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.
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), then 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 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 relative power of public randomness and quantum communication in the multiparty Simultaneous Message Passing (SMP) model. By analyzing a total Boolean function and a relational problem, we establish polynomial and exponential separations between classical public-coin protocols and quantum protocols without shared randomness or prior entanglement.
We first introduce XOR Equality, a total Boolean function in which each of k players receives an unrestricted string xj∈{0,1}n, and the referee must decide whether x1⊕⋯⊕xk=0n. For k≥8 and worst-case error at most 1/3, we prove quantum lower bounds of Ω(min{n,k2}) on total communication and Ω(min{n/k,k}) on maximum message length. A classical public-coin protocol uses two bits per player with error at most 1/4. For n=k2 and worst-case error at most 1/3, we prove that quantum protocols require maximum message length Ω(n) qubits, whereas two public-coin bits per player suffice.
We then turn to relations, for which we obtain exponential separations. Specifically, we study Multiparty Index Coordination, a k-player generalization of the two-player relation studied by Gavinsky, Kempe, Regev, and de Wolf (STOC 2006). We prove an unambiguous quantum lower bound of Ω(n1−1/k) on maximum message length, matched by a classical private-coin protocol. For any fixed error ε∈(0,1/8), we also prove a bounded-error quantum lower bound of Ω(n(k−1)/(k+1)) while we show that public-coin protocols solve the relation unambiguously with maximum message length O(logn), giving exponential separations in both regimes. The implicit constants are independent of k, and both quantum lower bounds become linear when k≥clogn for any fixed c>0.
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.
A major open problem in communication complexity is whether or not quantum protocols can be exponentially more efficient than classical ones for computing a {\em total} Boolean function in the two-party interactive model. Razborov's result ({\em Izvestiya: Mathematics}, 67(1):145--159, 2002) implies the conjectured negative answer for functions F of the following form: F(x,y)=fn(x1⋅y1,x2⋅y2,...,xn⋅yn), where fn is a {\em symmetric} Boolean function on n Boolean inputs, and xi, yi are the i'th bit of x and y, respectively. His proof critically depends on the symmetry of fn. We develop a lower-bound method that does not require symmetry and prove the conjecture for a broader class of functions. Each of those functions F is the ``block-composition'' of a ``building block'' gk:{0,1}k×{0,1}k→{0,1}, and an fn:{0,1}n→{0,1}, such that F(x,y)=fn(gk(x1,y1),gk(x2,y2),...,gk(xn,yn)), where xi and yi are the i'th k-bit block of x,y∈{0,1}nk, respectively. We show that as long as g_k itself is "hard'' enough, its block-composition with an arbitrary f_n has polynomially related quantum and classical communication complexities. For example, when g_k is the Inner Product function with k=\Omega(\log n), the deterministic communication complexity of its block-composition with any f_n is asymptotically at most the quantum complexity to the power of 7.
We consider a quantum and a classical version of a multi-party function computation problem with n players, where players 2,…,n need to communicate appropriate information to player 1 so that a “generalized” inner product function with an appropriate promise can be calculated. In the quantum version of the protocol, the players have access to entangled qudits but the communication is still classical. The communication complexity of a given protocol is the total number of classical bits that need to be communicated. When n is prime, and for our chosen function, we exhibit a quantum protocol (with complexity (n−1)(logn) bits) and a classical protocol (with complexity ((n−1)2(logn2) bits)). Furthermore, we present an integer linear programming formulation for determining a lower bound on the classical communication complexity. This demonstrates that our quantum protocol is strictly better than classical protocols.
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) . 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) 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.