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arXiv:2303.17700

Topological Quantum Teleportation and Superdense Coding -- Without Braiding

Sachin J. Valera

Published Mar 30, 2023• 0 citations
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Abstract

We present the teleportation and superdense coding protocols for a family of anyon theories coming from Tambara-Yamagami categories, of which the lowest rank theories describe Ising anyons. In contrast to the usual approach to anyonic computation, we relax the requirement that we should be able to braid anyons. This is motivated by the goal of designing basic protocols that require less control over quasiparticles, and which may therefore be amenable to realisation in near-term systems. Since these implementations are braid-free, they are also compatible with Majorana modes on a 1d quantum wire.

DOI
10.48550/arxiv.2303.17700

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We present the teleportation and superdense coding protocols for a family of anyon theories coming from Tambara-Yamagami categories, of which the lowest rank theories describe Ising anyons. In contrast to the usual approach to anyonic computation, we relax the requirement that we should be able to braid anyons. This is motivated by the goal of designing basic protocols that require less control over quasiparticles, and which may therefore be amenable to realisation in near-term systems. Since these implementations are braid-free, they are also compatible with Majorana modes on a 1d quantum wire.
Anyons have been extensively investigated as information carriers in topological quantum computation. However, how to characterize the information flow in quantum networks composed of anyons is less understood, which motivates us to study quantum communication protocols in anyonic systems. Here we propose a general topologically protected protocol for quantum teleportation based on the Ising anyon model and prove that with our protocol an unknown anyonic state of any number of Ising anyons can be teleported from Alice to Bob. Our protocol naturally generalizes quantum state teleportation from systems of locally distinguishable particles to systems of Ising anyons, which may promote our understandings of anyonic quantum entanglement as a quantum resource. In addition, our protocol is expected to be realized with the Majorana zero modes, one of the possible physical realizations for the Ising anyon in experiments.
Topological quantum computation based on anyons is a promising approach to achieve fault-tolerant quantum computing. The Majorana zero modes in the Kitaev chain are an example of non-Abelian anyons where braiding operations can be used to perform quantum gates. Here we perform a quantum simulation of topological quantum computing, by teleporting a qubit encoded in the Majorana zero modes of a Kitaev chain. The quantum simulation is performed by mapping the Kitaev chain to its equivalent spin version and realizing the ground states in a superconducting quantum processor. The teleportation transfers the quantum state encoded in the spin-mapped version of the Majorana zero mode states between two Kitaev chains. The teleportation circuit is realized using only braiding operations and can be achieved despite being restricted to Clifford gates for the Ising anyons. The Majorana encoding is a quantum error detecting code for phase-flip errors, which is used to improve the average fidelity of the teleportation for six distinct states from 70.76±0.35% to 84.60±0.11%, well beyond the classical bound in either case.
Quantum error correction is widely considered to be an essential ingredient for overcoming decoherence and achieving large-scale quantum computation. Topological quantum computation based on anyons is a promising approach to achieve fault-tolerant quantum computing. The Majorana zero modes in the Kitaev chain are an example of non-Abelian anyons where braiding operations can be used to perform quantum gates. Here we demonstrate in a superconducting quantum processor that the spin-mapped version of the Majorana zero modes can be used to perform quantum teleportation. The teleportation transfers the quantum state encoded on two-qubit Majorana zero mode states between two Kitaev chains, using only braiding operations. The Majorana encoding is a quantum-error-detecting code for phase flip errors, which is used to improve the average fidelity of the teleportation for six distinct states from $70.76 \pm 0.35 \% $ to 84.60±0.11%84.60 \pm 0.11 \%84.60±0.11%, well beyond the classical bound in either case.
The braiding of the worldlines of particles restricted to move on a network (graph) is governed by the graph braid group, which can be strikingly different from the standard braid group known from two-dimensional physics. It has been recently shown that imposing the compatibility of graph braiding with anyon fusion for anyons exchanging at a single wire junction leads to new types of anyon models with the braiding exchange operators stemming from solutions of certain generalised hexagon equations. In this work, we establish these graph-braided anyon fusion models for general wire networks. We show that the character of braiding strongly depends on the graph-theoretic connectivity of the given network. In particular, we prove that triconnected networks yield the same braiding exchange operators as the planar anyon models. In contrast, modular biconnected networks support independent braiding exchange operators in different modules. Consequently, such modular networks may lead to more efficient topological quantum computer circuits. Finally, we conjecture that the graph-braided anyon fusion models will possess the (generalised) coherence property where certain polygon equations determine the braiding exchange operators for an arbitrary number of anyons. We also extensively study solutions to these polygon equations for chosen low-rank multiplicity-free fusion rings, including the Ising theory, quantum double of \mathbb{Z}_{2} ℤ 2 , and Tambara-Yamagami models. We find numerous solutions that do not appear in the planar theory of anyons.
The braiding of the worldlines of particles restricted to move on a network (graph) is governed by the graph braid group, which can be strikingly different from the standard braid group known from two-dimensional physics. It has been recently shown that imposing the compatibility of graph braiding with anyon fusion for anyons exchanging at a single wire junction leads to new types of anyon models with the braiding exchange operators stemming from solutions of certain generalised hexagon equations. In this work, we establish these graph-braided anyon fusion models for general wire networks. We show that the character of braiding strongly depends on the graph-theoretic connectivity of the given network. In particular, we prove that triconnected networks yield the same braiding exchange operators as the planar anyon models. In contrast, modular biconnected networks support independent braiding exchange operators in different modules. Consequently, such modular networks may lead to more efficient topological quantum computer circuits. Finally, we conjecture that the graph-braided anyon fusion models will possess the (generalised) coherence property where certain polygon equations determine the braiding exchange operators for an arbitrary number of anyons. We also extensively study solutions to these polygon equations for chosen low-rank multiplicity-free fusion rings, including the Ising theory, quantum double of Z2, and Tambara-Yamagami models. We find numerous solutions that do not appear in the planar theory of anyons.