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arXiv:2203.10201

A Quantum Algorithm for Network Reliability

Stefan Pabst, Yunseong Nam

Published Mar 19, 2022• 3 citations
arXivHTMLPDF

Abstract

Building a network that is resilient to a component failure is vital. Our access to electricity and telecommunications or the internet of things all hinge on an uninterrupted service provided by a robust network. Calculating the network reliability $R$ is $\sharp$P-complete and intractable to calculate exactly for medium and large networks. Here, we present an explicit, circuit-level implementation of a quantum algorithm that computes $R$. Our algorithm requires $O(EV/ε)$ gate operations and $O(E)$ qubits, where $V$ and $E$ are the number of nodes and edges in the graph and $ε$ is the uncertainty in the reliability estimation. This constitutes a significant polynomial speedup over the best classical approaches currently known. We further provide quantum gate counts, relevant for both pre-fault-tolerant and fault-tolerant regimes, sufficient to compute $R$.

DOI
10.48550/arxiv.2203.10201

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Building a network that is resilient to a component failure is vital. Our access to electricity and telecommunications or the internet of things all hinge on an uninterrupted service provided by a robust network. Calculating the network reliability RRR is ♯\sharp♯P-complete and intractable to calculate exactly for medium and large networks. Here, we present an explicit, circuit-level implementation of a quantum algorithm that computes RRR. Our algorithm requires O(EV/ε)O(EV/ε)O(EV/ε) gate operations and O(E)O(E)O(E) qubits, where VVV and EEE are the number of nodes and edges in the graph and εεε is the uncertainty in the reliability estimation. This constitutes a significant polynomial speedup over the best classical approaches currently known. We further provide quantum gate counts, relevant for both pre-fault-tolerant and fault-tolerant regimes, sufficient to compute RRR.
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Quantum computing promises to solve some problems that are intractable by classical computers. Several quantum processors based on different technologies and consisting of a few tens of noisy qubits have already been developed. However, qubits are fragile as they tend to decohere extremely quickly and quantum operations are faulty, making reliable computation very difficult. Moreover, quantum processors have hardware constraints such as limited qubit connectivity and shared classical control, making quantum algorithms not directly executable. This thesis focuses on some of the challenges of the implementation of quantum algorithms on near-termintermediate-scale and future large-scale quantum processors. More precisely, it investigates how to perform reliable quantum computation using fault-tolerant protocols and how to execute quantum algorithms on hardware-constrained processors using circuit-mapping techniques.
We introduce the concept of quantum reliability as an extension of the concept of network reliability in the context of quantum networks. We show that this concept is intimately related to the concept of quantum reliability operator that we also introduce and show a Negami like splitting formula for it. Considering that the simple factorization formula for classical networks which is the basis of most of the calculation algorithms does not hold in the quantum context due to entanglement, a Negami like splitting for the quantum reliability operator becomes relevant.
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