Quantum Ballistic Evolution in Quantum Mechanics: Application to Quantum Computers

Quantum computers are important examples of processes whose evolution can be described in terms of iterations of single step operators or their adjoints. Based on this, Hamiltonian evolution of processes with associated step operators $T$ is investigated here. The main limitation of this paper is to processes which evolve quantum ballistically, i.e. motion restricted to a collection of nonintersecting or distinct paths on an arbitrary basis. The main goal of this paper is proof of a theorem which gives necessary and sufficient conditions that T must satisfy so that there exists a Hamiltonian description of quantum ballistic evolution for the process, namely, that T is a partial isometry and is orthogonality preserving and stable on some basis. Simple examples of quantum ballistic evolution for quantum Turing machines with one and with more than one type of elementary step are discussed. It is seen that for nondeterministic machines the basis set can be quite complex with much entanglement present. It is also proved that, given a step operator T for an arbitrary deterministic quantum Turing machine, it is decidable if T is stable and orthogonality preserving, and if quantum ballistic evolution is possible. The proof fails if T is a step operator for a nondeterministic machine. It is an open question if such a decision procedure exists for nondeterministic machines. This problem does not occur in classical mechanics.Comment: 37 pages Latexwith 2 postscript figures tar+gzip+uuencoded, to be published in Phys. Rev.

Tight Binding Hamiltonians and Quantum Turing Machines

This paper extends work done to date on quantum computation by associating potentials with different types of computation steps. Quantum Turing machine Hamiltonians, generalized to include potentials, correspond to sums over tight binding Hamiltonians each with a different potential distribution. Which distribution applies is determined by the initial state. An example, which enumerates the integers in succession as binary strings, is analyzed. It is seen that for some initial states the potential distributions have quasicrystalline properties and are similar to a substitution sequence.Comment: 4 pages Latex, 2 postscript figures, submitted to Phys Rev Letter

Quantum Robots and Environments

Quantum robots and their interactions with environments of quantum systems are described and their study justified. A quantum robot is a mobile quantum system that includes a quantum computer and needed ancillary systems on board. Quantum robots carry out tasks whose goals include specified changes in the state of the environment or carrying out measurements on the environment. Each task is a sequence of alternating computation and action phases. Computation phase activities include determination of the action to be carried out in the next phase and possible recording of information on neighborhood environmental system states. Action phase activities include motion of the quantum robot and changes of neighborhood environment system states. Models of quantum robots and their interactions with environments are described using discrete space and time. To each task is associated a unitary step operator T that gives the single time step dynamics. T = T_{a}+T_{c} is a sum of action phase and computation phase step operators. Conditions that T_{a} and T_{c} should satisfy are given along with a description of the evolution as a sum over paths of completed phase input and output states. A simple example of a task carrying out a measurement on a very simple environment is analyzed. A decision tree for the task is presented and discussed in terms of sums over phase paths. One sees that no definite times or durations are associated with the phase steps in the tree and that the tree describes the successive phase steps in each path in the sum.Comment: 30 Latex pages, 3 Postscript figures, Minor mathematical corrections, accepted for publication, Phys Rev

Spatial quantum search in a triangular network

The spatial search problem consists in minimizing the number of steps required to find a given site in a network, under the restriction that only oracle queries or translations to neighboring sites are allowed. We propose a quantum algorithm for the spatial search problem on a triangular lattice with N sites and torus-like boundary conditions. The proposed algortithm is a special case of the general framework for abstract search proposed by Ambainis, Kempe and Rivosh [AKR05] (AKR) and Tulsi [Tulsi08], applied to a triangular network. The AKR-Tulsi formalism was employed to show that the time complexity of the quantum search on the triangular lattice is O(sqrt(N logN)).Comment: 10 pages, 4 Postscript figures, uses sbc-template.sty, appeared in Annals of WECIQ 2010, III Workshop of Quantum Computation and Quantum Informatio

Cyclic networks of quantum gates

In this article initial steps in an analysis of cyclic networks of quantum logic gates is given. Cyclic networks are those in which the qubit lines are loops. Here we have studied one and two qubit systems plus two qubit cyclic systems connected to another qubit on an acyclic line. The analysis includes the group classification of networks and studies of the dynamics of the qubits in the cyclic network and of the perturbation effects of an acyclic qubit acting on a cyclic network. This is followed by a discussion of quantum algorithms and quantum information processing with cyclic networks of quantum gates, and a novel implementation of a cyclic network quantum memory. Quantum sensors via cyclic networks are also discussed.Comment: 14 pages including 11 figures, References adde

International Assistance for Low-Emission Development Planning: Coordinated Low Emissions Assistance Network (CLEAN) Inventory of Activities and Tools--Preliminary Trends

The Coordinated Low Emissions Assistance Network (CLEAN) is a voluntary network of international practitioners supporting low-emission planning in developing countries. The network seeks to improve quality of support through sharing project information, tools, best practices and lessons, and by fostering harmonized assistance. CLEAN has developed an inventory to track and analyze international technical support and tools for low-carbon planning activities in developing countries. This paper presents a preliminary analysis of the inventory to help identify trends in assistance activities and tools available to support developing countries with low-emission planning

Pattern formation in quantum Turing machines

We investigate the iteration of a sequence of local and pair unitary transformations, which can be interpreted to result from a Turing-head (pseudo-spin $S$) rotating along a closed Turing-tape ($M$ additional pseudo-spins). The dynamical evolution of the Bloch-vector of $S$, which can be decomposed into $2^{M}$ primitive pure state Turing-head trajectories, gives rise to fascinating geometrical patterns reflecting the entanglement between head and tape. These machines thus provide intuitive examples for quantum parallelism and, at the same time, means for local testing of quantum network dynamics.Comment: Accepted for publication in Phys.Rev.A, 3 figures, REVTEX fil

Efficient Implementation and the Product State Representation of Numbers

The relation between the requirement of efficient implementability and the product state representation of numbers is examined. Numbers are defined to be any model of the axioms of number theory or arithmetic. Efficient implementability (EI) means that the basic arithmetic operations are physically implementable and the space-time and thermodynamic resources needed to carry out the implementations are polynomial in the range of numbers considered. Different models of numbers are described to show the independence of both EI and the product state representation from the axioms. The relation between EI and the product state representation is examined. It is seen that the condition of a product state representation does not imply EI. Arguments used to refute the converse implication, EI implies a product state representation, seem reasonable; but they are not conclusive. Thus this implication remains an open question.Comment: Paragraph in page proof for Phys. Rev. A revise

Quantum search without entanglement

Entanglement of quantum variables is usually thought to be a prerequisite for obtaining quantum speed-ups of information processing tasks such as searching databases. This paper presents methods for quantum search that give a speed-up over classical methods, but that do not require entanglement. These methods rely instead on interference to provide a speed-up. Search without entanglement comes at a cost: although they outperform analogous classical devices, the quantum devices that perform the search are not universal quantum computers and require exponentially greater overhead than a quantum computer that operates using entanglement. Quantum search without entanglement is compared to classical search using waves.Comment: 9 pages, TeX, submitted to Physical Review Letter

Quantum computation by quantum-like systems

Using a quantumlike description for light propagation in nonhomogeneous optical fibers, quantum information processing can be implemented by optical means. Quantum-like bits (qulbits) are associated to light modes in the optical fiber and quantum gates to segments of the fiber providing an unitary transformation of the mode structure along a space direction. Simulation of nonlinear quantum effects is also discussed.Comment: 12 pages, Late