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Toric Code, Doubled Semion and Ising anyons

INTRODUCTION

Toric Code Recap

Doubled Semion Model

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Ising Anyons

Acknowledgement

References

Tushar Pandey*, Eugene Dumitrescu#

Texas A&M University*, Oak Ridge National Labs#

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Fig: String excitations, always created in pairs

Topological Quantum Computing (TQC) is an approach towards building a quantum computer based on topological materials with resistance to noise and errors. It’s potentially scalable because the anyons can be arranged on a 2D lattice. It’s related to Topological Quantum Field Theories (TQFTs) which have a one-one correspondence with Unitary Modular Tensor Categories (UMTC). For reaching universality using TQC, one needs non-abelian anyons.

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We start with a recap of Toric Code, then give an error correction algorithm for Doubled Semion. We realize that patches of DS could be used for error detection in TC. We provide insights into realization of Ising anyons on quantum circuit. We also look into generalization of the twists generating the non-abelian systems.

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  • Change operators in the TC3 and the constraints to realize “Ising” like behavior

 

Fig: Ising twist and logical operators

  1. K.J. Satzinger et. al. . Realizing topologically ordered states on a quantum processor. Science, 374(6572):1237–1241, dec 2021
  2. Tyler D. Ellison, Yu An Chen, Arpit Dua, Wilbur Shirley, Nathanan Tantivasadakarn, and Dominic J. Williamson. Pauli Stabilizer Models of Twisted Quantum Doubles. PRX Quantum, 3(1):010353, mar 2022.
  3. H. Bombin. Topological order with a twist: Ising anyons from an Abelian model. Physical Review Letters, 105(3):030403, jul 2010.
  4. T. Pandey, Topological phases of matter, Quantum Error Correction and Topological twists. arxiv/2208.08571, aug. 2022
  5. T. Pandey, E. Dumitrescu, Topological Characterization with a Twist, Condensation, and Reflection. arxiv/2209.11126, sep. 2022

This project was possible due to NSF, DOE, ORNL, ORAU and MSGRI Program

I would also like to thank NCSU for providing the opportunity to present a poster

Fig: Plavnik's presentation illustrating the mathematics of topological quantum computing. (Credit: Dr. Eric C. Rowell.)

Fig: (Left) Deconfined anyons loop closure (Right) Confined anyon error propagation

  • Error Correction Algorithm for deconfined Doubled Semion system

 

Fig: Different types of topological twists generating non-abelian particles