Kitaev honeycomb tensor networks: exact unitary circuits and applications

Abstract

The Kitaev honeycomb model is a paradigm of exactly-solvable models, showing non-trivial physical properties such as topological quantum order, abelian and non-abelian anyons, and chirality. Its solution is one of the most beautiful examples of the interplay of different mathematical techniques in condensed matter physics. In this paper, we show how to derive a tensor network (TN) description of the eigenstates of this spin-1/2 model in the thermodynamic limit, and in particular for its ground state. In our setting, eigenstates are naturally encoded by an exact 3d TN structure made of fermionic unitary operators, corresponding to the unitary quantum circuit building up the many-body quantum state. In our derivation we review how the different "solution ingredients" of the Kitaev honeycomb model can be accounted for in the TN language, namely: Jordan-Wigner transformation, braidings of Majorana modes, fermionic Fourier transformation, and Bogoliubov transformation. The TN built in this way allows for a clear understanding of several properties of the model. In particular, we show how the fidelity diagram is straightforward both at zero temperature and at finite temperature in the vortex-free sector. We also show how the properties of two-point correlation functions follow easily. Finally, we also discuss the pros and cons of contracting of our 3d TN down to a 2d Projected Entangled Pair State (PEPS) with finite bond dimension. The results in this paper can be extended to generalizations of the Kitaev model, e.g., to other lattices, spins, and dimensions.

26 pages, 31 figures, 1 table. Revised version, including a extended discussion on the Majorana braidings and results for correlation functions

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• Gapless Kitaev Spin Liquid to Classical String Gas through Tensor Networks
• Preparing Valence-Bond-Solid states on noisy intermediate-scale quantum computers
• Determining quantum phase diagrams of topological Kitaev-inspired models on NISQ quantum hardware
• Finite-temperature fidelity and von Neumann entropy in the honeycomb spin lattice with quantum Ising interaction
• Topological quantum error correction in the Kitaev honeycomb model
• Squeezing and quantum approximate optimization
• Benchmarking near-term quantum devices with the Variational Quantum Eigensolver and the Lipkin-Meshkov-Glick model
• Holographic encoding of universality in corner spectra
• Restricted Boltzmann machine representation for the groundstate and excited states of Kitaev Honeycomb model
• Geometric Entanglement and Quantum Phase Transition in Generalized Cluster-XY models
• Momentum space magic for the transverse field quantum Ising model
• Scale-Invariant Continuous Entanglement Renormalization of a Chern Insulator
• Simulation of the 1d XY model on a quantum computer
• A primer on Kitaev Model: Basic aspects, material realization, and recent experiments
• Maximal Entanglement: Applications in Quantum Information and Particle Physics
• Emergent Kitaev materials in synthetic Fermi-Hubbard bilayers