Controlling qubit networks in polynomial time

Future quantum devices often rely on favourable scaling with respect to the system components. To achieve desirable scaling, it is therefore crucial to implement unitary transformations in an efficient manner. We develop an upper bound for the minimum time required to implement a unitary transformation on a generic qubit network in which each of the qubits is subject to local time dependent controls. The set of gates is characterized that can be implemented in a time that scales at most polynomially in the number of qubits. Furthermore, we show how qubit systems can be concatenated through controllable two body interactions, making it possible to implement the gate set efficiently on the combined system. Finally a system is identified for which the gate set can be implemented with fewer controls. The considered model is particularly important, since it describes electron-nuclear spin interactions in NV centers

Cooperating or Fighting with Control Noise in the Optimal Manipulation of Quantum Dynamics

This paper investigates the impact of control field noise on the optimal manipulation of quantum dynamics. Simulations are performed on several multilevel quantum systems with the goal of population transfer in the presence of significant control noise. The noise enters as run-to-run variations in the control amplitude and phase with the observation being an ensemble average over many runs as is commonly done in the laboratory. A genetic algorithm with an improved elitism operator is used to find the optimal field that either fights against or cooperates with control field noise. When seeking a high control yield it is possible to find fields that successfully fight with the noise while attaining good quality stable results. When seeking modest control yields, fields can be found which are optimally shaped to cooperate with the noise and thereby drive the dynamics more efficiently. In general, noise reduces the coherence of the dynamics, but the results indicate that population transfer objectives can be met by appropriately either fighting or cooperating with noise, even when it is intense.Comment: Scientific Workplace Late

Dynamic Homotopy and Landscape Dynamical Set Topology in Quantum Control

We examine the topology of the subset of controls taking a given initial state to a given final state in quantum control, where "state" may mean a pure state |\psi>, an ensemble density matrix \rho, or a unitary propagator U(0,T). The analysis consists in showing that the endpoint map acting on control space is a Hurewicz fibration for a large class of affine control systems with vector controls. Exploiting the resulting fibration sequence and the long exact sequence of basepoint-preserving homotopy classes of maps, we show that the indicated subset of controls is homotopy equivalent to the loopspace of the state manifold. This not only allows us to understand the connectedness of "dynamical sets" realized as preimages of subsets of the state space through this endpoint map, but also provides a wealth of additional topological information about such subsets of control space.Comment: Minor clarifications, and added new appendix addressing scalar control of 2-level quantum system

Constitution or Conflict?

A self-enforcing constitution creates a political process that provides an alternative to civil conflict for resolving disputes among the constituent groups of the polity. This paper is concerned with discovering the conditions under which it is possible to design such a self-enforcing constitution. The paper is also concerned with discovering generic features of a self-enforcing constitution. The analysis yields the following theoretical propositions: If and only if (1) none of the parties to a dispute regards the dispute to be too important relative to the expected incremental cost of civil conflict and (2) no party has too big of an advantage in civil conflict, then the parties are able to resolve a dispute constitutionally. Also, under a constitution that is self enforcing the outcomes of constitutional contests for political power do not matter too much. The paper illustrates the relevance of the theoretical analysis by applying these propositions to two dramatic historical examples of constitutional failure: the secession of eleven Southern states from the Union in 1861 and the National Socialist revolution in Germany in 1933.