In the simultaneous message model, two parties holding n-bit integers x,y send messages to a third party, the referee, enabling him to compute a boolean function f(x,y). Buhrman et al [3] proved the remarkable result that, when f is the equality function, the referee can solve this problem by comparing short "quantum fingerprints" sent by the two parties, i.e., there exists a quantum protocol using only O(log n) bits. This is in contrast to the well-known classical case for which Ω(n1/2) bits are provably necessary for the same problem even with randomization. In this paper we show that short quantum fingerprints can be used to solve the problem for a much larger class of functions. Let R<??par line>,pub(f) denote the number of bits needed in the classical case, assuming in addition a common sequence of random bits is known to all parties (the public coin model). We prove that, if R<??par line>,pub(f)=O(1), then there exists a quantum protocol for f using only O(log n) bits. As an application we show that O(log n) quantum bits suffice for the bounded Hamming distance function, defined by f(x,y)=1 if and only if x and y have a constant Hamming distance d or less.
In the simultaneous message model, two parties holding n-bit integers x,y send messages to a third party, the referee, enabling him to compute a boolean function f(x,y). Buhrman et al [3] proved the remarkable result that, when f is the equality function, the referee can solve this problem by comparing short "quantum fingerprints" sent by the two parties, i.e., there exists a quantum protocol using only O(log n) bits. This is in contrast to the well-known classical case for which Ω(n1/2) bits are provably necessary for the same problem even with randomization. In this paper we show that short quantum fingerprints can be used to solve the problem for a much larger class of functions. Let R<??par line>,pub(f) denote the number of bits needed in the classical case, assuming in addition a common sequence of random bits is known to all parties (the public coin model). We prove that, if R<??par line>,pub(f)=O(1), then there exists a quantum protocol for f using only O(log n) bits. As an application we show that O(log n) quantum bits suffice for the bounded Hamming distance function, defined by f(x,y)=1 if and only if x and y have a constant Hamming distance d or less.
In the simultaneous message model, two parties holding n-bit integers x,y send messages to a third party, the referee, enabling him to compute a boolean function f(x,y). Buhrman et al [3] proved the remarkable result that, when f is the equality function, the referee can solve this problem by comparing short "quantum fingerprints" sent by the two parties, i.e., there exists a quantum protocol using only O(log n) bits. This is in contrast to the well-known classical case for which Ω(n1/2) bits are provably necessary for the same problem even with randomization. In this paper we show that short quantum fingerprints can be used to solve the problem for a much larger class of functions. Let R<??par line>,pub(f) denote the number of bits needed in the classical case, assuming in addition a common sequence of random bits is known to all parties (the public coin model). We prove that, if R<??par line>,pub(f)=O(1), then there exists a quantum protocol for f using only O(log n) bits. As an application we show that O(log n) quantum bits suffice for the bounded Hamming distance function, defined by f(x,y)=1 if and only if x and y have a constant Hamming distance d or less.
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.
Fingerprinting is a technique in communication complexity in which two parties (Alice and Bob) with large data sets send short messages to a third party (a referee), who attempts to compute some function of the larger data sets. For the equality function, the referee attempts to determine whether Alice's data and Bob's data are the same. In this paper, we consider the extreme scenario of performing fingerprinting where Alice and Bob both send either one bit (classically) or one qubit (in the quantum regime) messages to the referee for the equality problem. Restrictive bounds are demonstrated for the error probability of one-bit fingerprinting schemes, and show that it is easy to construct one-qubit fingerprinting schemes which can outperform any one-bit fingerprinting scheme. The author hopes that this analysis will provide results useful for performing physical experiments, which may help to advance implementations for more general quantum communication protocols.
Buhrman, Cleve, Watrous, and de Wolf (PRL 2001) discovered the quantum fingerprinting protocol, which is the quantum SMP protocol with O(logn) qubits communication for the equality problem. In the protocol, Alice and Bob create some quantum fingerprints of their inputs, and the referee conducts the SWAP tests for the quantum fingerprints. Since Ω(n) bits communication is required with the classical SMP scheme for the equality problem first shown by Newman and Szegedy (STOC 1996), there exists an exponential quantum advantage in the amount of communication. In this paper, we consider a setting in which the referee can do only incoherent measurements rather than coherent measurements including the SWAP tests. We first show that, in the case of one-way LOCC measurements, Ω(n) qubits communication is required. To prove the result, we derive a new method to replace quantum messages by classical messages and consider a reduction to the optimal lower bound in the hybrid SMP model where one message is quantum and the other is classical, which was first shown by Klauck and Podder (MFCS 2014). Our method uses the result of Oszmaniec, Guerini, Wittek, and Acín (PRL 2017), who showed that general POVM measurements can be simulated by randomized projective measurements with small ancilla qubits, and Newman's theorem. We further investigate the setting of quantum SMP protocols with two-way LOCC measurements, and derive a lower bound against some restricted two-way LOCC measurements. To prove it, we revisit the technique to replace quantum messages by classical deterministic messages introduced by Aaronson (ToC 2005) and generalized by Gavinsky, Regev, and de Wolf (CJTCS 2008), and show that, using the deterministic message, the referee can simulate the two-way LOCC measurements.
We study the simultaneous message passing model of communication complexity. Building on the quantum fingerprinting protocol of Buhrman et al., Yao recently showed that a large class of efficient classical public-coin protocols can be turned into efficient quantum protocols without public coin. This raises the question whether this can be done always, i.e. whether quantum communication can always replace a public coin in the SMP model. We answer this question in the negative, exhibiting a communication problem where classical communication with public coin is exponentially more efficient than quantum communication. Together with a separation in the other direction due to Bar-Yossef et al., this shows that the quantum SMP model is incomparable with the classical public-coin SMP model. In addition we give a characterization of the power of quantum fingerprinting by means of a connection to geometrical tools from machine learning, a quadratic improvement of Yao's simulation, and a nearly tight analysis of the Hamming distance problem from Yao's paper.
One of the standard communication complexity problems is deciding whether bit strings held by two separate parties Alice (A) and Bob (B) are equal or different. In the simultaneous message passing model, A and B send certain information to a Referee, who makes the decision. The overall goal is to minimize the amount of communication involved while not exceeding desired error probability. This is achieved by sending only fingerprints of the original n-bit strings. In the best classical protocols, these scale as O(√n) if A and B have no access to shared randomness [1]. The recently proposed [2] and demonstrated quantum fingerprinting [3], exploiting phase-modulated weak coherent pulses, provides O(log2n) scaling in the number of used qubits, but needs phase stability between A and B. Here we introduce and analyse a quantum fingerprinting protocol based on two-photon interference which does require a shared phase reference and works in scenarios when the global phase of transmitted signals becomes completely random. Interestingly, the performance of this protocol, quantified in terms of the Chernoff information [4], is enhanced through the use of non-classical single-photon states, even in the regime of high transmission loss.