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arXiv:0805.2714

Topological Quantum Computation by Manipulating Quantum Tunneling Effect of the Toric Codes

Su-Peng Kou

Published May 18, 2008• 0 citations
arXivHTMLPDF

Abstract

Quantum computers are predicted to utilize quantum states to perform memory and to process tasks far faster than those of conventional classical computers. In this paper we show a new road towards building fault tolerance quantum computer by tuning quantum tunneling effect of the degenerate quantum states in topological order, instead of by braiding anyons. Using a designer Hamiltonian - the Wen-Plaquette model as an example, we study its quantum tunneling effect of the toric codes and show how to control the toric code to realize topological quantum computation (TQC). In particular, we give a proposal to the measurement of TQC. In the end the realization of the Wen-Plaquette model in cold atoms is discussed.

DOI
10.48550/arxiv.0805.2714

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Quantum computers are predicted to utilize quantum states to perform memory and to process tasks far faster than those of conventional classical computers. In this paper we show a new road towards building fault tolerance quantum computer by tuning quantum tunneling effect of the degenerate quantum states in topological order, instead of by braiding anyons. Using a designer Hamiltonian - the Wen-Plaquette model as an example, we study its quantum tunneling effect of the toric codes and show how to control the toric code to realize topological quantum computation (TQC). In particular, we give a proposal to the measurement of TQC. In the end the realization of the Wen-Plaquette model in cold atoms is discussed.
Quantum computers are predicted to utilize quantum states to process tasks far faster than those of conventional classical computers. In this Letter we show an alternative approach towards building topological quantum computers by tuning the quantum tunneling effect of degenerate quantum states in topological order, instead of braiding anyons. Using a designer Hamiltonian-the Wen-Plaquette model as an example, we study its quantum tunneling effect of the toric codes and show how to control the toric codes to realize topological quantum computation. In particular, we give a proposal to the measurement of the toric codes from Aharonov-Bohm interferences of quasiparticles.
In this paper, the degenerate ground states of Z2{Z}_{2}Z2​ topological order on a plane with holes (the so-called planar codes) are used as the protected code subspace to build a topological quantum computer by tuning their quantum tunneling effect. Using a designer Hamiltonian---the Kitaev toric-code model as an example, we study quantum tunneling effects of the planar codes and obtain its effective theory. Finally, we show how to do topological quantum computation including the initialization, the unitary transformation and the measurement.
A paradigm to build a quantum computer, based on topological invariants is highlighted. The identities in the ensemble of knots, links and braids originally discovered in relation to topological quantum field theory are shown: how they define Artin braid group -- the mathematical basis of topological quantum computation (TQC). Vector spaces of TQC correspond to associated strings of particle interactions, and TQC operates its calculations on braided strings of special physical quasiparticles -- anyons -- with non-Abelian statistics. The physical platform of TQC is to use the topological quantum numbers of such small groups of anyons as qubits and to perform operations on these qubits by exchanging the anyons, both within the groups that form the qubits and, for multi-qubit gates, between groups. By braiding two or more anyons, they acquire up a topological phase or Berry phase similar to that found in the Aharonov-Bohm effect. Topological matter such as fractional quantum Hall systems and novel discovered topological insulators open the way to form system of anyons -- Majorana fermions -- with the unique property of encoding and processing quantum information in a naturally fault-tolerant way. In the topological insulators, due to its fundamental attribute of topological surface state occurrence of the bound, Majorana fermions are generated at its heterocontact with superconductors. One of the key operations of TQC -- braiding of non-Abelian anyons: it is illustrated how it can be implemented in one-dimensional topological isolator wire networks.
Topological quantum computation (TQC) is one of the most striking architectures that can realize fault-tolerant quantum computers. In TQC, the logical space and the quantum gates are topologically protected, i.e., robust against local disturbances. The topological protection, however, requires rather complicated lattice models and hard-to-manipulate dynamics; even the simplest system that can realize universal TQC--the Fibonacci anyon system--lacks a physical realization, let alone braiding the non-Abelian anyons. Here, we propose a disk model that can realize the Fibonacci anyon system, and construct the topologically protected logical spaces with the Fibonacci anyons. Via braiding the Fibonacci anyons, we can implement universal quantum gates on the logical space. Our proposal is platform-independent. As a demonstration, we implement a topological Hadamard gate on a logical qubit through a sequence of 151515 braiding operations of three Fibonacci anyons with merely 222 nuclear spin qubits. The gate fidelity reaches 97.18% by randomized benchmarking. We further prove by experiment that the logical space and Hadamard gate are topologically protected: local disturbances due to thermal fluctuations result in a global phase only. Our work is a proof of principle of TQC and paves the way towards fault-tolerant quantum computation.
Topological quantum computation (TQC) is one of the most striking architectures that can realize fault-tolerant quantum computers. In TQC, the logical space and the quantum gates are topologically protected, i.e., robust against local disturbances. The topological protection, however, requires complicated lattice models and hard-to-manipulate dynamics; even the simplest system that can realize universal TQC—the Fibonacci anyon system—lacks a physical realization, let alone braiding the non-Abelian anyons. Here, we propose a disk model that can simulate the Fibonacci anyon system and construct the topologically protected logical spaces with the Fibonacci anyons. Via braiding the Fibonacci anyons, we can implement universal quantum gates on the logical space. Our disk model merely requires two physical qubits to realize three Fibonacci anyons at the boundary. By 15 sequential braiding operations, we construct a topologically protected Hadamard gate, which is to date the least-resource requirement for TQC. To showcase, we implement a topological Hadamard gate with two nuclear spin qubits, which reaches 97.18 % fidelity by randomized benchmarking. We further prove by experiment that the logical space and Hadamard gate are topologically protected: local disturbances due to thermal fluctuations result in a global phase only. As a platform-independent proposal, our work is a proof of principle of TQC and paves the way toward fault-tolerant quantum computation.
This paper explores the design of a fault-tolerant encoding scheme specifically tailored for topological quantum computing. The central premise is to leverage the inherent robustness of topological protection against decoherence, a primary challenge in realizing practical quantum computers. We propose a scheme utilizing non-Abelian anyons, exotic quasiparticles found in topological superconductors, to construct a robust quantum code. The core concept involves encoding quantum information within the braiding operations of these anyons, where the resulting topological transformations inherently protect against local errors. The scheme's efficacy relies on the topological protection mechanism, effectively mitigating errors arising from environmental noise and imperfections in qubit control. This work provides a theoretical foundation for utilizing topological quantum computing to achieve high-fidelity quantum computation. The key mathematical elements explored include qubit states, braiding operations, and the mathematical description of the topological protection effect.