634,319 research outputs found
On the role of entanglement in quantum computational speed-up
For any quantum algorithm operating on pure states we prove that the presence
of multi-partite entanglement, with a number of parties that increases
unboundedly with input size, is necessary if the quantum algorithm is to offer
an exponential speed-up over classical computation. Furthermore we prove that
the algorithm can be classically efficiently simulated to within a prescribed
tolerance \eta even if a suitably small amount of global entanglement
(depending on \eta) is present. We explicitly identify the occurrence of
increasing multi-partite entanglement in Shor's algorithm. Our results do not
apply to quantum algorithms operating on mixed states in general and we discuss
the suggestion that an exponential computational speed-up might be possible
with mixed states in the total absence of entanglement. Finally, despite the
essential role of entanglement for pure state algorithms, we argue that it is
nevertheless misleading to view entanglement as a key resource for quantum
computational power.Comment: Main proofs simplified. A few further explanatory remarks added. 22
pages, plain late
Deriving Grover's lower bound from simple physical principles
Grover's algorithm constitutes the optimal quantum solution to the search
problem and provides a quadratic speed-up over all possible classical search
algorithms. Quantum interference between computational paths has been posited
as a key resource behind this computational speed-up. However there is a limit
to this interference, at most pairs of paths can ever interact in a fundamental
way. Could more interference imply more computational power? Sorkin has defined
a hierarchy of possible interference behaviours---currently under experimental
investigation---where classical theory is at the first level of the hierarchy
and quantum theory belongs to the second. Informally, the order in the
hierarchy corresponds to the number of paths that have an irreducible
interaction in a multi-slit experiment. In this work, we consider how Grover's
speed-up depends on the order of interference in a theory. Surprisingly, we
show that the quadratic lower bound holds regardless of the order of
interference. Thus, at least from the point of view of the search problem,
post-quantum interference does not imply a computational speed-up over quantum
theory.Comment: Updated title and exposition in response to referee comments. 6+2
pages, 5 figure
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