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Physical Review Letters

Topological Quantum Computation via the Quantum Tunneling Effect

Su-Peng Kou

Published Mar 27, 2009• 17 citations
Publisher

Abstract

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.

DOI
10.1103/physrevlett.102.120402

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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.
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.
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.
The discovery of topological order has revised the understanding of quantum matter and provided the theoretical foundation for many quantum error-correcting codes. Realizing topologically ordered states has proven to be challenging in both condensed matter and synthetic quantum systems. We prepared the ground state of the toric code Hamiltonian using an efficient quantum circuit on a superconducting quantum processor. We measured a topological entanglement entropy near the expected value of -ln2 and simulated anyon interferometry to extract the braiding statistics of the emergent excitations. Furthermore, we investigated key aspects of the surface code, including logical state injection and the decay of the nonlocal order parameter. Our results demonstrate the potential for quantum processors to provide insights into topological quantum matter and quantum error correction.
The discovery of topological order has revised the understanding of quantum matter and provided the theoretical foundation for many quantum error–correcting codes. Realizing topologically ordered states has proven to be challenging in both condensed matter and synthetic quantum systems. We prepared the ground state of the toric code Hamiltonian using an efficient quantum circuit on a superconducting quantum processor. We measured a topological entanglement entropy near the expected value of –ln2 and simulated anyon interferometry to extract the braiding statistics of the emergent excitations. Furthermore, we investigated key aspects of the surface code, including logical state injection and the decay of the nonlocal order parameter. Our results demonstrate the potential for quantum processors to provide insights into topological quantum matter and quantum error correction.
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.
Abstract A quantum computer can perform exponentially faster than its classical counterpart. It works on the principle of superposition. But due to the decoherence effect, the superposition of a quantum state gets destroyed by the interaction with the environment. It is a real challenge to completely isolate a quantum system to make it free of decoherence. This problem can be circumvented by the use of topological quantum phases of matter. These phases have quasiparticles excitations called anyons. The anyons are charge-flux composites and show exotic fractional statistics. When the order of exchange matters, then the anyons are called non-abelian anyons. Majorana fermions in topological superconductors and quasiparticles in some quantum Hall states are non-abelian anyons. Such topological phases of matter have a ground state degeneracy. The fusion of two or more non-abelian anyons can result in a superposition of several anyons. The topological quantum gates are implemented by braiding and fusion of the non-abelian anyons. The fault-tolerance is achieved through the topological degrees of freedom of anyons. Such degrees of freedom are non-local, hence inaccessible to the local perturbations. In this paper, the Hilbert space for a topological qubit is discussed. The Ising and Fibonacci anyonic models for binary gates are briefly given. Ternary logic gates are more compact than their binary counterparts and naturally arise in a type of anyonic model called the metaplectic anyons. The mathematical model, for the fusion and braiding matrices of metaplectic anyons, is the quantum deformation of the recoupling theory. We proposed that the existing quantum ternary arithmetic gates can be realized by braiding and topological charge measurement of the metaplectic anyons.