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Quantum Information and Computation

The computational power of the W and GHZ states

Ellie D’Hondt, Prakash Panangaden

Published Mar 1, 2006• 94 citations
Publisher

Abstract

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.

DOI
10.26421/qic6.2-3

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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.
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.
Quantum computing and distributed systems may enter a mutually beneficial partnership in the future. On the one hand, it is much easier to build a number of small quantum computers rather than a single large one. On the other hand, the best results concerning some of the fundamental problems in distributed computing can potentially be dramatically improved upon by taking advantage of the superior resources and processing power that quantum mechanics offers. This survey has the purpose to highlight both of these benefits. We first review the current results regarding the implementation of arbitrary quantum algorithms on distributed hardware. We then discuss existing proposals for quantum solutions of leader election - a fundamental problem from distributed computing. Quantum mechanics allows leader election to be solved with no communication, provided that certain pre-shared entanglement is already in place. Further, an impossibility result from classical distributed computing is circumvented by the quantum solution of anonymous leader election - a unique leader is elected in finite time with certainty. Finally, we discuss the viability of these proposals from a practical perspective. Although, theoretically, distributed quantum computing looks promising, it is still unclear how to build quantum hardware and how to create and maintain robust large-scale entangled states. Moreover, it is not clear whether the costs of creating entangled states and working with them are smaller than the costs of existing classical solutions.
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.
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.
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.