CS 860 Quantum Algorithms and Complexity Spring

CS860:Quantum Algorithms and Complexity Spring2006Lecture12—June13Lecturer:Andris Ambainis Scribe:Sarvagya Upadhyay1Adversary Lower BoundsThe last few lectures were devoted to adversary lower bound and its various generalizations. We started with two simple adversary lower bound methods due to Ambainis[2]and then generalized it.There are other variants of adversary lower bounds like weighted adversary method due to Ambainis[3],spectral adversary method due to Barnum,Saks and Szegedy[5], strong weighted adversary method due to Zhang[10]and minimax bound due to Laplante and Magniez[7]to name a few.All these bounds are proven to be equivalent[9].Adversary methods are quite useful lower bound methods for quantum query complexity of a problem and have been used to prove optimal lower bounds on searching andfinding local minimum [1].However,these bounds have some limitations.Beginning in this lecture,we will look into the trianglefinding problem and prove a lower bound on it using the simplest adversary method.We will then show that any adversary method will give us the same lower bound in two different ways:•By using the minimax lower bound method and equivalence of variants of adversary methods,we will give an upper bound on lower bounds that can be proven for the triangle problem.•We will show a general upper bound on minimax lower bound using certificate com-plexity that applies to many problems,in particular,the triangle problem.We then upper-bound the best lower bound obtained by any adversary method for several other problems discussed in previous lectures:finding a copy of a subgraph H on k-vertices in a given graph G and perfect matching problem.In the second part of the lecture,we show a semidefinite program(SDP)which gives a general lower bound method for quantum query complexity and subsumes the adversary lower bound method and its generalizations[5].That is,we will formulate a SDP P(f,t,ǫ) that is feasible if and only if f can be evaluated by a t-step quantum query algorithm within errorǫ.In the next section,we focus our attention of triangle problem and derive a lower bound using Ambainis adversary method.In section3,we will prove that the lower bound obtained is the best possible lower bound obtained by any adversary method.In section4,we give a general upper bound on any adversary lower bound method using certificate complexity. Finally,in the last section,we give a SDP which lower bounds the quantum query complexity.12-1mm′)queries to compute f.We construct the two sets A and B as follows.A will consist of(some)graphs with no triangle and B is will consist of(some)graphs with a triangle.Consider a complete bipartite graph over(⌊n/2⌋,⌈n/2⌉)vertices.This graph contains no triangle.Adding an edge to this graph will give a graph which contains exactly one triangle.These set of graphs will serve as a choice for our sets A and B,respectively,i.e.•A={K⌊n/2⌋,⌈n/2⌉}.•B is the set of all possible graphs which are obtained by adding an edge to K⌊n/2⌋,⌈n/2⌉.Clearly,the sets A and B also satisfy the constraints of the theorem.All we need to do is to calculate the value of m and m′.m is the cardinality of set B and hence m= n2 . Similarly,m′=|A|=1.Therefore,the lower bound we obtain by using the above values of m and m′is√ n2 =Ω(n)LB(f△)=Ω(We got a lower bound on the triangle problem using the simplest adversary lower bound method.Now few questions arise.12-2i:x i=y i i:x i=y i(i,j)= 31p G(i,j)p′G(i,j)=Ω 1p G(i,j)p′GC0(f)C1(f).2.For any partial Boolean function f,MM(f)≤p x(i)p y(i)≥ 1|S y|=1C0(f)C1(f)12-4i:x i=y i C0(f)C1(f)This proves the required upper bound for the total Boolean function.For partial Boolean function the above argument fails.Recall that we made a claim that there exists an i∈S x∩S y such that x i=y i.This claim is true if the Boolean function is total.For partial Boolean function,the proof of the claim is not always true because we may end up with a z for which f(z)is not defined.In this case,we define the same probability distribution on the input x satisfying f(x)=0with certificate S x.p x(i)= 1/|S x|if i∈S x0otherwiseFor inputs y such that f(y)=1,we define probability distribution as p y(i)=1p x(i)p y(i)≥ |S x|1nC0(f).Interchanging the probability distributions for the inputs x:f(x)=0and y:f(y)=1and following the same argument,we get MM(f)≤n min{C0(f),C1(f)}.Now,we show that the best lower bound obtained by any adversary method for triangle finding isΩ(n).For trianglefinding,C0(f△)≤ n2 =O(n2)and C1(f△)=3.Therefore, MM(f△)≤ 3·O(n2)=O(n).In fact,we canfind a best lower bound obtained by adversary methods for many other problems discussed in previous lectures.Suppose,our problem is tofind whether a graph G contains a subgraph H on k-vertices. The zero and one certificate complexities of the following problem areC0(f H⊆G)≤ n2 and C1(f H⊆G)≤ k2√Hence,the best lower bound is MM(f H⊆G)=O(n3=O(n3/2).12-5。