413 research outputs found
(Almost) tight bounds for randomized and quantum Local Search on hypercubes and grids
The Local Search problem, which finds a local minimum of a black-box function
on a given graph, is of both practical and theoretical importance to many areas
in computer science and natural sciences. In this paper, we show that for the
Boolean hypercube \B^n, the randomized query complexity of Local Search is
and the quantum query complexity is
. We also show that for the constant dimensional grid
, the randomized query complexity is for and the quantum query complexity is for . New
lower bounds for lower dimensional grids are also given. These improve the
previous results by Aaronson [STOC'04], and Santha and Szegedy [STOC'04].
Finally we show for a new upper bound of on the quantum query complexity, which implies that Local Search on
grids exhibits different properties at low dimensions.Comment: 18 pages, 1 figure. v2: introduction rewritten, references added. v3:
a line for grant added. v4: upper bound section rewritte
Efficient quantum protocols for XOR functions
We show that for any Boolean function f on {0,1}^n, the bounded-error quantum
communication complexity of XOR functions satisfies that
, where d is the F2-degree of f, and
.
This implies that the previous lower bound by Lee and Shraibman \cite{LS09} is tight
for f with low F2-degree. The result also confirms the quantum version of the
Log-rank Conjecture for low-degree XOR functions. In addition, we show that the
exact quantum communication complexity satisfies , where is the number of nonzero Fourier coefficients of
f. This matches the previous lower bound
by Buhrman and de Wolf \cite{BdW01} for low-degree XOR functions.Comment: 11 pages, no figur
Sensitivity Conjecture and Log-rank Conjecture for functions with small alternating numbers
The Sensitivity Conjecture and the Log-rank Conjecture are among the most
important and challenging problems in concrete complexity. Incidentally, the
Sensitivity Conjecture is known to hold for monotone functions, and so is the
Log-rank Conjecture for and with monotone
functions , where and are bit-wise AND and XOR,
respectively. In this paper, we extend these results to functions which
alternate values for a relatively small number of times on any monotone path
from to . These deepen our understandings of the two conjectures,
and contribute to the recent line of research on functions with small
alternating numbers
Quantum game players can have advantage without discord
The last two decades have witnessed a rapid development of quantum
information processing, a new paradigm which studies the power and limit of
"quantum advantages" in various information processing tasks. Problems such as
when quantum advantage exists, and if existing, how much it could be, are at a
central position of these studies. In a broad class of scenarios, there are,
implicitly or explicitly, at least two parties involved, who share a state, and
the correlation in this shared state is the key factor to the efficiency under
concern. In these scenarios, the shared \emph{entanglement} or \emph{discord}
is usually what accounts for quantum advantage. In this paper, we examine a
fundamental problem of this nature from the perspective of game theory, a
branch of applied mathematics studying selfish behaviors of two or more
players. We exhibit a natural zero-sum game, in which the chance for any player
to win the game depends only on the ending correlation. We show that in a
certain classical equilibrium, a situation in which no player can further
increase her payoff by any local classical operation, whoever first uses a
quantum computer has a big advantage over its classical opponent. The
equilibrium is fair to both players and, as a shared correlation, it does not
contain any discord, yet a quantum advantage still exists. This indicates that
at least in game theory, the previous notion of discord as a measure of
non-classical correlation needs to be reexamined, when there are two players
with different objectives.Comment: 15 page
A single-shot measurement of the energy of product states in a translation invariant spin chain can replace any quantum computation
In measurement-based quantum computation, quantum algorithms are implemented
via sequences of measurements. We describe a translationally invariant
finite-range interaction on a one-dimensional qudit chain and prove that a
single-shot measurement of the energy of an appropriate computational basis
state with respect to this Hamiltonian provides the output of any quantum
circuit. The required measurement accuracy scales inverse polynomially with the
size of the simulated quantum circuit. This shows that the implementation of
energy measurements on generic qudit chains is as hard as the realization of
quantum computation. Here a ''measurement'' is any procedure that samples from
the spectral measure induced by the observable and the state under
consideration. As opposed to measurement-based quantum computation, the
post-measurement state is irrelevant.Comment: 19 pages, transition rules for the CA correcte
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