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arXiv:1406.0798 · Quantum Information and Computation

W-like states are not necessary for totally correct quantum anonymous leader election

Alexander Norton

Published May 1, 2015• 0 citations
arXivHTMLPDF

Abstract

I show that $W$-like entangled quantum states are not a necessary quantum resource for totally correct anonymous leader election protocols. This is proven by defining a symmetric quantum state that is $n$-partite SLOCC inequivalent to the $W$ state, and then constructing a totally correct anonymous leader election protocol using this state. This result, which contradicts the previous necessity result of D'Hondt and Panangaden, furthers our understanding of how non-local quantum states can be used as a resource for distributed computation.

DOI
10.26421/qic15.7-8-8

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I show that WWW-like entangled quantum states are not a necessary quantum resource for totally correct anonymous leader election protocols. This is proven by defining a symmetric quantum state that is nnn-partite SLOCC inequivalent to the WWW state, and then constructing a totally correct anonymous leader election protocol using this state. This result, which contradicts the previous necessity result of D'Hondt and Panangaden, furthers our understanding of how non-local quantum states can be used as a resource for distributed computation.
This article gives a separation between quantum and classical models in pure (i.e., noncryptographic) computing abilities with no restriction on the amount of available computing resources, by considering the exact solvability of the leader election problem in anonymous networks, a celebrated unsolvable problem in classical distributed computing. The goal of the leader election problem is to elect a unique leader from among distributed parties. In an anonymous network, all parties with the same number of communication links are identical. It is well-known that no classical algorithm can exactly solve (i.e., in bounded time without error) the leader election problem in anonymous networks, even if the number of parties is given. This article devises a quantum algorithm that, if the number of parties is given, exactly solves the problem for any network topology in polynomial rounds with polynomial communication/time complexity with respect to the number of parties, when the parties are connected with quantum communication links and they have the ability of quantum computing. Our algorithm works even when only an upper bound of the number of parties is given. In such a case, no classical algorithm can solve the problem even under the zero-error setting, the setting in which error is not allowed but running time may be unbounded.
This paper gives the first separation of quantum and classical pure (i.e., non-cryptographic) computing abilities with no restriction on the amount of available computing resources, by considering the exact solvability of a celebrated unsolvable problem in classical distributed computing, the ``leader election problem'' on anonymous networks. The goal of the leader election problem is to elect a unique leader from among distributed parties. The paper considers this problem for anonymous networks, in which each party has the same identifier. It is well-known that no classical algorithm can solve exactly (i.e., in bounded time without error) the leader election problem in anonymous networks, even if it is given the number of parties. This paper gives two quantum algorithms that, given the number of parties, can exactly solve the problem for any network topology in polynomial rounds and polynomial communication/time complexity with respect to the number of parties, when the parties are connected by quantum communication links.
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.
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.
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.
The anonymous network model was introduced by Angluin in [STOC '80, to understand the fundamental properties of distributed computing by examining how much each party in a network needs to know about its own identity and the identities of other parties.In this model, all parties with the same number of communication links are identical.When studying the model, the problem of electing a unique leader is the central problem in the sense that once a unique leader is elected, he can assign a unique identity to each party.It was proved in the literature, however, that it is impossible to deterministically solve the leader election problem with any finite amount of communication.This contrasts sharply with the situation in the quantum setting: the present authors gave quantum algorithms that exactly solve the problem on anonymous quantum networks, where quantum communication and computation can be performed [TOCT, vol.4,no.1, article 1].The core of their algorithm consists of somewhat tricky unitary operators, which eliminate the possibility of ties during (quantum) coin flipping and thereby reduce the number of leader candidates with certainty.This paper presents a new quantum leader election algorithm that is based on quantum reduction via exact amplitude amplification to a classically solvable problem, computing a certain symmetric function, which provides more intuitive reasoning behind the existence of exact quantum algorithms for leader election.The algorithm first achieves a round complexity that is linear in the number of parties, i.e., the largest possible diameter plus one of the underlying graph of the network.