3,316 research outputs found
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 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.
One-Tape Turing Machine Variants and Language Recognition
We present two restricted versions of one-tape Turing machines. Both
characterize the class of context-free languages. In the first version,
proposed by Hibbard in 1967 and called limited automata, each tape cell can be
rewritten only in the first visits, for a fixed constant .
Furthermore, for deterministic limited automata are equivalent to
deterministic pushdown automata, namely they characterize deterministic
context-free languages. Further restricting the possible operations, we
consider strongly limited automata. These models still characterize
context-free languages. However, the deterministic version is less powerful
than the deterministic version of limited automata. In fact, there exist
deterministic context-free languages that are not accepted by any deterministic
strongly limited automaton.Comment: 20 pages. This article will appear in the Complexity Theory Column of
the September 2015 issue of SIGACT New
Two-Way Automata Making Choices Only at the Endmarkers
The question of the state-size cost for simulation of two-way
nondeterministic automata (2NFAs) by two-way deterministic automata (2DFAs) was
raised in 1978 and, despite many attempts, it is still open. Subsequently, the
problem was attacked by restricting the power of 2DFAs (e.g., using a
restricted input head movement) to the degree for which it was already possible
to derive some exponential gaps between the weaker model and the standard
2NFAs. Here we use an opposite approach, increasing the power of 2DFAs to the
degree for which it is still possible to obtain a subexponential conversion
from the stronger model to the standard 2DFAs. In particular, it turns out that
subexponential conversion is possible for two-way automata that make
nondeterministic choices only when the input head scans one of the input tape
endmarkers. However, there is no restriction on the input head movement. This
implies that an exponential gap between 2NFAs and 2DFAs can be obtained only
for unrestricted 2NFAs using capabilities beyond the proposed new model. As an
additional bonus, conversion into a machine for the complement of the original
language is polynomial in this model. The same holds for making such machines
self-verifying, halting, or unambiguous. Finally, any superpolynomial lower
bound for the simulation of such machines by standard 2DFAs would imply LNL.
In the same way, the alternating version of these machines is related to L =?
NL =? P, the classical computational complexity problems.Comment: 23 page
A Casual Tour Around a Circuit Complexity Bound
I will discuss the recent proof that the complexity class NEXP
(nondeterministic exponential time) lacks nonuniform ACC circuits of polynomial
size. The proof will be described from the perspective of someone trying to
discover it.Comment: 21 pages, 2 figures. An earlier version appeared in SIGACT News,
September 201
Inductive Definition and Domain Theoretic Properties of Fully Abstract
A construction of fully abstract typed models for PCF and PCF^+ (i.e., PCF +
"parallel conditional function"), respectively, is presented. It is based on
general notions of sequential computational strategies and wittingly consistent
non-deterministic strategies introduced by the author in the seventies.
Although these notions of strategies are old, the definition of the fully
abstract models is new, in that it is given level-by-level in the finite type
hierarchy. To prove full abstraction and non-dcpo domain theoretic properties
of these models, a theory of computational strategies is developed. This is
also an alternative and, in a sense, an analogue to the later game strategy
semantics approaches of Abramsky, Jagadeesan, and Malacaria; Hyland and Ong;
and Nickau. In both cases of PCF and PCF^+ there are definable universal
(surjective) functionals from numerical functions to any given type,
respectively, which also makes each of these models unique up to isomorphism.
Although such models are non-omega-complete and therefore not continuous in the
traditional terminology, they are also proved to be sequentially complete (a
weakened form of omega-completeness), "naturally" continuous (with respect to
existing directed "pointwise", or "natural" lubs) and also "naturally"
omega-algebraic and "naturally" bounded complete -- appropriate generalisation
of the ordinary notions of domain theory to the case of non-dcpos.Comment: 50 page
Automata with Nested Pebbles Capture First-Order Logic with Transitive Closure
String languages recognizable in (deterministic) log-space are characterized
either by two-way (deterministic) multi-head automata, or following Immerman,
by first-order logic with (deterministic) transitive closure. Here we elaborate
this result, and match the number of heads to the arity of the transitive
closure. More precisely, first-order logic with k-ary deterministic transitive
closure has the same power as deterministic automata walking on their input
with k heads, additionally using a finite set of nested pebbles. This result is
valid for strings, ordered trees, and in general for families of graphs having
a fixed automaton that can be used to traverse the nodes of each of the graphs
in the family. Other examples of such families are grids, toruses, and
rectangular mazes. For nondeterministic automata, the logic is restricted to
positive occurrences of transitive closure.
The special case of k=1 for trees, shows that single-head deterministic
tree-walking automata with nested pebbles are characterized by first-order
logic with unary deterministic transitive closure. This refines our earlier
result that placed these automata between first-order and monadic second-order
logic on trees.Comment: Paper for Logical Methods in Computer Science, 27 pages, 1 figur
Requirement for quantum computation
We identify "proper quantum computation" with computational processes that
cannot be efficiently simulated on a classical computer. For optical quantum
computation, we establish "no-go" theorems for classes of quantum optical
experiments that cannot yield proper quantum computation, and we identify
requirements for optical proper quantum computation that correspond to
violations of assumptions underpinning the no-go theorems.Comment: 11 pages, no figure
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