2,081 research outputs found
Closed Timelike Curves Make Quantum and Classical Computing Equivalent
While closed timelike curves (CTCs) are not known to exist, studying their
consequences has led to nontrivial insights in general relativity, quantum
information, and other areas. In this paper we show that if CTCs existed, then
quantum computers would be no more powerful than classical computers: both
would have the (extremely large) power of the complexity class PSPACE,
consisting of all problems solvable by a conventional computer using a
polynomial amount of memory. This solves an open problem proposed by one of us
in 2005, and gives an essentially complete understanding of computational
complexity in the presence of CTCs. Following the work of Deutsch, we treat a
CTC as simply a region of spacetime where a "causal consistency" condition is
imposed, meaning that Nature has to produce a (probabilistic or quantum)
fixed-point of some evolution operator. Our conclusion is then a consequence of
the following theorem: given any quantum circuit (not necessarily unitary), a
fixed-point of the circuit can be (implicitly) computed in polynomial space.
This theorem might have independent applications in quantum information.Comment: 15 page
Perfect state distinguishability and computational speedups with postselected closed timelike curves
Bennett and Schumacher's postselected quantum teleportation is a model of
closed timelike curves (CTCs) that leads to results physically different from
Deutsch's model. We show that even a single qubit passing through a
postselected CTC (P-CTC) is sufficient to do any postselected quantum
measurement, and we discuss an important difference between "Deutschian" CTCs
(D-CTCs) and P-CTCs in which the future existence of a P-CTC might affect the
present outcome of an experiment. Then, based on a suggestion of Bennett and
Smith, we explicitly show how a party assisted by P-CTCs can distinguish a set
of linearly independent quantum states, and we prove that it is not possible
for such a party to distinguish a set of linearly dependent states. The power
of P-CTCs is thus weaker than that of D-CTCs because the Holevo bound still
applies to circuits using them regardless of their ability to conspire in
violating the uncertainty principle. We then discuss how different notions of a
quantum mixture that are indistinguishable in linear quantum mechanics lead to
dramatically differing conclusions in a nonlinear quantum mechanics involving
P-CTCs. Finally, we give explicit circuit constructions that can efficiently
factor integers, efficiently solve any decision problem in the intersection of
NP and coNP, and probabilistically solve any decision problem in NP. These
circuits accomplish these tasks with just one qubit traveling back in time, and
they exploit the ability of postselected closed timelike curves to create
grandfather paradoxes for invalid answers.Comment: 15 pages, 4 figures; Foundations of Physics (2011
Simulations of closed timelike curves
Proposed models of closed timelike curves (CTCs) have been shown to enable
powerful information-processing protocols. We examine the simulation of models
of CTCs both by other models of CTCs and by physical systems without access to
CTCs. We prove that the recently proposed transition probability CTCs (T-CTCs)
are physically equivalent to postselection CTCs (P-CTCs), in the sense that one
model can simulate the other with reasonable overhead. As a consequence, their
information-processing capabilities are equivalent. We also describe a method
for quantum computers to simulate Deutschian CTCs (but with a reasonable
overhead only in some cases). In cases for which the overhead is reasonable, it
might be possible to perform the simulation in a table-top experiment. This
approach has the benefit of resolving some ambiguities associated with the
equivalent circuit model of Ralph et al. Furthermore, we provide an explicit
form for the state of the CTC system such that it is a maximum-entropy state,
as prescribed by Deutsch.Comment: 15 pages, 1 figure, accepted for publication in Foundations of
Physic
Quantum Computational Complexity in the Presence of Closed Timelike Curves
Quantum computation with quantum data that can traverse closed timelike
curves represents a new physical model of computation. We argue that a model of
quantum computation in the presence of closed timelike curves can be formulated
which represents a valid quantification of resources given the ability to
construct compact regions of closed timelike curves. The notion of
self-consistent evolution for quantum computers whose components follow closed
timelike curves, as pointed out by Deutsch [Phys. Rev. D {\bf 44}, 3197
(1991)], implies that the evolution of the chronology respecting components
which interact with the closed timelike curve components is nonlinear. We
demonstrate that this nonlinearity can be used to efficiently solve
computational problems which are generally thought to be intractable. In
particular we demonstrate that a quantum computer which has access to closed
timelike curve qubits can solve NP-complete problems with only a polynomial
number of quantum gates.Comment: 8 pages, 2 figures. Minor changes and typos fixed. Reference adde
Non-causal computation
Computation models such as circuits describe sequences of computation steps
that are carried out one after the other. In other words, algorithm design is
traditionally subject to the restriction imposed by a fixed causal order. We
address a novel computing paradigm beyond quantum computing, replacing this
assumption by mere logical consistency: We study non-causal circuits, where a
fixed time structure within a gate is locally assumed whilst the global causal
structure between the gates is dropped. We present examples of logically
consistent non- causal circuits outperforming all causal ones; they imply that
suppressing loops entirely is more restrictive than just avoiding the
contradictions they can give rise to. That fact is already known for
correlations as well as for communication, and we here extend it to
computation.Comment: 6 pages, 4 figure
The Computational Power of Minkowski Spacetime
The Lorentzian length of a timelike curve connecting both endpoints of a
classical computation is a function of the path taken through Minkowski
spacetime. The associated runtime difference is due to time-dilation: the
phenomenon whereby an observer finds that another's physically identical ideal
clock has ticked at a different rate than their own clock. Using ideas
appearing in the framework of computational complexity theory, time-dilation is
quantified as an algorithmic resource by relating relativistic energy to an
th order polynomial time reduction at the completion of an observer's
journey. These results enable a comparison between the optimal quadratic
\emph{Grover speedup} from quantum computing and an speedup using
classical computers and relativistic effects. The goal is not to propose a
practical model of computation, but to probe the ultimate limits physics places
on computation.Comment: 6 pages, LaTeX, feedback welcom
The Quantum Propagator for a Nonrelativistic Particle in the Vicinity of a Time Machine
We study the propagator of a non-relativistic, non-interacting particle in
any non-relativistic ``time-machine'' spacetime of the type shown in Fig.~1: an
external, flat spacetime in which two spatial regions, at time and
at time , are connected by two temporal wormholes, one leading from
the past side of to t the future side of and the other from the
past side of to the future side of . We express the propagator
explicitly in terms of those for ordinary, flat spacetime and for the two
wormholes; and from that expression we show that the propagator satisfies
completeness and unitarity in the initial and final ``chronal regions''
(regions without closed timelike curves) and its propagation from the initial
region to the final region is unitary. However, within the time machine it
satisfies neither completeness nor unitarity. We also give an alternative proof
of initial-region-to-final-region unitarity based on a conserved current and
Gauss's theorem. This proof can be carried over without change to most any
non-relativistic time-machine spacetime; it is the non-relativistic version of
a theorem by Friedman, Papastamatiou and Simon, which says that for a free
scalar field, quantum mechanical unitarity follows from the fact that the
classical evolution preserves the Klein-Gordon inner product
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