اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدQuantum and Randomized Lower Bounds for Local Search on Vertex-Transitive Graphs
146
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We study the problem of emph{local search} on a graph. Given a real-valued black-box function f on the graphs vertices, this is the problem of determining a local minimum of f--a vertex v for which f(v) is no more than f evaluated at any of vs neighbors. In 1983, Aldous gave the first strong lower bounds for the problem, showing that any randomized algorithm requires $Omega(2^{n/2 - o(1)})$ queries to determine a local minima on the n-dimensional hypercube. The next major step forward was not until 2004 when Aaronson, introducing a new method for query complexity bounds, both strengthened this lower bound to $Omega(2^{n/2}/n^2)$ and gave an analogous lower bound on the quantum query complexity. While these bounds are very strong, they are known only for narrow families of graphs (hypercubes and grids). We show how to generalize Aaronsons techniques in order to give randomized (and quantum) lower bounds on the query complexity of local search for the family of vertex-transitive graphs. In particular, we show that for any vertex-transitive graph G of N vertices and diameter d, the randomized and quantum query complexities for local search on G are $Omega(N^{1/2}/dlog N)$ and $Omega(N^{1/4}/sqrt{dlog N})$, respectively.
قيم البحث
اقرأ أيضاً
While recent work suggests that quantum computers can speed up the solution of semidefinite programs, little is known about the quantum complexity of more general convex optimization. We present a quantum algorithm that can optimize a convex function
over an $n$-dimensional convex body using $tilde{O}(n)$ queries to oracles that evaluate the objective function and determine membership in the convex body. This represents a quadratic improvement over the best-known classical algorithm. We also study limitations on the power of quantum computers for general convex optimization, showing that it requires $tilde{Omega}(sqrt n)$ evaluation queries and $Omega(sqrt{n})$ membership queries.
We prove lower bounds on the error probability of a quantum algorithm for searching through an unordered list of N items, as a function of the number T of queries it makes. In particular, if T=O(sqrt{N}) then the error is lower bounded by a constant.
If we want error <1/2^N then we need T=Omega(N) queries. We apply this to show that a quantum computer cannot do much better than a classical computer when amplifying the success probability of an RP-machine. A classical computer can achieve error <=1/2^k using k applications of the RP-machine, a quantum computer still needs at least ck applications for this (when treating the machine as a black-box), where c>0 is a constant independent of k. Furthermore, we prove a lower bound of Omega(sqrt{log N}/loglog N) queries for quantum bounded-error search of an ordered list of N items.
The goal of the ordered search problem is to find a particular item in an ordered list of n items. Using the adversary method, Hoyer, Neerbek, and Shi proved a quantum lower bound for this problem of (1/pi) ln n + Theta(1). Here, we find the exact va
lue of the best possible quantum adversary lower bound for a symmetrized version of ordered search (whose query complexity differs from that of the original problem by at most 1). Thus we show that the best lower bound for ordered search that can be proved by the adversary method is (1/pi) ln n + O(1). Furthermore, we show that this remains true for the generalized adversary method allowing negative weights.
We generalise the standard constructions of a Cayley graph in terms of a group presentation by allowing some vertices to obey different relators than others. The resulting notion of presentation allows us to represent every vertex transitive graph. A
s an intermediate step, we prove that every countably infinite, connected, vertex transitive graph has a perfect matching. Incidentally, we construct an example of a 2-ended cubic vertex transitive graph which is not a Cayley graph, answering a question of Watkins from 1990.
Shors and Grovers 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 spec
ifically how to prove limits on their computational power. We cover the main known techniques for proving lower bounds, and exemplify and compare the methods.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
