New algorithms for testing monotonicity Alexander Belov CWI Eric - PowerPoint PPT Presentation

New algorithms for testing monotonicity Alexander Belov CWI Eric Blais University of Waterloo

Monotone functions Definition (Monotone functions; M ) f : { 0 , 1 } n → { 0 , 1 } is monotone if for every x � y ∈ { 0 , 1 } n , it satisfies f ( x ) ≤ f ( y ). 1 / 16

Functions that are far from monotone Definition (Functions far from monotone; M ǫ ) f : { 0 , 1 } n → { 0 , 1 } is ǫ -far from monotone if for every monotone function g , we have |{ x : f ( x ) � = g ( x ) }| ≥ ǫ 2 n . 2 / 16

Testing monotonicity vs. How many queries does a bounded-error randomized algorithm need to distinguish monotone functions from functions that are ǫ -far from monotone? 3 / 16

Edge tester Definition (Goldreich, Goldwasser, Lehman, Ron ’98) The edge tester selects edges ( x, y ) of the hypercube uniformly at random and checks that f ( x ) ≤ f ( y ). 4 / 16

Pair testers Definition (Dodis, Goldreich, Lehman, Raskhodnikova, Ron, Samorodnitsky ’99) A pair tester selects comparable pairs x � y ∈ { 0 , 1 } n from some distribution and checks that f ( x ) ≤ f ( y ). 5 / 16

Another view of pair testers The query complexity of pair testers can also be viewed as the solution to the following optimization problem. � minimize φ x,y x � y � subject to φ x,y ≥ 1 ∀ f ∈ M ǫ x � y : f ( x ) >f ( y ) ∀ x � y ∈ { 0 , 1 } n φ x,y ≥ 0 6 / 16

A different optimization problem 2   � � minimize max φ x,y ( f )  f ∈M∪M ǫ x y � x   � �  = 1 subject to φ x,y ( f ) · φ x,y ( g ) ∀ f ∈ M , g ∈ M ǫ . y � x x : f ( x ) � = g ( x ) 7 / 16

A different optimization problem 2   � � minimize max φ x,y ( f )  f ∈M∪M ǫ x y � x   � �  = 1 subject to φ x,y ( f ) · φ x,y ( g ) ∀ f ∈ M , g ∈ M ǫ . y � x x : f ( x ) � = g ( x ) Corollary (to the Dual adversary bound Theorem) Every feasible solution to this problem gives an upper bound on the quantum query complexity for testing monotonicity. 7 / 16

The dual adversary bound Theorem (Dual adversary bound) The quantum query complexity for distinguishing X and Y is the solution to the optimization problem � max X x [ f, f ] minimize f ∈X∪Y x � X x [ f, g ] = 1 ∀ f ∈ X , g ∈ Y subject to x : f ( x ) � = g ( x ) ∀ x ∈ { 0 , 1 } n X x � 0 8 / 16

Simplifying the optimization problem 2   �  � minimize max φ x,j ( f )  f ∈M∪M ǫ x j ∈ [ n ] � � s.t. φ x,j ( f ) · φ x,j ( g ) = 1 ∀ f ∈ M , g ∈ M ǫ . x : f ( x ) � = g ( x ) j ∈ [ n ] vs. 9 / 16

First quantum monotonicity tester For f ∈ M , define  1 /L if x j = 0 and f ( x ) = 0   or x j = 1 and f ( x ) = f ( x ⊕ j ) = 1 φ x,j ( f ) =  0 otherwise .  For g ∈ M ǫ , define � if ( x, x ⊕ j ) ∈ E g L/ | E g | φ x,j ( g ) = 0 otherwise where E g is the set of edges of the hypercube on which g is anti-monotone and L is a constant to be fixed later. 10 / 16

First quantum tester: Correctness vs. φ x,j ( f ) · φ x,j ( g ) = | E g | · ( 1 L � � L · | E g | ) = 1 . x : f ( x ) � = g ( x ) j ∈ [ n ] 11 / 16

First quantum tester: Complexity I For f ∈ M , the objective value of the optimization is 2   = n 2 n �  � φ x,j ( f )  L 2 x j ∈ [ n ] And for g ∈ M ǫ , it is 2   | E g | 2 = 2 L L �  � φ x,j ( g ) = 2 | E g | | E g | .  x j ∈ [ n ] 12 / 16

First quantum tester: Complexity II When L = √ nǫ · 2 n − 1 , the objective value of the optimization problem is 2 n √ nǫ �� � max n/ǫ, max . | E g | g ∈M ǫ 13 / 16

First quantum tester: Complexity II When L = √ nǫ · 2 n − 1 , the objective value of the optimization problem is 2 n √ nǫ �� � max n/ǫ, max . | E g | g ∈M ǫ Lemma (Goldreich, Goldwasser, Lehman, Ron, Samorodnitsky ’00) For every g ∈ M ǫ , | E g | ≥ ǫ 2 n . � So the quantum query complexity of the first tester is n/ǫ . 13 / 16

A more flexible optimization problem 2   � � min. max  ψ x ( f ) + φ x,j ( f )  f ∈M∪M ǫ x j ∈ [ n ]   � �  = 1 ∀ ... s.t.  ψ x ( f ) · ψ x ( g ) + φ x,j ( f ) · φ x,j ( g ) x : f ( x ) � = g ( x ) j ∈ [ n ] vs. 14 / 16

Second quantum monotonicity tester Theorem (Belovs, B. ’15) There is a feasible solution to this optimization problem with objective value 2 n √ ǫ � ∆( G g ) � + n 1 / 4 log n | E g | n 1 / 4 where G g is any subgraph of the (1 , 0) -graph of g , ∆( G g ) is its maximum degree, and E g is the set of non-monotone edges in G g . 15 / 16

Second quantum monotonicity tester Theorem (Belovs, B. ’15) There is a feasible solution to this optimization problem with objective value 2 n √ ǫ � ∆( G g ) � + n 1 / 4 log n | E g | n 1 / 4 where G g is any subgraph of the (1 , 0) -graph of g , ∆( G g ) is its maximum degree, and E g is the set of non-monotone edges in G g . Theorem (Khot, Minzer, Safra ’15) For every g ∈ M ǫ , there exists a such a subgraph G g that satisfies � � ǫ 2 n � ∆( G g ) | E g | = Ω . log 2 n 15 / 16

Conclusions O ( n 1 / 4 / √ ǫ ) quantum queries. ◮ We can test monotonicity with ˜ ◮ The design of quantum testers can be done by considering natural optimization problems. ◮ The analysis of quantum monotonicity testers uncovers the key inequalities that are also required to analyze classical monotonicity testers. ◮ Are there other property testing problems where considering quantum testers may yield insights on promising directions? 16 / 16

Thank you! For all the details, see A. Belovs and E.B. Quantum Algorithm for Monotonicity Testing on the Hypercube. Theory of Computing 11(16), 2015.