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International Conference on Smart Generation Computing, Communication and Networking (SMART GENCON)

Classification of Quantum Graph State Codes using Decision Trees

Vishnu Shetty Belanje, Ankur Raina

Published Dec 23, 2022• 0 citations
Publisher

Abstract

Graph states are multipartite quantum states that exhibit rich entanglement because of their topology. We consider graph states for studying a family of quantum error correcting codes such as$[[n,\ 1,\ d]]$encoding one logical qubit into$n$physical qubits of the graph state. We investigate the impact of edge connectivity on the minimum$d$of the resulting graph state codes (GSCs).

DOI
10.1109/smartgencon56628.2022.10084179

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Graph states are multipartite quantum states that exhibit rich entanglement because of their topology. We consider graph states for studying a family of quantum error correcting codes such as[[n, 1, d]][[n,\ 1,\ d]][[n, 1, d]]encoding one logical qubit intonnnphysical qubits of the graph state. We investigate the impact of edge connectivity on the minimumdddof the resulting graph state codes (GSCs).
Deciding if a given family of quantum states is topologically ordered is an important but nontrivial problem in condensed matter physics and quantum information theory. We derive necessary and sufficient conditions for a family of graph states to be in TQO-1, which is a class of quantum error-correction code states whose code distance scales macroscopically with the number of physical qubits. Using these criteria, we consider a number of specific graph families, including the star and complete graphs, and the line graphs of complete and completely bipartite graphs, and discuss which are topologically ordered and how to construct the codewords. The formalism is then employed to construct several codes with macroscopic distance, including a three-dimensional topological code generated by local stabilizers that also has a macroscopic number of encoded logical qubits. The results indicate that graph states provide a fruitful approach to the construction and characterization of topological stabilizer quantum error-correction codes.
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In a previous paper a straight forward construction method for quantum error correcting codes, based on graphs, has been presented. These graph codes are directly related to cluster states which have been introduced by Briegel and Raussendorf. We show that the preparation of a cluster state as well as the coding operation for a graph code, can be implemented by a logical network. Concerning the qubit case each vertex corresponds to an Hadamard gate and each edge corresponds to a controlled not gate.
Entanglement is a key resource in quantum communication, but bipartite schemes are often insufficient for advanced protocols like quantum secret sharing or distributed computing. Graph states offer a flexible way to represent and manage multipartite entanglement in quantum networks, enabling logical connectivity through local operations and classical communication (LOCC). In this work, we extend existing approaches based on bi-star configurations to more complex multi-star topologies. We analyze the maximum connectivity that can be achieved in networks ofmmmswitches, each connected tonnnclients, including asymmetric cases where the number of clients varies per switch. We also propose methods to enable logical communication between distant nodes. Our results support the development of scalable quantum networks with rich connectivity beyond traditional bipartite structures.
Graph states are multiparticle entangled states that correspond to mathematical graphs, where the vertices of the graph take the role of quantum spin systems and edges represent Ising interactions. They are many-body spin states of distributed quantum systems that play a significant role in quantum error correction, multiparty quantum communication, and quantum computation within the framework of the one-way quantum computer. We characterize and quantify the genuine multiparticle entanglement of such graph states in terms of the Schmidt measure, to which we provide upper and lower bounds in graph theoretical terms. Several examples and classes of graphs will be discussed, where these bounds coincide. These examples include trees, cluster states of different dimensions, graphs that occur in quantum error correction, such as the concatenated [7,1,3]-CSS code, and a graph associated with the quantum Fourier transform in the one-way computer. We also present general transformation rules for graphs when local Pauli measurements are applied, and give criteria for the equivalence of two graphs up to local unitary transformations, employing the stabilizer formalism. For graphs of up to seven vertices we provide complete characterization modulo local unitary transformations and graph isomorphisms.