86 research outputs found
ROM-based computation: quantum versus classical
We introduce a model of computation based on read only memory (ROM), which
allows us to compare the space-efficiency of reversible, error-free classical
computation with reversible, error-free quantum computation. We show that a
ROM-based quantum computer with one writable qubit is universal, whilst two
writable bits are required for a universal classical ROM-based computer. We
also comment on the time-efficiency advantages of quantum computation within
this model.Comment: 12 pages, 3 figures, minor corrections + section 5 substantially
change
Substituting a qubit for an arbitrarily large number of classical bits
We show that a qubit can be used to substitute for an arbitrarily large
number of classical bits. We consider a physical system S interacting locally
with a classical field phi(x) as it travels directly from point A to point B.
The field has the property that its integrated value is an integer multiple of
some constant. The problem is to determine whether the integer is odd or even.
This task can be performed perfectly if S is a qubit. On the otherhand, if S is
a classical system then we show that it must carry an arbitrarily large amount
of classical information. We identify the physical reason for such a huge
quantum advantage, and show that it also implies a large difference between the
size of quantum and classical memories necessary for some computations. We also
present a simple proof that no finite amount of one-way classical communication
can perfectly simulate the effect of quantum entanglement.Comment: 8 pages, LaTeX, no figures. v2: added result on entanglement
simulation with classical communication; v3: minor correction to main proof,
change of title, added referenc
Preparing encoded states in an oscillator
Recently a scheme has been proposed for constructing quantum error-correcting
codes that embed a finite-dimensional code space in the infinite-dimensional
Hilbert space of a system described by continuous quantum variables. One of the
difficult steps in this scheme is the preparation of the encoded states. We
show how these states can be generated by coupling a continuous quantum
variable to a single qubit. An ion trap quantum computer provides a natural
setting for a continuous system coupled to a qubit. We discuss how encoded
states may be generated in an ion trap.Comment: 5 pages, 4 figures, RevTe
Quantum random walks with decoherent coins
The quantum random walk has been much studied recently, largely due to its
highly nonclassical behavior. In this paper, we study one possible route to
classical behavior for the discrete quantum walk on the line: the presence of
decoherence in the quantum ``coin'' which drives the walk. We find exact
analytical expressions for the time dependence of the first two moments of
position, and show that in the long-time limit the variance grows linearly with
time, unlike the unitary walk. We compare this to the results of direct
numerical simulation, and see how the form of the position distribution changes
from the unitary to the usual classical result as we increase the strength of
the decoherence.Comment: Minor revisions, especially in introduction. Published versio
Quantum random walk on the line as a markovian process
We analyze in detail the discrete--time quantum walk on the line by
separating the quantum evolution equation into Markovian and interference
terms. As a result of this separation, it is possible to show analytically that
the quadratic increase in the variance of the quantum walker's position with
time is a direct consequence of the coherence of the quantum evolution. If the
evolution is decoherent, as in the classical case, the variance is shown to
increase linearly with time, as expected. Furthermore we show that this system
has an evolution operator analogous to that of a resonant quantum kicked rotor.
As this rotator may be described through a quantum computational algorithm, one
may employ this algorithm to describe the time evolution of the quantum walker.Comment: few typos corrected, 13 pages, 2 figures, to appear in Physica
Improved algorithm for quantum separability and entanglement detection
Determining whether a quantum state is separable or entangled is a problem of
fundamental importance in quantum information science. It has recently been
shown that this problem is NP-hard. There is a highly inefficient `basic
algorithm' for solving the quantum separability problem which follows from the
definition of a separable state. By exploiting specific properties of the set
of separable states, we introduce a new classical algorithm that solves the
problem significantly faster than the `basic algorithm', allowing a feasible
separability test where none previously existed e.g. in 3-by-3-dimensional
systems. Our algorithm also provides a novel tool in the experimental detection
of entanglement.Comment: 4 pages, revtex4, no figure
Quantum Walks driven by many coins
Quantum random walks have been much studied recently, largely due to their
highly nonclassical behavior. In this paper, we study one possible route to
classical behavior for the discrete quantum random walk on the line: the use of
multiple quantum ``coins'' in order to diminish the effects of interference
between paths. We find solutions to this system in terms of the single coin
random walk, and compare the asymptotic limit of these solutions to numerical
simulations. We find exact analytical expressions for the time-dependence of
the first two moments, and show that in the long time limit the ``quantum
mechanical'' behavior of the one-coin walk persists. We further show that this
is generic for a very broad class of possible walks, and that this behavior
disappears only in the limit of a new coin for every step of the walk.Comment: 36 pages RevTeX 4.0 + 5 figures (encapsulated Postscript). Submitted
to Physical Review
Generalized Quantum Walk in Momentum Space
We consider a new model of quantum walk on a one-dimensional momentum space
that includes both discrete jumps and continuous drift. Its time evolution has
two stages; a Markov diffusion followed by localized dynamics. As in the well
known quantum kicked rotor, this model can be mapped into a localized
one-dimensional Anderson model. For exceptional (rational) values of its scale
parameter, the system exhibits resonant behavior and reduce to the usual
discrete time quantum walk on the line.Comment: 11 pages, 5 figure
Simulation of quantum random walks using interference of classical field
We suggest a theoretical scheme for the simulation of quantum random walks on
a line using beam splitters, phase shifters and photodetectors. Our model
enables us to simulate a quantum random walk with use of the wave nature of
classical light fields. Furthermore, the proposed set-up allows the analysis of
the effects of decoherence. The transition from a pure mean photon-number
distribution to a classical one is studied varying the decoherence parameters.Comment: extensively revised version; title changed; to appear on Phys. Rev.
Generation of eigenstates using the phase-estimation algorithm
The phase estimation algorithm is so named because it allows the estimation
of the eigenvalues associated with an operator. However it has been proposed
that the algorithm can also be used to generate eigenstates. Here we extend
this proposal for small quantum systems, identifying the conditions under which
the phase estimation algorithm can successfully generate eigenstates. We then
propose an implementation scheme based on an ion trap quantum computer. This
scheme allows us to illustrate two simple examples, one in which the algorithm
effectively generates eigenstates, and one in which it does not.Comment: 5 pages, 3 Figures, RevTeX4 Introduction expanded, typos correcte
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