Separating quantum communication and approximate rank Anurag Anshu a , Shalev Ben-David b , Ankit Garg c , Rahul Jain a , d , e , Robin Kothari b , Troy Lee a , e , f a CQT, National University of Singapore b Massachusetts Institute of Technology c Microsoft Research, New England d Dept. of CS, National University of Singapore e MajuLab, UMI 3654, Singapore f SPMS, Nanyang Technological University July 8, 2017 Anurag Anshu a , Shalev Ben-David b , Ankit Garg c , Rahul Jain a , d , e , Robin Kothari b , Troy Lee a , e , f (CQT) Separations in communication complexity July 8, 2017 1 / 22
Roadmap Some background 1 Separating quantum communication and approximate rank 2 Anurag Anshu a , Shalev Ben-David b , Ankit Garg c , Rahul Jain a , d , e , Robin Kothari b , Troy Lee a , e , f (CQT) Separations in communication complexity July 8, 2017 2 / 22
Models of query complexity For a function F , Randomized (two-sided error of ε ) query complexity R dt ε ( F ), Quantum (two sided error of ε ) query complexity Q dt ε ( F ). Quadratic separation: using Grover’s search algorithm [Grov95] and its variant proved in [BBHT96]. OR: { 0 , 1 } n → { 0 , 1 } outputs 1 if the input contains at least one 1. Q dt 1 / 3 R dt 2 [BBHT96] 1 / 3 Anurag Anshu a , Shalev Ben-David b , Ankit Garg c , Rahul Jain a , d , e , Robin Kothari b , Troy Lee a , e , f (CQT) Separations in communication complexity July 8, 2017 3 / 22
Lower bounds on quantum query complexity For a function F , approximate polynomial degree deg ε ( F ) is the minimum among the degrees of all polynomials p ( x ) satisfying | p ( x ) − F ( x ) | ≤ ε , for all x . It lower bounds quantum query complexity [Beals, Buhrman, Cleve, ε ( F ) ≥ 1 Mosca, de Wolf 1998]: Q dt 2 deg ε ( F ). Example: deg 1 / 3 ( OR ) = Θ( √ n ). Other well known bounds: Adversary bound [Ambainis 2000], Negative weights adversary bound [Hoyer, Lee, Spalek 2005]. Anurag Anshu a , Shalev Ben-David b , Ankit Garg c , Rahul Jain a , d , e , Robin Kothari b , Troy Lee a , e , f (CQT) Separations in communication complexity July 8, 2017 4 / 22
Degree not a tight lower bound 1 / 3 ( F ) = O ( deg 1 / 3 ( F )) 6 . It is known that Q dt Moreover, there exists a function F , such that 1 / 3 ( F ) = Θ( deg 1 / 3 ( F )) 1 . 3219 [Ambainis 2003]. Q dt Is this the best possible separation? Anurag Anshu a , Shalev Ben-David b , Ankit Garg c , Rahul Jain a , d , e , Robin Kothari b , Troy Lee a , e , f (CQT) Separations in communication complexity July 8, 2017 5 / 22
Cheat sheets Aaronson, Ben-David and Kothari [2016] introduced the technique of cheat sheet. Follow up to the works G¨ o¨ os, Pitassi and Watson [2015] and Ambainis, Balodis, Belovs, Lee, Santha and Smotrovs [2015]. A transformation from F → F cs . Q dt 1 / 3 R dt 2 . 5 [ABK16] 1 / 3 Anurag Anshu a , Shalev Ben-David b , Ankit Garg c , Rahul Jain a , d , e , Robin Kothari b , Troy Lee a , e , f (CQT) Separations in communication complexity July 8, 2017 6 / 22
Cheat sheets Aaronson, Ben-David and Kothari [2015] introduced the technique of cheat sheet. Follow up to the works G¨ o¨ os, Pitassi and Watson [2015] and Ambainis, Balodis, Belovs, Lee, Santha and Smotrovs [2015]. A transformation from F → F cs . deg 1 / 3 Q dt 4 [ABK16] 1 / 3 Anurag Anshu a , Shalev Ben-David b , Ankit Garg c , Rahul Jain a , d , e , Robin Kothari b , Troy Lee a , e , f (CQT) Separations in communication complexity July 8, 2017 7 / 22
Cheat sheet review F cs has two components: ‘ c ’ copies of a parent function F and a cheat sheet cs . Compute based on inputs to functions and content at ‘ decimal ( b )’. b = F 1 , . . . F c F 1 F c 2 c 1 2 Anurag Anshu a , Shalev Ben-David b , Ankit Garg c , Rahul Jain a , d , e , Robin Kothari b , Troy Lee a , e , f (CQT) Separations in communication complexity July 8, 2017 8 / 22
Communication complexity F y x Randomized communication complexity R 1 / 3 ( F ): number of bits communicated in a randomized protocol. Quantum communication complexity Q 1 / 3 ( F ): number of qubits communicated in an entanglement assisted quantum protocol. Anurag Anshu a , Shalev Ben-David b , Ankit Garg c , Rahul Jain a , d , e , Robin Kothari b , Troy Lee a , e , f (CQT) Separations in communication complexity July 8, 2017 9 / 22
Lower bound on quantum communication complexity Approximate rank for F , rk ε ( F ) = min F ′ { rk ( F ′ ) : | F ′ ( x , y ) − F ( x , y ) | ≤ ε } . Lower bound on quantum communication complexity [Buhrman and de Wolf 2001, Lee and Shraibman 2008]: For F : { 0 , 1 } n × { 0 , 1 } n → { 0 , 1 } , Q 1 / 3 ( F ) ≥ Ω(log rk 1 / 3 ( F ) − log n ) . Quantum log-rank conjecture: are Q 1 / 3 ( F ) and log rk 1 / 3 ( M F ) polynomially related? Anurag Anshu a , Shalev Ben-David b , Ankit Garg c , Rahul Jain a , d , e , Robin Kothari b , Troy Lee a , e , f (CQT) Separations in communication complexity July 8, 2017 10 / 22
Lower bound on quantum communication complexity Approximate rank for F , rk ε ( F ) = min F ′ { rk ( F ′ ) : | F ′ ( x , y ) − F ( x , y ) | ≤ ε } . Lower bound on quantum communication complexity [Buhrman and de Wolf 2001, Lee and Shraibman 2008]: For F : { 0 , 1 } n × { 0 , 1 } n → { 0 , 1 } , Q 1 / 3 ( F ) ≥ Ω(log rk 1 / 3 ( F ) − log n ) . Quantum log-rank conjecture: are Q 1 / 3 ( F ) and log rk 1 / 3 ( F ) polynomially related? Other lower bound: quantum information complexity ([Touchette 2015]). Anurag Anshu a , Shalev Ben-David b , Ankit Garg c , Rahul Jain a , d , e , Robin Kothari b , Troy Lee a , e , f (CQT) Separations in communication complexity July 8, 2017 11 / 22
Cheat sheets in communication complexity Notion of cheat sheet extended to communication complexity in A., Belovs, Ben-David, G¨ o¨ os, Jain, Kothari, Lee and Santha [2016]. A similar transformation: F → F G , called look-up function. Super-quadratic separation between R 1 / 3 ( F ) and Q 1 / 3 ( F ). Anurag Anshu a , Shalev Ben-David b , Ankit Garg c , Rahul Jain a , d , e , Robin Kothari b , Troy Lee a , e , f (CQT) Separations in communication complexity July 8, 2017 12 / 22
Look-up function F G F 1 F : X ⊗ Y → { 0 , 1 } x 1 y 1 F 1 , F 2 . . . F c ≡ F F c x c y c u 0 v 0 u 1 v 1 G : X ⊗ c ⊗ Y ⊗ c ⊗ W → { 0 , 1 } u 0 , v 0 , u 1 , v 1 . . . u 2 c , v 2 c ∈ W W is set of strings u 2 c v 2 c Anurag Anshu a , Shalev Ben-David b , Ankit Garg c , Rahul Jain a , d , e , Robin Kothari b , Troy Lee a , e , f (CQT) Separations in communication complexity July 8, 2017 13 / 22
Look-up function F G F 1 x 1 y 1 compute b = ( F 1 , F 2 , . . . F c ) F c x c y c u 0 v 0 u 1 v 1 u 2 c v 2 c Anurag Anshu a , Shalev Ben-David b , Ankit Garg c , Rahul Jain a , d , e , Robin Kothari b , Troy Lee a , e , f (CQT) Separations in communication complexity July 8, 2017 14 / 22
Look-up function F G F 1 x 1 y 1 goto block number decimal( b ) F c x c y c u 0 v 0 u 1 v 1 u 2 c v 2 c Anurag Anshu a , Shalev Ben-David b , Ankit Garg c , Rahul Jain a , d , e , Robin Kothari b , Troy Lee a , e , f (CQT) Separations in communication complexity July 8, 2017 15 / 22
Look-up function F G F 1 x 1 y 1 F c x c y c u 0 v 0 u 1 v 1 F G = 1 Iff G ( u b ⊕ v b , x 1 , y 1 . . . x c , y c ) = 1 u 2 c v 2 c Anurag Anshu a , Shalev Ben-David b , Ankit Garg c , Rahul Jain a , d , e , Robin Kothari b , Troy Lee a , e , f (CQT) Separations in communication complexity July 8, 2017 16 / 22
Quantum communication complexity of look-up function For reasonably non-trivial function G , we show the following. Theorem 1 Q 1 / 3 ( F G ) = Ω(log disc ( F )) . disc ( F ) is the discrepancy of F . Anurag Anshu a , Shalev Ben-David b , Ankit Garg c , Rahul Jain a , d , e , Robin Kothari b , Troy Lee a , e , f (CQT) Separations in communication complexity July 8, 2017 17 / 22
An outline of proof We show that for any r -round protocol Π for F G that makes an error 3 , there exists a protocol Π ′ for F that makes an error of 1 of 1 2 − 1 r 2 and communicates the same as in Π. disc ( F ) − log r 2 ). 1 So, Q 1 / 3 ( F G ) = Ω( Q 1 r 2 ( F )) = Ω(log 2 − 1 Anurag Anshu a , Shalev Ben-David b , Ankit Garg c , Rahul Jain a , d , e , Robin Kothari b , Troy Lee a , e , f (CQT) Separations in communication complexity July 8, 2017 18 / 22
An outline of proof Key idea: Quantum cut and paste theorem [Jain, Radhakrishnan and Sen 2003, Nayak and Touchette 2016]. In a protocol where each player has low information about content of the correct location of other player’s ‘look up part’, output cannot be correct. Anurag Anshu a , Shalev Ben-David b , Ankit Garg c , Rahul Jain a , d , e , Robin Kothari b , Troy Lee a , e , f (CQT) Separations in communication complexity July 8, 2017 19 / 22
Choice of G Recall: in cheat sheet of Aaronson, Ben-David and Kothari, correct cheat sheet location must certify the evaluation of F 1 , F 2 , . . . F c on their inputs. Fix a circuit C for F , with number of gates size ( F ). We require that u b ⊕ v b certifies the evaluation of inputs (to F 1 , F 2 , . . . F c ) on C . Anurag Anshu a , Shalev Ben-David b , Ankit Garg c , Rahul Jain a , d , e , Robin Kothari b , Troy Lee a , e , f (CQT) Separations in communication complexity July 8, 2017 20 / 22
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