150 research outputs found
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
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
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
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 ) rotating along a closed Turing-tape ( additional
pseudo-spins). The dynamical evolution of the Bloch-vector of , which can be
decomposed into 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
Transmission and Spectral Aspects of Tight Binding Hamiltonians for the Counting Quantum Turing Machine
It was recently shown that a generalization of quantum Turing machines
(QTMs), in which potentials are associated with elementary steps or transitions
of the computation, generates potential distributions along computation paths
of states in some basis B. The distributions are computable and are thus
periodic or have deterministic disorder. These generalized machines (GQTMs) can
be used to investigate the effect of potentials in causing reflections and
reducing the completion probability of computations. This work is extended here
by determination of the spectral and transmission properties of an example GQTM
which enumerates the integers as binary strings. A potential is associated with
just one type of step. For many computation paths the potential distributions
are initial segments of a quasiperiodic distribution that corresponds to a
substitution sequence. The energy band spectra and Landauer Resistance (LR) are
calculated for energies below the barrier height by use of transfer matrices.
The LR fluctuates rapidly with momentum with minima close to or at band-gap
edges. For several values of the parameters, there is good transmission over
some momentum regions.Comment: 22 pages Latex, 13 postscript figures, Submitted to Phys. Rev.
The Representation of Natural Numbers in Quantum Mechanics
This paper represents one approach to making explicit some of the assumptions
and conditions implied in the widespread representation of numbers by composite
quantum systems. Any nonempty set and associated operations is a set of natural
numbers or a model of arithmetic if the set and operations satisfy the axioms
of number theory or arithmetic. This work is limited to k-ary representations
of length L and to the axioms for arithmetic modulo k^{L}. A model of the
axioms is described based on states in and operators on an abstract L fold
tensor product Hilbert space H^{arith}. Unitary maps of this space onto a
physical parameter based product space H^{phy} are then described. Each of
these maps makes states in H^{phy}, and the induced operators, a model of the
axioms. Consequences of the existence of many of these maps are discussed along
with the dependence of Grover's and Shor's Algorithms on these maps. The
importance of the main physical requirement, that the basic arithmetic
operations are efficiently implementable, is discussed. This conditions states
that there exist physically realizable Hamiltonians that can implement the
basic arithmetic operations and that the space-time and thermodynamic resources
required are polynomial in L.Comment: Much rewrite, including response to comments. To Appear in Phys. Rev.
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
Spin-1/2 particles moving on a 2D lattice with nearest-neighbor interactions can realize an autonomous quantum computer
What is the simplest Hamiltonian which can implement quantum computation
without requiring any control operations during the computation process? In a
previous paper we have constructed a 10-local finite-range interaction among
qubits on a 2D lattice having this property. Here we show that
pair-interactions among qutrits on a 2D lattice are sufficient, too, and can
also implement an ergodic computer where the result can be read out from the
time average state after some post-selection with high success probability.
Two of the 3 qutrit states are given by the two levels of a spin-1/2 particle
located at a specific lattice site, the third state is its absence. Usual
hopping terms together with an attractive force among adjacent particles induce
a coupled quantum walk where the particle spins are subjected to spatially
inhomogeneous interactions implementing holonomic quantum computing. The
holonomic method ensures that the implemented circuit does not depend on the
time needed for the walk.
Even though the implementation of the required type of spin-spin interactions
is currently unclear, the model shows that quite simple Hamiltonians are
powerful enough to allow for universal quantum computing in a closed physical
system.Comment: More detailed explanations including description of a programmable
version. 44 pages, 12 figures, latex. To appear in PR
- …