We study shared randomness in the context of multi-party number-in-hand communication protocols in the simultaneous message passing model. We show that with three or more players, shared randomness exhibits new interesting properties that have no direct analogues in the two-party case.
First, we demonstrate a hierarchy of modes of shared randomness, with the usual shared randomness where all parties access the same random string as the form in the hierarchy. We show exponential separations between its levels, and some of our bounds may be of independent interest. For example, we show that the equality function can be solved by a protocol of constant length using the weakest form of shared randomness, which we call XOR-shared
Second, we show that quantum communication cannot replace shared randomness in the k-party case, where k >= 3 is any constant. We demonstrate a promise function GP_k that can be computed by a classical protocol of constant length when (the form of) shared randomness is available, but any quantum protocol without shared randomness must send n^Omega(1) qubits to compute it. Moreover, the quantum complexity of GP_k remains n^Omega(1) even if the second strongest mode of shared randomness is available. While a somewhat similar separation was already known in the two-party case, in the multi-party case our statement is qualitatively stronger:
* In the two-party case, only a relational communication problem with similar properties is known.
* In the two-party case, the gap between the two complexities of a problem can be at most exponential, as it is known that 2^(O(c)) log n qubits can always replace shared randomness in any c-bit protocol. Our bounds imply that with quantum communication alone, in general, it is not possible to simulate efficiently even a three-bit three-party classical protocol that uses shared randomness.
We study shared randomness in the context of multi-party number-in-hand communication protocols in the simultaneous message passing model. We show that with three or more players, shared randomness exhibits new interesting properties that have no direct analogues in the two-party case.
First, we demonstrate a hierarchy of modes of shared randomness, with the usual shared randomness where all parties access the same random string as the form in the hierarchy. We show exponential separations between its levels, and some of our bounds may be of independent interest. For example, we show that the equality function can be solved by a protocol of constant length using the weakest form of shared randomness, which we call XOR-shared
Second, we show that quantum communication cannot replace shared randomness in the k-party case, where k >= 3 is any constant. We demonstrate a promise function GP_k that can be computed by a classical protocol of constant length when (the form of) shared randomness is available, but any quantum protocol without shared randomness must send n^Omega(1) qubits to compute it. Moreover, the quantum complexity of GP_k remains n^Omega(1) even if the second strongest mode of shared randomness is available. While a somewhat similar separation was already known in the two-party case, in the multi-party case our statement is qualitatively stronger:
In the two-party case, only a relational communication problem with similar properties is known.
In the two-party case, the gap between the two complexities of a problem can be at most exponential, as it is known that 2^(O(c)) log n qubits can always replace shared randomness in any c-bit protocol. Our bounds imply that with quantum communication alone, in general, it is not possible to simulate efficiently even a three-bit three-party classical protocol that uses shared randomness.
We study shared randomness in the context of multi-party number-in-hand communication protocols in the simultaneous message passing model. We show that with three or more players, shared randomness exhibits new interesting properties that have no direct analogues in the two-party case. First, we demonstrate a hierarchy of modes of shared randomness, with the usual shared randomness where all parties access the same random string as the strongest form in the hierarchy. We show exponential separations between its levels, and some of our bounds may be of independent interest. For example, we show that the equality function can be solved by a protocol of constant length using the weakest form of shared randomness, which we call "XOR-shared randomness." Second, we show that quantum communication cannot replace shared randomness in the k-party case, where k >= 3 is any constant. We demonstrate a promise function GP_k that can be computed by a classical protocol of constant length when (the strongest form of) shared randomness is available, but any quantum protocol without shared randomness must send n^Omega(1) qubits to compute it. Moreover, the quantum complexity of GP_k remains n^Omega(1) even if the "second strongest" mode of shared randomness is available. While a somewhat similar separation was already known in the two-party case, in the multi-party case our statement is qualitatively stronger: * In the two-party case, only a relational communication problem with similar properties is known. * In the two-party case, the gap between the two complexities of a problem can be at most exponential, as it is known that 2^(O(c)) log n qubits can always replace shared randomness in any c-bit protocol. Our bounds imply that with quantum communication alone, in general, it is not possible to simulate efficiently even a three-bit three-party classical protocol that uses shared randomness.
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.
We study shared randomness in the context of multi-party number-in-hand communication protocols in the simultaneous message passing model. We show that with three or more players, shared randomness exhibits new interesting properties that have no direct analogues in the two-party case. First, we demonstrate a hierarchy of modes of shared randomness, with the usual shared randomness where all parties access the same random string as the strongest form in the hierarchy. We show exponential separations between its levels, and some of our bounds may be of independent interest. For example, we show that the equality function can be solved by a protocol of constant length using the weakest form of shared randomness, which we call XOR-shared randomness. Second, we show that quantum communication cannot replace shared randomness in the k-party case, where k ≥ 3 is any constant. We demonstrate a promise function GPkthat can be computed by a classical protocol of constant length when (the strongest form of) shared randomness is available, but any quantum protocol without shared randomness must send nΩ(1)qubits to compute it. Moreover, the quantum complexity of GPk remains nΩ(1)even if the “second strongest” mode of shared randomness is available. While a somewhat similar separation was already known in the two-party case, in the multi-party case our statement is qualitatively stronger: · In the two-party case, only a relational communication problem with similar properties is known. · In the two-party case, the gap between the two complexities of a problem can be at most exponential, as it is known that 2O(c)log n qubits can always replace shared randomness in any c-bit protocol. Our bounds imply that with quantum communication alone, in general, it is not possible to simulate efficiently even a three-bit three-party classical protocol that uses shared randomness.
This paper addresses two problems in the context of two-party communication complexity of functions. First, it concludes the line of research which can be viewed as demonstrating qualitative advantage of quantum communication in the three most common communication “layouts”: two-way interactive communication, one-way communication and simultaneous message passing (SMP). I demonstrate a functional (problem c̅E̅q̅T̅, whose communication complexity is O (log n)2) in the quantum version of the SMP and Ω̃ (√n) in the classical (randomized) version of SMP. Second, this paper contributes to understanding the power of the weakest commonly studied regime of quantum communication-SMP with quantum messages and without shared randomness (the latter restriction can be viewed as a somewhat artificial way of making the quantum model “as weak as possible”). Our function c̅E̅q̅T̅ has an efficient solution in this regime as well, which means that even lacking shared randomness, quantum SMP can be exponentially stronger than its classical counterpart with shared randomness.
We study the simultaneous message passing (SMP) model of communication complexity, for the case where one party is quantum and the other is classical. We show that in an SMP protocol that computes some function with the first party sending q qubits and the second sending c classical bits, the quantum message can be replaced by a randomized message of O(qc) classical bits, as well as by a deterministic message of O(q c log q) classical bits. Our proofs rely heavily on earlier results due to Scott Aaronson. In particular, our results imply that quantum-classical protocols need to send Omega(sqrt{n/log n}) bits/qubits to compute Equality on n-bit strings, and hence are not significantly better than classical-classical protocols (and are much worse than quantum-quantum protocols such as quantum fingerprinting). This essentially answers a recent question of Wim van Dam. Our results also imply, more generally, that there are no superpolynomial separations between quantum-classical and classical-classical SMP protocols for functional problems. This contrasts with the situation for relational problems, where exponential gaps between quantum-classical and classical-classical SMP protocols are known. We show that this surprising situation cannot arise in purely classical models: there, an exponential separation for a relational problem can be converted into an exponential separation for a functional problem.
Despite the apparent similarity between shared randomness and shared entanglement in the context of Communication Complexity, our understanding of the latter is not as good as of the former. In particular, there is no known "entanglement analogue" for the famous theorem by Newman, saying that the number of shared random bits required for solving any communication problem can be at most logarithmic in the input length (i.e., using more than O(log n) shared random bits would not reduce the complexity of an optimal solution). In this paper we prove that the same is not true for entanglement. We establish a wide range of tight (up to a polylogarithmic factor) entanglement vs. communication tradeoffs for relational problems. The low end is: for any t>2, reducing shared entanglement from log^t(n) to o(log^{t-2}(n)) qubits can increase the communication required for solving a problem almost exponentially, from O(log^t(n)) to Ω(\sqrt n). The high end is: for any \eps>0, reducing shared entanglement from n^{1-\eps}log(n) to o(n^{1-\eps}/log(n)) can increase the required communication from O(n^{1-\eps}log(n)) to Ω(n^{1-\eps/2}/log(n)). The upper bounds are demonstrated via protocols which are exact and work in the \e{simultaneous message passing model}, while the lower bounds hold for \e{bounded-error protocols}, even in the more powerful \e{model of 1-way communication}. Our protocols use shared EPR pairs while the lower bounds apply to any sort of prior entanglement. We base the lower bounds on a strong direct product theorem for communication complexity of a certain class of relational problems. We believe that the theorem might have applications outside the scope of this work.