Introduction Quantum algorithm Lower bounds On “Testing group commutativity” by F.Magniez and A.Nayak Laura Mancinska University of Waterloo, Department of C&O April 3, 2008
Introduction Quantum algorithm Lower bounds Introduction
Introduction Quantum algorithm Lower bounds Black box groups Black box group model Elements of the group are encoded as words over a finite alphabet
Introduction Quantum algorithm Lower bounds Black box groups Black box group model Elements of the group are encoded as words over a finite alphabet Group operation is performed by a black box containing oracles O G and O − 1 G O G | g, h � = | g, gh � O − 1 � � g, g − 1 h � G | g, h � =
Introduction Quantum algorithm Lower bounds Black box groups Black box group model Elements of the group are encoded as words over a finite alphabet Group operation is performed by a black box containing oracles O G and O − 1 G O G | g, h � = | g, gh � O − 1 � � g, g − 1 h � G | g, h � = When do we use black box groups?
Introduction Quantum algorithm Lower bounds Group Commutativity Problem Input: Generators g 1 , . . . , g k of G (specified as n − bit strings)
Introduction Quantum algorithm Lower bounds Group Commutativity Problem Input: Generators g 1 , . . . , g k of G (specified as n − bit strings) Black box: Oracles O G and O − 1 G
Introduction Quantum algorithm Lower bounds Group Commutativity Problem Input: Generators g 1 , . . . , g k of G (specified as n − bit strings) Black box: Oracles O G and O − 1 G Task: Determine whether G is abelian
Introduction Quantum algorithm Lower bounds Classical algorithms for Group commutativity Naive algorithm with query complexity Θ( k 2 ) . This is optimal deterministic algorithm up to a constant [I.Pak, 2000].
Introduction Quantum algorithm Lower bounds Classical algorithms for Group commutativity Naive algorithm with query complexity Θ( k 2 ) . This is optimal deterministic algorithm up to a constant [I.Pak, 2000]. Randomized algorithm with query complexity Θ( k ) [I.Pak, 2000]. This is optimal randomized algorithm up to a constant [F.Magniez, A.Nayak, 2005]
Introduction Quantum algorithm Lower bounds Randomized algorithm for group commutativity Definition. Define random subproduct as 1 . . . g a k h = g a 1 k , where a i ∈ { 0 , 1 } are determined by independent tosses of a fair coin.
Introduction Quantum algorithm Lower bounds Randomized algorithm for group commutativity Definition. Define random subproduct as 1 . . . g a k h = g a 1 k , where a i ∈ { 0 , 1 } are determined by independent tosses of a fair coin. Algorithm: 1 Take two random subproducts h 1 , h 2
Introduction Quantum algorithm Lower bounds Randomized algorithm for group commutativity Definition. Define random subproduct as 1 . . . g a k h = g a 1 k , where a i ∈ { 0 , 1 } are determined by independent tosses of a fair coin. Algorithm: 1 Take two random subproducts h 1 , h 2 2 Test whether h 1 h 2 = h 2 h 1
Introduction Quantum algorithm Lower bounds Randomized algorithm for group commutativity Definition. Define random subproduct as 1 . . . g a k h = g a 1 k , where a i ∈ { 0 , 1 } are determined by independent tosses of a fair coin. Algorithm: 1 Take two random subproducts h 1 , h 2 2 Test whether h 1 h 2 = h 2 h 1 3 Repeat steps 1,2 for c times (to give correct answer with � 3 � c ) probability at least 1 − 4
Introduction Quantum algorithm Lower bounds Randomized algorithm for group commutativity Definition. Define random subproduct as 1 . . . g a k h = g a 1 k , where a i ∈ { 0 , 1 } are determined by independent tosses of a fair coin. Algorithm: 1 Take two random subproducts h 1 , h 2 2 Test whether h 1 h 2 = h 2 h 1 3 Repeat steps 1,2 for c times (to give correct answer with � 3 � c ) probability at least 1 − 4 4 Answer that G is abelian if the tested subproducts commuted
Introduction Quantum algorithm Lower bounds Randomized algorithm for group commutativity Definition. Define random subproduct as 1 . . . g a k h = g a 1 k , where a i ∈ { 0 , 1 } are determined by independent tosses of a fair coin. Algorithm: 1 Take two random subproducts h 1 , h 2 ( ≤ 2 k queries) 2 Test whether h 1 h 2 = h 2 h 1 3 Repeat steps 1,2 for c times (to give correct answer with � 3 � c ) probability at least 1 − 4 4 Answer that G is abelian if the tested subproducts commuted
Introduction Quantum algorithm Lower bounds Randomized algorithm for group commutativity Definition. Define random subproduct as 1 . . . g a k h = g a 1 k , where a i ∈ { 0 , 1 } are determined by independent tosses of a fair coin. Algorithm: 1 Take two random subproducts h 1 , h 2 ( ≤ 2 k queries) 2 Test whether h 1 h 2 = h 2 h 1 (2 queries) 3 Repeat steps 1,2 for c times (to give correct answer with � 3 � c ) probability at least 1 − 4 4 Answer that G is abelian if the tested subproducts commuted
Introduction Quantum algorithm Lower bounds Quantum algorithm
Introduction Quantum algorithm Lower bounds Main steps Construct a random walk on a graph
Introduction Quantum algorithm Lower bounds Main steps Construct a random walk on a graph Quantize the random walk using Szegedy’s approach
Introduction Quantum algorithm Lower bounds Main steps Construct a random walk on a graph Quantize the random walk using Szegedy’s approach Evaluate the quantities in 1 S + √ ( U + C ) δε
Introduction Quantum algorithm Lower bounds Constructing random walk S l – the set of all l -tuples of distinct elements from { 1 , . . . , k }
Introduction Quantum algorithm Lower bounds Constructing random walk S l – the set of all l -tuples of distinct elements from { 1 , . . . , k } g u := g u 1 . . . g u l , where u = ( u 1 , . . . , u l ) ∈ S l
Introduction Quantum algorithm Lower bounds Constructing random walk S l – the set of all l -tuples of distinct elements from { 1 , . . . , k } g u := g u 1 . . . g u l , where u = ( u 1 , . . . , u l ) ∈ S l t u – balanced binary tree with generators g u 1 , . . . , g u l as leaves
Introduction Quantum algorithm Lower bounds Constructing random walk S l – the set of all l -tuples of distinct elements from { 1 , . . . , k } g u := g u 1 . . . g u l , where u = ( u 1 , . . . , u l ) ∈ S l t u – balanced binary tree with generators g u 1 , . . . , g u l as leaves Example. Let l = 4 , k = 20 , u = { 3 , 5 , 10 , 4 } ∈ S 4 .
Introduction Quantum algorithm Lower bounds Constructing random walk S l – the set of all l -tuples of distinct elements from { 1 , . . . , k } g u := g u 1 . . . g u l , where u = ( u 1 , . . . , u l ) ∈ S l t u – balanced binary tree with generators g u 1 , . . . , g u l as leaves Example. Let l = 4 , k = 20 , u = { 3 , 5 , 10 , 4 } ∈ S 4 . Then g u = g 3 · g 5 · g 10 · g 4 and t u looks as follows g u g 3 � g 5 g 10 � g 4 g 3 g 5 g 10 g 4
Introduction Quantum algorithm Lower bounds Constructing random walk Random walk on S l States are trees t u , u ∈ S l
Introduction Quantum algorithm Lower bounds Constructing random walk Random walk on S l States are trees t u , u ∈ S l Transitions from each t u are as follows With probability 1 / 2 stay at t u
Introduction Quantum algorithm Lower bounds Constructing random walk Random walk on S l States are trees t u , u ∈ S l Transitions from each t u are as follows With probability 1 / 2 stay at t u With probability 1 / 2 do Pick a random leave position i ∈ { 1 , · · · , l } and a random 1 generator index j ∈ { 1 , · · · , k }
Introduction Quantum algorithm Lower bounds Constructing random walk Random walk on S l States are trees t u , u ∈ S l Transitions from each t u are as follows With probability 1 / 2 stay at t u With probability 1 / 2 do Pick a random leave position i ∈ { 1 , · · · , l } and a random 1 generator index j ∈ { 1 , · · · , k } If j = u m for some m , exchange u i and u m , else set u i = j 2
Introduction Quantum algorithm Lower bounds Constructing random walk Random walk on S l States are trees t u , u ∈ S l Transitions from each t u are as follows With probability 1 / 2 stay at t u With probability 1 / 2 do Pick a random leave position i ∈ { 1 , · · · , l } and a random 1 generator index j ∈ { 1 , · · · , k } If j = u m for some m , exchange u i and u m , else set u i = j 2 Update tree t u 3
Introduction Quantum algorithm Lower bounds Constructing quantum walk We quantize a random walk consisting of two independent random walks on S l
Introduction Quantum algorithm Lower bounds Constructing quantum walk We quantize a random walk consisting of two independent random walks on S l States are pairs of trees ( t u , t v ) , where u, v ∈ S l
Introduction Quantum algorithm Lower bounds Constructing quantum walk We quantize a random walk consisting of two independent random walks on S l States are pairs of trees ( t u , t v ) , where u, v ∈ S l If transition matrix of the walk on S l was P , then the new transition matrix is P ⊗ P
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