Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

IEEE Conference on Computational Complexity (CCC)

Lower Bounds on Quantum Multiparty Communication Complexity

Troy Lee, Gideon Schechtman, Adi Shraibman

Published Jul 1, 2009• 21 citations
Publisher

Abstract

A major open question in communication complexity is if randomized and quantum communication are polynomially related for all total functions. So far, no gap larger than a power of two is known, despite significant efforts. We examine this question in the number-on-the-forehead model of multiparty communication complexity. We show that essentially all lower bounds known on randomized complexity in this model also hold for quantum communication. This includes bounds of size Omega(n/2k) for the k-party complexity of explicit functions, bounds for the generalized inner product function, and recent work on the multiparty complexity of disjointness. To the best of our knowledge, these are the first lower bounds of any kind on quantum communication in the general number-on-the-forehead model. We show this result in the following way. In the two-party case, there is a lower bound on quantum communication complexity in terms of a norm gamma2, which is known to subsume nearly all other techniques in the literature. For randomized complexity there is another natural bound in terms of a different norm mu which is also one of the strongest techniques available. A deep theorem in functional analysis, Grothendieck's inequality, implies that gamma2and mu are equivalent up to a constant factor. This connection is one of the major obstacles to showing a larger gap between randomized and quantum communication complexity in the two-party case. The lower bound technique in terms of the norm mu was recently extended to the multiparty number-on-the-forehead model. Here we show how the gamma2norm can be also extended to lower bound quantum multiparty complexity. Surprisingly, even in this general setting the two lower bounds, on quantum and classical communication, are still very closely related. This implies that separating quantum and classical communication in this setting will require the development of new techniques. The relation between these extensions of mu and gamma2is proved by a multi-dimensional version of Grothendieck's inequality.

DOI
10.1109/ccc.2009.24

Related papers

Multiparty Communication Complexity of Disjointness

Arkadev Chattopadhyay, Anil Ada

We obtain a lower bound of n^Omega(1) on the k-party randomized communication complexity of the Disjointness function in the `Number on the Forehead' model of multiparty communication when k is a constant. For k=o(loglog n), the bounds remain super-polylogarithmic i.e. (log n)^omega(1). The previous best lower bound for three players until recently was Omega(log n). Our bound separates the communication complexity classes NP^{CC}_k and BPP^{CC}_k for k=o(loglog n). Furthermore, by the results of Beame, Pitassi and Segerlind \cite{BPS07}, our bound implies proof size lower bounds for tree-like, degree k-1 threshold systems and superpolynomial size lower bounds for Lovasz-Schrijver proofs. Sherstov \cite{She07b} recently developed a novel technique to obtain lower bounds on two-party communication using the approximate polynomial degree of boolean functions. We obtain our results by extending his technique to the multi-party setting using ideas from Chattopadhyay \cite{Cha07}. A similar bound for Disjointness has been recently and independently obtained by Lee and Shraibman.

arXiv:0801.3624•Jan 23, 2008•63 citations

Approximation norms and duality for communication complexity lower bounds

Troy Lee, Adi Shraibman

Abstract: We will discuss a general norm based framework for showing lower bounds on communication complexity. An advantage of this approach is that one can use duality theory to obtain a lower bound quantity phrased as a maximization problem, which can be more convenient to work with in showing lower bounds. We discuss two applications of this approach. 1. The approximation rank of a matrix A is the minimum rank of a matrix close to A in ell_infty norm. The logarithm of approximation rank lower bounds quantum communication complexity and is one of the most powerful techniques available, albeit difficult to compute in practice. We show that an approximation norm known as gamma_2 is polynomially related to approximation rank. This results in a polynomial time algorithm to approximate approximation rank, and also shows that the logarithm of approximation rank lower bounds quantum communication complexity even with entanglement which was previously not known. 2. By means of an approximation norm which lower bounds multiparty number-on-the-forehead complexity, we show non-trivial lower bounds on the complexity of the disjointness function for up to c log log n players, c <1.

Dagstuhl Seminar Proceedings•Jan 1, 2008•0 citations

Quantum multiparty communication complexity and circuit lower bounds

Iordanis Kerenidis

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 $Ω(n/2^k)$, then its classical k-party communication is $Ω(n/2^{k/2})$. 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.

arXiv:quant-ph/0504087•Apr 12, 2005•1 citation

Quantum multiparty communication complexity and circuit lower bounds

Iordanis Kerenidis

We define a quantum model for multiparty communication complexity and prove a simulation theorem between the classical and quantum models. As a result, we show that if the quantum k-party communication complexity of a function f is Ω(n/2k), its classical k-party communication is Ω(n/2k/2). Finding such an f would allow us to prove strong classical lower bounds for k ≥ log n players and make progress towards solving a major open question about symmetric circuits.

Mathematical Structures in Computer Science•Feb 1, 2009•6 citations

Communication Lower Bounds Using Directional Derivatives

Alexander A. Sherstov

We study the set disjointness problem in the most powerful model of bounded-error communication, the k -party randomized number-on-the-forehead model. We show that set disjointness requires Ω(√n/2 k k ) bits of communication, where n is the size of the universe. Our lower bound generalizes to quantum communication, where it is essentially optimal. Proving this bound was a longstanding open problem even in restricted settings, such as one-way classical protocols with k =4 parties [Wigderson 1997]. The proof contributes a novel technique for lower bounds on multiparty communication, based on directional derivatives of protocols over the reals.

Journal of the ACM•Dec 17, 2014•54 citations

Multiparty Communication Complexity of AC 0

Paul W. Beame, Dang-Trinh Huynh-Ngoc

We prove n Ω(1) lower bounds on the multiparty communication complexity of AC 0 functions in the number-on-forehead (NOF) model for up to Θ(log n) players. These are the first lower bounds for any AC 0 function for ω(log log n) players. In particular we show that there are families of depth 3 read-once AC 0 formulas having k-player randomized multiparty NOF communication complexity n Ω(1) /2 O(k). We show similar lower bounds for depth 4 read-once AC 0 formulas that have nondeterministic communication complexity O(log 2 n), yielding exponential separations between k-party nondeterministic and randomized communication complexity for AC 0 functions. As a consequence of the latter bound, we obtain an n Ω(1/k) /2 O(k) lower bound on the k-party NOF communication complexity of set disjointness. This is non-trivial for up to Θ (√ log n) players which is significantly larger than the up to Θ(log log n) players allowed in the best previous lower bounds for multiparty set disjointness given by Lee and Shraibman [LS08] and Chattopadhyay and Ada [CA08] (though our complexity bounds themselves are not as strong as those in [LS08, CA08] for o(log log n) players). We derive these results by extending the k-party generalization in [CA08, LS08] of the pattern matrix method of Sherstov [She07, She08]. Using this technique, we derive a new sufficient criterion for strong communication complexity lower bounds based on functions having many diverse subfunctions that do not have good low-degree polynomial approximations. This criterion guarantees that such functions have orthogonalizing distributions that are “max-smooth” as opposed to the “min-smooth” orthogonalizing distributions used by Razborov and Sherstov [RS08] to analyze the sign-rank of AC⁰.

Electronic colloquium on computational complexity•Jan 1, 2008•10 citations
A major open question in communication complexity is if randomized and quantum communication are polynomially related for all total functions. So far, no gap larger than a power of two is known, despite significant efforts. We examine this question in the number-on-the-forehead model of multiparty communication complexity. We show that essentially all lower bounds known on randomized complexity in this model also hold for quantum communication. This includes bounds of size Omega(n/2k) for the k-party complexity of explicit functions, bounds for the generalized inner product function, and recent work on the multiparty complexity of disjointness. To the best of our knowledge, these are the first lower bounds of any kind on quantum communication in the general number-on-the-forehead model. We show this result in the following way. In the two-party case, there is a lower bound on quantum communication complexity in terms of a norm gamma2, which is known to subsume nearly all other techniques in the literature. For randomized complexity there is another natural bound in terms of a different norm mu which is also one of the strongest techniques available. A deep theorem in functional analysis, Grothendieck's inequality, implies that gamma2and mu are equivalent up to a constant factor. This connection is one of the major obstacles to showing a larger gap between randomized and quantum communication complexity in the two-party case. The lower bound technique in terms of the norm mu was recently extended to the multiparty number-on-the-forehead model. Here we show how the gamma2norm can be also extended to lower bound quantum multiparty complexity. Surprisingly, even in this general setting the two lower bounds, on quantum and classical communication, are still very closely related. This implies that separating quantum and classical communication in this setting will require the development of new techniques. The relation between these extensions of mu and gamma2is proved by a multi-dimensional version of Grothendieck's inequality.
We obtain a lower bound of n^Omega(1) on the k-party randomized communication complexity of the Disjointness function in the `Number on the Forehead' model of multiparty communication when k is a constant. For k=o(loglog n), the bounds remain super-polylogarithmic i.e. (log n)^omega(1). The previous best lower bound for three players until recently was Omega(log n). Our bound separates the communication complexity classes NP^{CC}_k and BPP^{CC}_k for k=o(loglog n). Furthermore, by the results of Beame, Pitassi and Segerlind \cite{BPS07}, our bound implies proof size lower bounds for tree-like, degree k-1 threshold systems and superpolynomial size lower bounds for Lovasz-Schrijver proofs. Sherstov \cite{She07b} recently developed a novel technique to obtain lower bounds on two-party communication using the approximate polynomial degree of boolean functions. We obtain our results by extending his technique to the multi-party setting using ideas from Chattopadhyay \cite{Cha07}. A similar bound for Disjointness has been recently and independently obtained by Lee and Shraibman.
Abstract: We will discuss a general norm based framework for showing lower bounds on communication complexity. An advantage of this approach is that one can use duality theory to obtain a lower bound quantity phrased as a maximization problem, which can be more convenient to work with in showing lower bounds. We discuss two applications of this approach. 1. The approximation rank of a matrix A is the minimum rank of a matrix close to A in ell_infty norm. The logarithm of approximation rank lower bounds quantum communication complexity and is one of the most powerful techniques available, albeit difficult to compute in practice. We show that an approximation norm known as gamma_2 is polynomially related to approximation rank. This results in a polynomial time algorithm to approximate approximation rank, and also shows that the logarithm of approximation rank lower bounds quantum communication complexity even with entanglement which was previously not known. 2. By means of an approximation norm which lower bounds multiparty number-on-the-forehead complexity, we show non-trivial lower bounds on the complexity of the disjointness function for up to c log log n players, c <1.
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 study the set disjointness problem in the most powerful model of bounded-error communication, the k -party randomized number-on-the-forehead model. We show that set disjointness requires Ω(√n/2 k k ) bits of communication, where n is the size of the universe. Our lower bound generalizes to quantum communication, where it is essentially optimal. Proving this bound was a longstanding open problem even in restricted settings, such as one-way classical protocols with k =4 parties [Wigderson 1997]. The proof contributes a novel technique for lower bounds on multiparty communication, based on directional derivatives of protocols over the reals.
We prove n Ω(1) lower bounds on the multiparty communication complexity of AC 0 functions in the number-on-forehead (NOF) model for up to Θ(log n) players. These are the first lower bounds for any AC 0 function for ω(log log n) players. In particular we show that there are families of depth 3 read-once AC 0 formulas having k-player randomized multiparty NOF communication complexity n Ω(1) /2 O(k). We show similar lower bounds for depth 4 read-once AC 0 formulas that have nondeterministic communication complexity O(log 2 n), yielding exponential separations between k-party nondeterministic and randomized communication complexity for AC 0 functions. As a consequence of the latter bound, we obtain an n Ω(1/k) /2 O(k) lower bound on the k-party NOF communication complexity of set disjointness. This is non-trivial for up to Θ (√ log n) players which is significantly larger than the up to Θ(log log n) players allowed in the best previous lower bounds for multiparty set disjointness given by Lee and Shraibman [LS08] and Chattopadhyay and Ada [CA08] (though our complexity bounds themselves are not as strong as those in [LS08, CA08] for o(log log n) players). We derive these results by extending the k-party generalization in [CA08, LS08] of the pattern matrix method of Sherstov [She07, She08]. Using this technique, we derive a new sufficient criterion for strong communication complexity lower bounds based on functions having many diverse subfunctions that do not have good low-degree polynomial approximations. This criterion guarantees that such functions have orthogonalizing distributions that are “max-smooth” as opposed to the “min-smooth” orthogonalizing distributions used by Razborov and Sherstov [RS08] to analyze the sign-rank of AC⁰.