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arXiv:1308.2450

Exponential Quantum-Classical Gaps in Multiparty Nondeterministic Communication Complexity

Xiaoming Sun, Marcos Villagra

Published Aug 12, 2013• 0 citations
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Abstract

There are three different types of nondeterminism in quantum communication: i) $\nqp$-communication, ii) $\qma$-communication, and iii) $\qcma$-communication. In this \redout{paper} we show that multiparty $\nqp$-communication can be exponentially stronger than $\qcma$-communication. This also implies an exponential separation with respect to classical multiparty nondeterministic communication complexity. We argue that there exists a total function that is hard for $\qcma$-communication and easy for $\nqp$-communication. The proof of it involves an application of the pattern tensor method and a new lower bound for polynomial threshold degree. Another important consequence of this result is that nondeterministic rank can be exponentially lower than the discrepancy bound.

DOI
10.48550/arxiv.1308.2450

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There are three different types of nondeterminism in quantum communication: i) \nqp\nqp\nqp-communication, ii) \qma\qma\qma-communication, and iii) \qcma\qcma\qcma-communication. In this \redout{paper} we show that multiparty \nqp\nqp\nqp-communication can be exponentially stronger than \qcma\qcma\qcma-communication. This also implies an exponential separation with respect to classical multiparty nondeterministic communication complexity. We argue that there exists a total function that is hard for \qcma\qcma\qcma-communication and easy for \nqp\nqp\nqp-communication. The proof of it involves an application of the pattern tensor method and a new lower bound for polynomial threshold degree. Another important consequence of this result is that nondeterministic rank can be exponentially lower than the discrepancy bound.
In this paper we study quantum nondeterminism in multiparty communication. There are three (possibly) different types of nondeterminism in quantum computation: i) strong, ii) weak with classical proofs, and iii) weak with quantum proofs. Here we focus on the first one. A strong quantum nondeterministic protocol accepts a correct input with positive probability and rejects an incorrect input with probability 1. In this work we relate strong quantum nondeterministic multiparty communication complexity to the rank of the communication tensor in the Number-On-Forehead and Number-In-Hand models. In particular, by extending the definition proposed by de Wolf to nondeterministic tensor-rank (nrank), we show that for any boolean function f when there is no prior shared entanglement between the players, 1) in the Number-On-Forehead model the cost is upper-bounded by the logarithm of nrank(f); 2) in the Number-In-Hand model the cost is lower-bounded by the logarithm of nrank(f). Furthermore, we show that when the number of players is o(log log n), we have NQP⊈BQPNQP\nsubseteq BQPNQP⊈BQP for Number-On-Forehead communication.
We study nondeterministic quantum algorithms for Boolean functions f. Such algorithms have positive acceptance probability on input x iff f(x)=1. In the setting of query complexity, we show that the nondeterministic quantum complexity of a Boolean function is equal to its ``nondeterministic polynomial'' degree. We also prove a quantum-vs-classical gap of 1 vs n for nondeterministic query complexity for a total function. In the setting of communication complexity, we show that the nondeterministic quantum complexity of a two-party function is equal to the logarithm of the rank of a nondeterministic version of the communication matrix. This implies that the quantum communication complexities of the equality and disjointness functions are n+1 if we do not allow any error probability. We also exhibit a total function in which the nondeterministic quantum communication complexity is exponentially smaller than its classical counterpart.
We study nondeterministic quantum algorithms for Boolean functions f. Such algorithms have positive acceptance probability on input x iff f(x)=1. In the setting of query complexity, we show that the nondeterministic quantum complexity of a Boolean function is equal to its "nondeterministic polynomial" degree. We also prove a quantum-vs.-classical gap of 1 vs. n for nondeterministic query complexity for a total function. In the setting of communication complexity, we show that the nondeterministic quantum complexity of a two-party function is equal to the logarithm of the rank of a nondeterministic version of the communication matrix. This implies that the quantum communication complexities of the equality and disjointness functions are n+1 if we do not allow any error probability. We also exhibit a total function in which the nondeterministic quantum communication complexity is exponentially smaller than its classical counterpart.
We show several results related to interactive proof modes of communication complexity. First we show lower bounds for the QMA-communication complexity of the functions Inner Product and Disjointness. We describe a general method to prove lower bounds for QMA-communication complexity, and show how one can 'transfer' hardness under an analogous measure in the query complexity model to the communication model using Sherstov's pattern matrix method. Combining a result by Vereshchagin and the pattern matrix method we find a communication problem with AM-communication complexity O(log⁡n)O(\log n)O(logn), PP-communication complexity Ω(n1/3)Ω(n^{1/3})Ω(n1/3), and QMA-communication complexity Ω(n1/6)Ω(n^{1/6})Ω(n1/6). Hence in the world of communication complexity noninteractive quantum proof systems are not able to efficiently simulate co-nondeterminism or interaction. These results imply that the related questions in Turing machine complexity theory cannot be resolved by 'algebrizing' techniques. Finally we show that in MA-protocols there is an exponential gap between one-way protocols and two-way protocols (this refers to the interaction between Alice and Bob). This is in contrast to nondeterministic, AM-, and QMA-protocols, where one-way communication is essentially optimal.
We define a quantum model for multiparty communication complexity and prove a simulation theorem between the classical and quantum models. As a result of our simulation, we show that if the quantum k-party communication complexity of a function f is Ω(n/2k)Ω(n/2^k)Ω(n/2k), then its classical k-party communication is Ω(n/2k/2)Ω(n/2^{k/2})Ω(n/2k/2). Finding such an f would allow us to prove strong classical lower bounds for (k>log n) players and hence resolve a main open question about symmetric circuits. Furthermore, we prove that for the Generalized Inner Product (GIP) function, the quantum model is exponentially more efficient than the classical one. This provides the first exponential separation for a total function between any quantum and public coin randomized communication model.
We prove lower bounds on the bounded error quantum communication complexity. Our methods are based on the Fourier transform of the considered functions. First we generalize a method for proving classical communication complexity lower bounds developed by Raz [Comput. Complexity, 5 (1995), pp. 205–221] to the quantum case. Applying this method, we give an exponential separation between bounded error quantum communication complexity and nondeterministic quantum communication complexity. We develop several other lower bound methods based on the Fourier transform, notably showing that sˉ(f)/log⁡n\sqrt{\bar{s}(f)/\log n}sˉ(f)/logn​, for the average sensitivity sˉ(f)\bar{s}(f)sˉ(f) of a function f, yields a lower bound on the bounded error quantum communication complexity of f((x∧y)⊕z)f((x \wedge y)\oplus z)f((x∧y)⊕z), where x is a Boolean word held by Alice and y,zy,zy,z are Boolean words held by Bob. We then prove the first large lower bounds on the bounded error quantum communication complexity of functions, for which a polynomial quantum speedup is possible. For all the functions we investigate, the only previously applied general lower bound method based on discrepancy yields bounds that are O(log⁡n)O(\log n)O(logn).