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SIAM Journal on Computing

Exponential Separation of Quantum and Classical One-Way Communication Complexity

Ziv Bar-Yossef, T. S. Jayram, Iordanis Kerenidis

Published Jan 1, 2008• 51 citations
Publisher

Abstract

We give the first exponential separation between quantum and bounded-error randomized one-way communication complexity. Specifically, we define the Hidden Matching Problem HM$_n$: Alice gets as input a string ${\bf x}\in\{0, 1\}^n$, and Bob gets a perfect matching M on the n coordinates. Bob's goal is to output a tuple $\langle i,j,b \rangle$ such that the edge $(i,j)$ belongs to the matching M and $b=x_i\oplus x_j$. We prove that the quantum one-way communication complexity of HM$_n$ is $O(\log n)$, yet any randomized one-way protocol with bounded error must use $\Omega({\sqrt{n}})$ bits of communication. No asymptotic gap for one-way communication was previously known. Our bounds also hold in the model of Simultaneous Messages (SM), and hence we provide the first exponential separation between quantum SM and randomized SM with public coins. For a Boolean decision version of HM$_n$, we show that the quantum one-way communication complexity remains $O(\log n)$ and that the 0-error randomized one-way communication complexity is $\Omega(n)$. We prove that any randomized linear one-way protocol with bounded error for this problem requires $\Omega(\sqrt[3]{n \log n})$ bits of communication.

DOI
10.1137/060651835

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We give the first exponential separation between quantum and bounded-error randomized one-way communication complexity. Specifically, we define the Hidden Matching Problem HMn_nn​: Alice gets as input a string x∈{0,1}n{\bf x}\in\{0, 1\}^nx∈{0,1}n, and Bob gets a perfect matching M on the n coordinates. Bob's goal is to output a tuple ⟨i,j,b⟩\langle i,j,b \rangle⟨i,j,b⟩ such that the edge (i,j)(i,j)(i,j) belongs to the matching M and b=xi⊕xjb=x_i\oplus x_jb=xi​⊕xj​. We prove that the quantum one-way communication complexity of HMn_nn​ is O(log⁡n)O(\log n)O(logn), yet any randomized one-way protocol with bounded error must use Ω(n)\Omega({\sqrt{n}})Ω(n​) bits of communication. No asymptotic gap for one-way communication was previously known. Our bounds also hold in the model of Simultaneous Messages (SM), and hence we provide the first exponential separation between quantum SM and randomized SM with public coins. For a Boolean decision version of HMn_nn​, we show that the quantum one-way communication complexity remains O(log⁡n)O(\log n)O(logn) and that the 0-error randomized one-way communication complexity is Ω(n)\Omega(n)Ω(n). We prove that any randomized linear one-way protocol with bounded error for this problem requires Ω(nlog⁡n3)\Omega(\sqrt[3]{n \log n})Ω(3nlogn​) bits of communication.
We give the first exponential separation between quantum and bounded-error randomized one-way communication complexity. Specifically, we define the Hidden Matching Problem HMn: Alice gets as input a string x ∈ (0, 1)n and Bob gets a perfect matching M on the n coordinates. Bob's goal is to output a tuple [i,j,b] such that the edge (i,j) belongs to the matching M and b = xi ⊕ xj. We prove that the quantum one-way communication complexity of HMn is O(log n), yet any randomized one-way protocol with bounded error must use Ω(√n) bits of communication. No asymptotic gap for one-way communication was previously known. Our bounds also hold in the model of Simultaneous Messages (SM) and hence we provide the first exponential separation between quantum SM and randomized SM with public coins.For a Boolean decision version of HMn, we show that the quantum one-way communication complexity remains O(log n) and that the 0-error randomized one-way communication complexity is Ω(n). We prove that any randomized linear one-way protocol with bounded error for this problem requires Ω(√[3] n log n) bits of communication.
Abstract. We give the first exponential separation between quantum and bounded-error randomized one-way communication complexity. Specifically, we define the Hidden Matching Problem HMn: Alice gets as input a string x ∈ {0,1} n, and Bob gets a perfect matching M on the n coordinates. Bob’s goal is to output a tuple 〈i, j, b 〉 such that the edge (i,j) belongs to the matching M and b = xi ⊕ xj. We prove that the quantum one-way communication complexity of HMn is O(logn), yet any randomized one-way protocol with bounded error must use Ω ( √ n) bits of communication. No asymptotic gap for one-way communication was previously known. Our bounds also hold in the model of Simultaneous Messages (SM), and hence we provide the first exponential separation between quantum SM and randomized SM with public coins. For a Boolean decision version of HMn, we show that the quantum one-way communication complexity remains O(logn) and that the 0-error randomized one-way communication complexity is Ω(n). We prove that any randomized linear one-way protocol with bounded error for this problem requires Ω ( 3 √ nlog n) bits of communication.
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 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: f can be computed by a simultaneous-message quantum protocol with communication complexity polylog( N ) (where at the beginning of the protocol Alice and Bob also have polylog( N ) entangled EPR pairs). Any classical randomized protocol for f , with any number of rounds, has communication complexity at least Ω~(N1/4)\tilde{\Omega}\left(N^{1/4}\right)Ω~(N1/4) . 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 (2016). Item (3) is new. (Our result is incomparable to the one of Gavinsky. While he obtained a quantitatively better lower bound of Ω(N1/2)\Omega\left(N^{1/2}\right)Ω(N1/2) 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 Göös et al. (2017), Chattopadhyay et al. (2019a). However, these separations seem to require quantum protocols with at least two rounds of communication, so they imply a separation of two-way quantum and classical communication complexity, but they do not give the stronger separations of simultaneous-message quantum communication complexity vs. two-way classical communication complexity (or even one-way quantum communication complexity vs. two-way classical communication complexity). Our proof technique is completely new, in the context of communication complexity, and is based on techniques from Raz & Tal (2019). Our function f is based on a lift of the forrelation problem, using xor as a gadget.
We consider the following problem of bounded-error quantum state identification: Given either state α0\alpha_0α0​ or state α1\alpha_1α1​, we are required to output “0”, “1”, or “?” (“don't know"), such that conditioned on outputting “0” or “1”, our guess is correct with high probability. The goal is to maximize the probability of not outputting “?”. We prove the following direct product theorem: If we are given two such problems, with optimal probabilities a and b, respectively, and the states in the first problem are pure, then the optimal probability for the joint bounded-error state identification problem is O(ab)O(ab)O(ab). Our proof is based on semidefinite programming duality. Using this result, we present two exponential separations in the simultaneous message passing model of communication complexity. First, we describe a relation that can be computed with O(log⁡n)O(\log n)O(logn) classical bits of communication in the presence of shared randomness, but needs Ω(n1/3)\Omega(n^{1/3})Ω(n1/3) communication if the parties don't share randomness, even if communication is quantum. This shows the optimality of Yao's recent exponential simulation of shared-randomness protocols by quantum protocols without shared randomness. Combined with an earlier separation in the other direction due to Bar-Yossef, Jayram, and Kerenidis, this shows that the quantum simultaneous message passing (SMP) model is incomparable with the classical shared-randomness SMP model. Second, we describe a relation that can be computed with O(log⁡n)O(\log n)O(logn) classical bits of communication in the presence of shared entanglement, but needs Ω((n/log⁡n)1/3)\Omega((n/\log n)^{1/3})Ω((n/logn)1/3) communication if the parties share randomness but no entanglement, even if communication is quantum. This is the first example in communication complexity of a situation where entanglement buys much more than quantum communication.
Numbers-on-Forehead (NOF) communication model is a central model in communication complexity. As a restricted variant, one-way NOF model is of particular interest. Establishing strong one-way NOF lower bounds would imply circuit lower bounds, resolve well-known problems in additive combinatorics, and yield wide-ranging applications in areas such as cryptography and distributed computing. However, proving strong lower bounds in one-way NOF communication remains highly challenging; many fundamental questions in one-way NOF communication remain wide open. One of the fundamental questions, proposed by Gavinsky and Pudlák (CCC 2008), is to establish an explicit exponential separation between quantum and classical one-way NOF communication. In this paper, we resolve this open problem by establishing the first exponential separation between quantum and randomized communication complexity in one-way NOF model. Specifically, we define a lifted variant of the Hidden Matching problem of Bar-Yossef, Jayram, and Kerenidis (STOC 2004) and show that it admits an (O(log⁡n)O(\log n)O(logn))-cost quantum protocol in the one-way NOF setting. By contrast, we prove that any kkk-party one-way randomized protocol for this problem requires communication Ω(n1/32k/3)Ω(\frac{n^{1/3}}{2^{k/3}})Ω(2k/3n1/3​). Notably, our separation applies even to a generalization of kkk-player one-way communication, where the first player speaks once, and all other k−1k-1k−1 players can communicate freely.
We give a tight lower bound of Omega(\sqrt{n}) for the randomized one-way communication complexity of the Boolean Hidden Matching Problem [BJK04]. Since there is a quantum one-way communication complexity protocol of O(\log n) qubits for this problem, we obtain an exponential separation of quantum and classical one-way communication complexity for partial functions. A similar result was independently obtained by Gavinsky, Kempe, de Wolf [GKdW06]. Our lower bound is obtained by Fourier analysis, using the Fourier coefficients inequality of Kahn Kalai and Linial [KKL88].