Near-Optimal Bounds on Bounded-Round Quantum Communication Complexity of Disjointness

Abstract

We prove a near optimal round-communication tradeoff for the two-party quantum communication complexity of disjointness. For protocols with r rounds, we prove a lower bound of Ω(n/r + r) on the communication required for computing disjointness of input size n, which is optimal up to logarithmic factors. The previous best lower bound was Ω(n/r 2 + r) due to Jain, Radhakrishnan and Sen [JRS03]. Along the way, we develop several tools for quantum information complexity, one of which is a lower bound for quantum information complexity in terms of the generalized discrepancy method. As a corollary, we get that the quantum communication complexity of any boolean function f is at most 2 O(QIC(f )) , where QIC(f ) is the prior-free quantum information complexity of f (with error 1/3). Research supported in part by an FRQNT B2 Doctoral Research Scholarship and by CryptoWorks21.6 From AND to Disj 31 7 Proof of the main result 34 8 Low information protocol for AND 35 2 1 Introduction We prove near-optimal bounds on the bounded-round quantum communication complexity of disjointness. Quantum communication complexity, introduced by Yao [Yao93], studies the amount of quantum communication that two parties, Alice and Bob, need to exchange in order to compute a function (usually boolean) of their private inputs. It is the natural quantum extension of classical communication complexity [Yao79]. While the inputs are classical and the end result is classical, the players are allowed to use quantum resources while communicating. The motivation for the introduction of quantum communication was to study questions in quantum computation. For example, in [Yao93], Yao used it to prove that the majority function does not have any linear size quantum formulas. While quantum communication (with entanglement) offers only a factor of 2 savings when transmitting n bits of classical information [Hol73, BW92, CvDNT98], it can still offer superconstant savings (and sometimes exponential) in communication if the goal is just to compute a boolean function of the inputs. For total boolean functions, the best-known separation between classical and quantum communication is quadratic, for the disjointness function [KS92, Raz92, Gro96, BCW98, AA03]. It is, in fact, a major open problem whether classical and quantum communication are polynomially related for all total boolean functions. For partial functions, exponential separations are known even between one-way quantum communication and arbitrary classical communication [Raz99, KR11]. For disjointness with input size n, Grover's search [Gro96, BBHT98] can be used to obtain a quantum communication protocol (with probability of error 1/3) with communication cost O( √ n log n) [BCW98]. The bound was later improved to O( √ n) in [AA03]. The protocols attaining 2 Proof overview and discussion High-level strategy. At a high-level, the proof builds on the connection between quantum information complexity and quantum communication complexity of the disjointness function DISJ m with various values of m. There ar...

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