Download Separations in communication complexity using cheat sheet ... and more Slides Algorithms and Programming in PDF only on Docsity!
Separations in communication complexity using cheat
sheet and information complexity
Anurag Anshua, Aleksandrs Belovsb, Shalev Ben-Davidc^ , Mika G¨o¨osd^ , Rahul Jaina,e,f^ , Robin Kotharic^ , Troy Leea,f^ ,g^ , Miklos Santhaa,h a (^) CQT, National University of Singapore b (^) University of Latvia c (^) Massachusetts Institute of Technology d (^) SEAS, Harvard University e (^) Dept. of CS, National University of Singapore f (^) MajuLab, UMI 3654, Singapore g (^) SPMS, Nanyang Technological University h (^) IRIF, Universit´e Paris Diderot, CNRS
January 16, 2017
Roadmap
(^1) Some background
(^2) New separations in communication complexity
Communication complexity
F
x y
Randomized communication complexity R(F ): number of bits communicated in a randomized protocol. Quantum communication complexity Q(F ): number of qubits communicated in an entanglement assisted quantum protocol. Information complexity IC (F ): amount of information about input that must be revealed (to other party) to compute the function.
Porting query separations to communication
A quantum query algorithm for a function gives rise to a quantum communication protocol for a related function [BCW98]. Disjointness function DISJ inputs two subsets x, y of the set { 1 , 2 ,... n} and outputs 0 if the subsets are disjoint. DISJ(x, y ) = OR(x 1 AND y 1 , x 2 AND y 2 ,... , xn AND yn) !!
R
Q
[BCW98]
[KS87],[Raz91]
Separating exact quantum and randomized
Exact quantum query complexity of F , denoted QdtE (F ), is number of quantum queries needed to compute F with zero error. Similarly we define QE (F ) for communication complexity.
R
Q QE
dt com^ dt com
[ABK16]
[Amb12]
[Amb12]
Separating exact quantum and randomized
Exact quantum query complexity of F , denoted QdtE (F ), is number of quantum queries needed to compute F with zero error. Similarly we define QE (F ) for communication complexity.
R
Q QE
dt com^ dt com
[ABK16]
[ABK16]
[Amb12]
Partition and Randomized
Unambiguous certificate complexity UNdt^ is a lower bound on deterministic query complexity. Analogously Partition number UN in communication complexity. Goos, Pitassi, Watson [2015] presented first super linear separation between UNdt^ and deterministic query complexity. Similar result in communication complexity.
R
Q QE UN
dt com^ dt com^ dt com
[ABK16]
[ABK16]
[Amb12]
[AKK16]
[GJPW]
Super-Disjointness in communication world?
Can we somehow lift these query results to communication? What gadgets should be used? AND is not a good: AND(x 1 AND y 1 ,... , xn AND yn) is easy. Inner Product lifts a lower bound (junta degree) on Rdt^ (F ) to a lower bound on communication complexity R(F ) (smooth rectangle bound) [GLMWZ, 2015]. But we have no idea what is junta degree for cheat sheet function.
Look-up function FG
F 1
x 1 y 1
Fc xc yc
compute b = (F 1 , F 2 ,... Fc )
u 0 v 0
u 1 v 1
u 2 c^ v 2 c
Look-up function FG
F 1
x 1 y 1
Fc xc yc
goto block number decimal(b)
u 0 v 0
u 1 v 1
u 2 c^ v 2 c
Lower bound on communication complexity of look-up
function
For reasonably non-trivial function G, we show the following.
Theorem
R(FG ) = Ω(R(F )/c^2 ) and IC (FG ) = Ω(IC (F )/c^3 ).
An idea of the proof: pointer function
F : X ⊗ Y → { 0 , 1 }
F 1 , F 2... Fc ≡ F
F 1
x 1 y 1
Fc xc yc
u 0 v 0
u 1 v 1
u 2 c^ v 2 c
An idea of the proof: pointer function
F : X ⊗ Y → { 0 , 1 }
F 1 , F 2... Fc ≡ F
F 1
x 1 y 1
Fc xc yc
u 0 v 0
u 1 v 1
u 2 c^ v 2 c
Output ub ⊕ vb
An idea of the proof: pointer function
F : X ⊗ Y → { 0 , 1 }
F 1 , F 2... Fc ≡ F
F 1
x 1 y 1
Fc xc yc
u 0 v 0
u 1 v 1
u 2 c^ v 2 c
Hard distribution for F: μ Distribution for pointer: μ⊗c^ ⊗ uniformUV
