adopt convenient forms for computational algorithms, in the practice, their design depends on specific quantum systems and stuff being used. Gates design is restricted to properties and limitations of interactions and physical elements being involved, where Quantum Control plays a deep role. Quantum complexity of multipartite systems and their interactions requires a tight control to manipulate their quantum states, either local and non-local ones, but still a reducibility procedure should be addressed. This work shows how a general 2d-partite two level spin system in SU(2d) could be decomposed in 2n−1 subsystems on SU(2), letting establish control operations. In particular, it is shown that Bell gems basis is a set of natural states on which decomposition happen naturally under some interaction restrictions. Thus, alternating the direction of local interaction terms in the Hamiltonian, this procedure states a universal exchange semantics on those basis. The structure developed could be understood as a splitting of the 2d information channels into 22d−1 pairs of 2 level information subsystems.
Quantum Information is a quantum resource being advised as a useful tool to set up information processing. Despite physical components being considered are normally two-level systems, still the combination of some of them together with their entangling interactions (another key property in the quantum information processing) become in a complex dynamics needing be addressed and modeled under precise control to set programmed quantum processing tasks. Universal quantum gates are simple controlled evolutions resembling some classical computation gates. Despite their simple forms, not always become easy fit the quantum evolution to them. SU(2) decomposition is a mechanism to reduce the dynamics on SU(2) operations in composed quantum processing systems. It lets an easier control of evolution into the structure required by those gates by the adequate election of the basis for the computation grammar. In this arena, SU(2) decomposition has been studied under piecewise magnetic field pulses. Despite, it is completely applicable for time-dependent pulses, which are more affordable technologically, could be continuous and then possibly free of resonant effects. In this work, we combine the SU(2) reduction with linear and quadratic numerical approaches in the solving of time-dependent Schr\"odinger equation to model and to solve the controlled dynamics for two-qubits, the basic block for composite quantum systems being analyzed under the SU(2) reduction. A comparative benchmark of both approaches is presented together with some useful outcomes for the dynamics in the context of quantum information processing operations.
Quantum Information is a discipline derived from Quantum Mechanics which uses quantum systems to exploit their states as information recipients.Normally, these states are conformed by two-level systems to reproduce the binary nature underlying the classical computation structure.Quantum evolution is then controlled to reproduce convenient information processing operations.Evolution could be hard to be controlled.SU(2) decomposition procedure lets to set a binary structure of processing when a convenient basis is selected to set the dynamics description.In this work, we exploit this procedure for a generic Hamiltonian in order to set the process to reduce arbitrary states into simplest ones.For this work, we use customary SU(2) operations on local and entangled states.These operations are described in the development.They involve 1, 2 and 4-local operations meaning the number of quantum parties involved, in agreement with the decomposition procedure scope.This task is complex in spite the difficulty to set a general way to manipulate the entanglement in the system.We are particularly interested on the application of stochastic procedures based in SU(2) decomposition operations to achieve that goal.In order to have a measure of the advancement of the last task, we use the Rényi-Ingarden-Urbanik entropy to describe the whole spectrum of entanglement in the large systems through the assembling/disassembling of the state.
Quantum information processing should be generated through control of quantum evolution for physical systems being used as resources, such as superconducting circuits, spinspin couplings in ions and artificial anyons in electronic gases. They have a quantum dynamics which should be translated into more natural languages for quantum information processing. On this terrain, this language should let to establish manipulation operations on the associated quantum information states as classical information processing does. This work shows how a kind of processing operations can be settled and implemented for quantum states design and quantum processing for systems fulfilling a SU (2) reduction in their dynamics.
A quantum computer is similar to an ordinary computer, except that instead of bits it uses two-level quantum systems (qubits), and the usual logical operations on bits are replaced with noiseless quantum dynamics in the form of unitary gates or Hamiltonian interactions. Many sets of resources that are universal for quantum computation (that is, capable of implementing any quantum computation) have been found, but general conditions for universality are still only partially understood. Improving our understanding of these conditions contributes to two broad goals. On the one hand, understanding what class of physical systems is suitable for quantum computation allows us choose the systems that are most easily controlled or built in the laboratory. On the other hand, identifying the unifying characteristics of such a class provides insight into the fundamental physics responsible for the much greater computational power of quantum computers. The original results presented in this thesis are a contribution to our understanding of the physical requirements for universal quantum computation. These results lead to the identification of a new, large class of universal dynamics. More specifically, the main problem that is addressed is the following: consider a physical system that is made up of a finite number of qubits that are subject to a natural, fixed interaction described by a two-body Hamiltonian. Suppose we have the ability to control the system only by applying fast one-qubit operations. Under what conditions is this system capable of universal quantum computation? The first original result of the thesis is to prove that universal quantum computation is possible under these conditions given the single extra constraint that the Hamiltonian must entangle, or connect, all of the qubits. In other words, since a two-body Hamiltonian is expressible as a sum of coupling terms each of which acts nontrivially on a pair of qubits, consider the graph whose vertices correspond to qubits and whose edges connect qubits that are coupled by a term in the Hamiltonian. Then this Hamiltonian, together with fast one-qubit unitaries, is universal for quantum computation precisely when the corresponding graph is connected. This result suggests many potential generalizations, three of which are explored in this thesis. The first generalization is to show that the same result holds for a two-body Hamiltonian that acts on D-dimensional quantum systems (qudits), given fast one-qudit unitaries. A second extension is to the case of a many-body Hamiltonian, that is, a Hamiltonian that has coupling terms that act on more than two qubits. Again, we find that under mild restrictions, such Hamiltonians are universal provided they are connected and that arbitrary one-qubit unitaries are available. We also discover and explore some surprising structure in many-body Hamiltonians that is associated with the parity of the couplings. A third related problem that is addressed is the question of what two-qubit unitaries are universal for quantum computation. This problem was originally solved by J.-L. and R. Brylinski [BB02], who gave a technical proof of the universality of any two-qudit gate that is not the swap gate or a product of one-qudit unitaries, given one-qudit unitaries as a free resource. We develop an elementary and constructive proof for the case of qubits, and prove that our construction is very close to optimal in the number of uses of the two-qubit gate. Given a qualitative understanding of what dynamical resources are universal for quantum computation, it is natural to consider developing a quantitative measure of the usefulness of quantum dynamics for quantum information processing. This is done in the final chapter of the thesis, in which the beginnings of a quantitative theory of quantum dynamics are developed.