92 research outputs found
The power of quantum systems on a line
We study the computational strength of quantum particles (each of finite
dimensionality) arranged on a line. First, we prove that it is possible to
perform universal adiabatic quantum computation using a one-dimensional quantum
system (with 9 states per particle). This might have practical implications for
experimentalists interested in constructing an adiabatic quantum computer.
Building on the same construction, but with some additional technical effort
and 12 states per particle, we show that the problem of approximating the
ground state energy of a system composed of a line of quantum particles is
QMA-complete; QMA is a quantum analogue of NP. This is in striking contrast to
the fact that the analogous classical problem, namely, one-dimensional
MAX-2-SAT with nearest neighbor constraints, is in P. The proof of the
QMA-completeness result requires an additional idea beyond the usual techniques
in the area: Not all illegal configurations can be ruled out by local checks,
so instead we rule out such illegal configurations because they would, in the
future, evolve into a state which can be seen locally to be illegal. Our
construction implies (assuming the quantum Church-Turing thesis and that
quantum computers cannot efficiently solve QMA-complete problems) that there
are one-dimensional systems which take an exponential time to relax to their
ground states at any temperature, making them candidates for being
one-dimensional spin glasses.Comment: 21 pages. v2 has numerous corrections and clarifications, and most
importantly a new author, merged from arXiv:0705.4067. v3 is the published
version, with additional clarifications, publisher's version available at
http://www.springerlink.co
Rewindable Quantum Computation and Its Equivalence to Cloning and Adaptive Postselection
We define rewinding operators that invert quantum measurements. Then, we
define complexity classes , , and as
sets of decision problems solvable by polynomial-size quantum circuits with a
polynomial number of rewinding operators, cloning operators, and adaptive
postselections, respectively. Our main result is that . As a
byproduct of this result, we show that any problem in can be
solved with only postselections of outputs whose probabilities are polynomially
close to one. Under the strongly believed assumption that , or the shortest independent vectors problem cannot be
efficiently solved with quantum computers, we also show that a single rewinding
operator is sufficient to achieve tasks that are intractable for quantum
computation. In addition, we consider rewindable Clifford and instantaneous
quantum polynomial time circuits.Comment: 29 pages, 3 figures, v2: Added Result 3 and improved Result
Oracles and query lower bounds in generalised probabilistic theories
We investigate the connection between interference and computational power
within the operationally defined framework of generalised probabilistic
theories. To compare the computational abilities of different theories within
this framework we show that any theory satisfying three natural physical
principles possess a well-defined oracle model. Indeed, we prove a subroutine
theorem for oracles in such theories which is a necessary condition for the
oracle to be well-defined. The three principles are: causality (roughly, no
signalling from the future), purification (each mixed state arises as the
marginal of a pure state of a larger system), and strong symmetry existence of
non-trivial reversible transformations). Sorkin has defined a hierarchy of
conceivable interference behaviours, where the order in the hierarchy
corresponds to the number of paths that have an irreducible interaction in a
multi-slit experiment. Given our oracle model, we show that if a classical
computer requires at least n queries to solve a learning problem, then the
corresponding lower bound in theories lying at the kth level of Sorkin's
hierarchy is n/k. Hence, lower bounds on the number of queries to a quantum
oracle needed to solve certain problems are not optimal in the space of all
generalised probabilistic theories, although it is not yet known whether the
optimal bounds are achievable in general. Hence searches for higher-order
interference are not only foundationally motivated, but constitute a search for
a computational resource beyond that offered by quantum computation.Comment: 17+7 pages. Comments Welcome. Published in special issue
"Foundational Aspects of Quantum Information" in Foundations of Physic
Kinetic Geodesic Voronoi Diagrams in a Simple Polygon
We study the geodesic Voronoi diagram of a set S of n linearly moving sites inside a static simple polygon P with m vertices. We identify all events where the structure of the Voronoi diagram changes, bound the number of such events, and then develop a kinetic data structure (KDS) that maintains the geodesic Voronoi diagram as the sites move. To this end, we first analyze how often a single bisector, defined by two sites, or a single Voronoi center, defined by three sites, can change. For both these structures we prove that the number of such changes is at most O(m³), and that this is tight in the worst case. Moreover, we develop compact, responsive, local, and efficient kinetic data structures for both structures. Our data structures use linear space and process a worst-case optimal number of events. Our bisector KDS handles each event in O(log m) time, and our Voronoi center handles each event in O(log² m) time. Both structures can be extended to efficiently support updating the movement of the sites as well. Using these data structures as building blocks we obtain a compact KDS for maintaining the full geodesic Voronoi diagram
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