22,893 research outputs found
Lower Bounds on Quantum Query Complexity
Shor's and Grover's famous quantum algorithms for factoring and searching
show that quantum computers can solve certain computational problems
significantly faster than any classical computer. We discuss here what quantum
computers_cannot_ do, and specifically how to prove limits on their
computational power. We cover the main known techniques for proving lower
bounds, and exemplify and compare the methods.Comment: survey, 23 page
(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
Adiabatic optimization without local minima
Several previous works have investigated the circumstances under which
quantum adiabatic optimization algorithms can tunnel out of local energy minima
that trap simulated annealing or other classical local search algorithms. Here
we investigate the even more basic question of whether adiabatic optimization
algorithms always succeed in polynomial time for trivial optimization problems
in which there are no local energy minima other than the global minimum.
Surprisingly, we find a counterexample in which the potential is a single basin
on a graph, but the eigenvalue gap is exponentially small as a function of the
number of vertices. In this counterexample, the ground state wavefunction
consists of two "lobes" separated by a region of exponentially small amplitude.
Conversely, we prove if the ground state wavefunction is single-peaked then the
eigenvalue gap scales at worst as one over the square of the number of
vertices.Comment: 20 pages, 1 figure. Journal versio
Bounds on Quantum Correlations in Bell Inequality Experiments
Bell inequality violation is one of the most widely known manifestations of
entanglement in quantum mechanics; indicating that experiments on physically
separated quantum mechanical systems cannot be given a local realistic
description. However, despite the importance of Bell inequalities, it is not
known in general how to determine whether a given entangled state will violate
a Bell inequality. This is because one can choose to make many different
measurements on a quantum system to test any given Bell inequality and the
optimization over measurements is a high-dimensional variational problem. In
order to better understand this problem we present algorithms that provide, for
a given quantum state, both a lower bound and an upper bound on the maximal
expectation value of a Bell operator. Both bounds apply techniques from convex
optimization and the methodology for creating upper bounds allows them to be
systematically improved. In many cases these bounds determine measurements that
would demonstrate violation of the Bell inequality or provide a bound that
rules out the possibility of a violation. Examples are given to illustrate how
these algorithms can be used to conclude definitively if some quantum states
violate a given Bell inequality.Comment: 13 pages, 1 table, 2 figures. Updated version as published in PR
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