Abstract
We study a new type of separations between quantum and classical communication complexity, separations that are obtained using quantum protocols where all parties are efficient, in the sense that they can be implemented by small quantum circuits, with oracle access to their inputs. Our main result qualitatively matches the strongest known separation between quantum and classical communication complexity [Dmitry Gavinsky, 2016] and is obtained using a quantum protocol where all parties are efficient. More precisely, we give an explicit partial Boolean function f over inputs of length N, such that:
(1) f can be computed by a simultaneousmessage quantum protocol with communication complexity polylog(N) (where at the beginning of the protocol Alice and Bob also have polylog(N) entangled EPR pairs).
(2) Any classical randomized protocol for f, with any number of rounds, has communication complexity at least Ω̃(N^{1/4}).
(3) All parties in the quantum protocol of Item (1) (Alice, Bob and the referee) can be implemented by quantum circuits of size polylog(N) (where Alice and Bob have oracle access to their inputs).
Items (1), (2) qualitatively match the strongest known separation between quantum and classical communication complexity, proved by Gavinsky [Dmitry Gavinsky, 2016]. Item (3) is new. (Our result is incomparable to the one of Gavinsky. While he obtained a quantitatively better lower bound of Ω(N^{1/2}) in the classical case, the referee in his quantum protocol is inefficient).
Exponential separations of quantum and classical communication complexity have been studied in numerous previous works, but to the best of our knowledge the efficiency of the parties in the quantum protocol has not been addressed, and in most previous separations the quantum parties seem to be inefficient. The only separations that we know of that have efficient quantum parties are the recent separations that are based on lifting [Arkadev Chattopadhyay et al., 2019; Arkadev Chattopadhyay et al., 2019]. However, these separations seem to require quantum protocols with at least two rounds of communication, so they imply a separation of twoway quantum and classical communication complexity but they do not give the stronger separations of simultaneousmessage quantum communication complexity vs. twoway classical communication complexity (or even oneway quantum communication complexity vs. twoway classical communication complexity).
Our proof technique is completely new, in the context of communication complexity, and is based on techniques from [Ran Raz and Avishay Tal, 2019]. Our function f is based on a lift of the forrelation problem, using xor as a gadget.
BibTeX  Entry
@InProceedings{girish_et_al:LIPIcs.ITCS.2021.54,
author = {Uma Girish and Ran Raz and Avishay Tal},
title = {{Quantum Versus Randomized Communication Complexity, with Efficient Players}},
booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
pages = {54:154:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771771},
ISSN = {18688969},
year = {2021},
volume = {185},
editor = {James R. Lee},
publisher = {Schloss DagstuhlLeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/13593},
URN = {urn:nbn:de:0030drops135932},
doi = {10.4230/LIPIcs.ITCS.2021.54},
annote = {Keywords: Exponential Separation, Quantum, Randomized, Communication, Complexity, Forrelation}
}
Keywords: 

Exponential Separation, Quantum, Randomized, Communication, Complexity, Forrelation 
Collection: 

12th Innovations in Theoretical Computer Science Conference (ITCS 2021) 
Issue Date: 

2021 
Date of publication: 

04.02.2021 