Solitude verification is arguably one of the simplest fundamental problems in distributed computing, where the goal is to verify that there is a unique contender in a network. This paper devises a quantum algorithm that exactly solves the problem on an anonymous network, which is known as a network model with minimal assumptions [Angluin, STOC'80]. The algorithm runs in @math rounds if every party initially has the common knowledge of an upper bound @math on the number of parties. This implies that all solvable problems can be solved in @math rounds on average without error (i.e., with zero-sided error) on the network. As a generalization, a quantum algorithm that works in @math rounds is obtained for the problem of exactly computing any symmetric Boolean function, over @math distributed input bits, which is constant over all the @math bits whose sum is larger than @math for @math . All these algorithms work with the bit complexities bounded by a polynomial in @math .
Pal, Singh, and Kumar @cite_19 and D'Hondt and Panangaden @cite_6 dealt with @math and the GHZ-state sharing problem in a different setting, where pre-shared entanglement is assumed but only classical communication is allowed. The relation between several network models that differ in available quantum resources has been discussed by Gavoille, Kosowski, and Markiewicz @cite_17 . Recently, @cite_13 proved that quantum communication cannot substantially speed up algorithms for some fundamental problems, such as the minimum spanning tree, compared to the classical setting. For fault-tolerant distributed quantum computing, the Byzantine agreement problem and the consensus problem were studied by Ben-Or and Hassidim @cite_28 and Chlebus, Kowalski, and Strojnowski @cite_8 , respectively. In the cryptographic context where there are cheating parties, Refs. @cite_21 @cite_15 devises quantum algorithms that elect a unique leader with a small bias. Some quantum distributed protocols were experimentally demonstrated by @cite_30 and @cite_3 .
@cite_30 @cite_8 @cite_28 @cite_21 @cite_6 @cite_3 @cite_19 @cite_15 @cite_13 @cite_17
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It is well understood that the use of quantum entanglement significantly enhances the computational power of systems. Much of the attention has focused on Bell states and their multipartite generalizations. However, in the multipartite case it is known that there are several inequivalent classes of states, such as those represented by the W-state and the GHZ-state. Our main contribution is a demonstration of the special computational power of these states in the context of paradigmatic problems from classical distributed computing. Concretely, we show that the W-state is the only pure state that can be used to exactly solve the problem of leader election in anonymous quantum networks. Similarly we show that the GHZ-state is the only one that can be used to solve the problem of distributed consensus when no classical post-processing is considered. These results generalize to a family of W- and GHZ-like states. At the heart of the proofs of these impossibility results lie symmetry arguments. We study the problem of k-pair communication (or multiple unicast problem) of quantum information in networks of quantum channels. We consider the asymptotic rates of high fidelity quantum communication between specific sender-receiver pairs. Four scenarios of classical communication assistance (none, forward, backward, and two-way) are considered. (I) We obtain outer and inner bounds of the achievable rate regions in the most general directed networks. (II) For two particular networks (including the butterfly network), routing is proved optimal, and the free assisting classical communication can at best be used to modify the directions of quantum channels in the network. Consequently, the achievable rate regions are given by counting edge avoiding paths, and precise achievable rate regions in all four assisting scenarios can be obtained. (III) Optimality of routing can also be proved in classes of networks. The first class consists of directed unassisted networks in which (1) the receivers are information sinks, (2) the maximum distance from senders to receivers is small, and (3) a certain type of 4-cycles are absent, but without further constraints (such as on the number of communicating and intermediate parties). The second class consists of arbitrary backward-assisted networks with two sender-receiver pairs. (IV) Beyond the k-pair communication problem, observations are made on quantum multicasting and a static version of network communication related to the entanglement of assistance. In this paper we show that sufficient multi-partite quantum entanglement helps in fair and unbiased election of a leader in a distributed network of processors with only linear classical communication complexity. We show that a total of @math distinct multi-partite maximally entanglement sets (ebits) are capable of supporting such a protocol in the presence of nodes that may lie and thus be biased. Here, @math is the number of nodes in the network. We also demonstrate the difficulty of performing unbiased and fair election of a leader with linear classical communication complexity in the absence of quantum entanglement even if all nodes have perfect random bit generators. We show that the presence of a sufficient number @math of biased agents leads to a non-zero limiting probability of biased election of the leader, whereas, the presence of a smaller number @math of biased agents matters little. We define two new related complexity classes motivated by the our leader election problem and discuss a few open questions. 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.