Topological Quantum Computation by Manipulating Toric Code

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 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.

arXiv:0805.2714May 18, 20080 citations

Unconventional $\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_2$ topological order in the kagome XY toric codeUnconventional $\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_2$ topological order in the kagome XY toric code

C. Kölbl, M. Vieweg, K. P. Schmidt

We investigate the quantum phase diagram of the XY toric code (XYTC) on the kagome lattice consisting of $XY$ hexagonal and triangular plaquette operators and conserved star operators. We demonstrate analytically by exploiting an exact local $\mathbb{Z}_2$ -symmetry on dodecagons that the kagome XYTC realizes a $\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_2$ topologically ordered phase in the limit of large hexagonal \mbox{plaquette} operators. This unconventional topological phase involves 64 quasi-particles - Abelian anyons - with restricted mobility on three colored sublattices establishing a quantum dimension of eight. The large number of topological superselection sectors originates from six independent Wilson loop operators, acting as emergent one-form symmetries. The resulting anyon structure is richer than that of established topological codes like the toric and the color code. A corresponding CSS topological stabilizer code is formulated, opening novel possibilities for the encoding and manipulation of topological quantum information.We investigate the quantum phase diagram of the XY toric code (XYTC) on the kagome lattice consisting of $XY$ hexagonal and triangular plaquette operators and conserved star operators. We demonstrate analytically by exploiting an exact local $\mathbb{Z}_2$-symmetry on dodecagons that the kagome XYTC realizes a $\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_2$ topologically ordered phase in the limit of large hexagonal \mbox{plaquette} operators. This unconventional topological phase involves 64 quasi-particles - Abelian anyons - with restricted mobility on three colored sublattices establishing a quantum dimension of eight. The large number of topological superselection sectors originates from six independent Wilson loop operators, acting as emergent one-form symmetries. The resulting anyon structure is richer than that of established topological codes like the toric and the color code. A corresponding CSS topological stabilizer code is formulated, opening novel possibilities for the encoding and manipulation of topological quantum information.

cond-mat.str-elCross-listedarXiv:2607.22296Jul 27, 20260 citations

Topological Quantum Computation via the Quantum Tunneling Effect

Su-Peng Kou

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 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.

Physical Review LettersMar 27, 200917 citations

Quantum Computation with Topological Codes

Keisuke Fujii

SpringerBriefs in mathematical physicsJan 1, 201559 citations

Topological Quantum Computation

Jiannis K. Pachos

Lecture notes in computer scienceJan 1, 20137 citations

Unconventional

\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_2

topological order in the kagome XY toric code

We investigate the quantum phase diagram of the XY toric code (XYTC) on the kagome lattice consisting of

XY

hexagonal and triangular plaquette operators and conserved star operators. We demonstrate analytically by exploiting an exact local

\mathbb{Z}_2

-symmetry on dodecagons that the kagome XYTC realizes a

\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_2

topologically ordered phase in the limit of large hexagonal \mbox{plaquette} operators. This unconventional topological phase involves 64 quasi-particles - Abelian anyons - with restricted mobility on three colored sublattices establishing a quantum dimension of eight. The large number of topological superselection sectors originates from six independent Wilson loop operators, acting as emergent one-form symmetries. The resulting anyon structure is richer than that of established topological codes like the toric and the color code. A corresponding CSS topological stabilizer code is formulated, opening novel possibilities for the encoding and manipulation of topological quantum information.