Noise-tolerant signature of ZN topological order in quantum many-body states

Noise-tolerant signature of ZN topological order in quantum many-body states
Xi Chen,1,2 Shi-Ju Ran,3,4,* Shuo Yang,5 Maciej Lewenstein,4,6 and Gang Su 1,2,†
1School of Physical Sciences, University of Chinese Academy of Sciences, P.O. Box 4588, Beijing 100049, China
2Kavli Institute for Theoretical Sciences, and CAS Center of Excellence in Topological Quantum Computation,
University of Chinese Academy of Sciences, Beijing 100190, China
3Department of Physics, Capital Normal University, Beijing 100048, China
4ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
5State Key Laboratory of Low-Dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing 100084, China
6ICREA, Passeig Lluis Companys 23, 08010 Barcelona, Spain
(Received 9 October 2018; revised manuscript received 25 March 2019; published 1 May 2019)

Topologically ordered states are fundamentally important in theoretical physics, which are also suggested as promising candidates to build fault-tolerant quantum devices. However, it is still elusive how topological orders can be affected or detected under noises. In this work, we find a quantity, termed as the ring degeneracy D, which is robust under pure noise to detect both trivial and intrinsic topological orders. The ring degeneracy is defined as the degeneracy of the solutions of the self-consistent equations that encode the contraction of the corresponding tensor network (TN). For the ZN orders, we find that the ring degeneracy satisfies a simple relation D = (N + 1)/2 + d, with d = 0 for odd N and d = 1/2 for even N. Simulations on several nontrivial states (two-dimensional Ising model, ZN topological states, and resonating valence bond states) show that the ring degeneracy can tolerate noises up to a strength associated to the gap of the TN boundary theory.