Topological Quantum Phases Through Majorana Wires Stablized by Many-Body Localization

Fig. 1: Kitaev chain with strong and weak pairing between same site and adjacent site Majoranas. (Source: M. Radzihovsky, after Alicea. [1])

Quantum computing has been vigorously pursued because of its promise of a number of revolutionary applications ranging from cryptography (e.g., Shor's exponential speedup of prime factorization) to solutions of many challenging problems of quantum chemistry and material science. Popular platform candidates include photon time-bin encoded qubits, superconducting qubits, ion trap qubits, among others. However, the emergence of topological qubits theoretically provides a foundation for more robust qubits that store information non-locally and are therefore robust to deleterious effects of local noise. In this paper, we investigate the theory behind Majorana fermions, as appearing at the ends of gated wires of spinless p-wave superconductors, and discuss their utilization for topological quantum computing. We also explore the possibility of expanding the range of stability of the corresponding Majorana phases in the presence of quenched disorder, viewing it through its Jordan-Wigner mapping onto the random transverse field Ising model.

The Model

The Transverse Field Ising model describes how interacting spins on different sites create an ordered phase (ferromagnet), an unordered phase (paramagnet) and even a gapless phase at a critical point between these phases. This model is described by the extensively studied Hamiltonian:

As we detail in the paper, using the Jordan-Wigner transformation, the 1D TFIM transforms into a 1D p-wave superconductor described by a fermionic Hamiltonian:

This Hamiltonian is the celebrated p-wave 1D superconductor which we solve exactly finding its spectrum and eigenstates for periodic boundary conditions. For open boundary conditions, representing each fermionic operator by its real and imaginary parts i.e., Majorana fermions, this Hamiltonian can be transformed into the famous Kitaev Majorana chain. This chain made up of Majoranas γ_A,i and γ _B,i as shown in Fig. 1, can be tuned to have a single unpaired Majorana at each end of the chain. These Majorana end modes can be used to make up a physical fermionic operator whose empty or filled state corresponds to a non-local topological qubit, |0>, |1> that can be used to encode non-local information.

We review Majorana fermions as a platform toward realization of topological quantum computation, highlight the relation to the transverse field Ising model (TFIM) and the Kitaev chain, and analyze the properties of this 1D p-wave superconductor, finding its spectrum and quasi-particles for periodic boundary conditions. We review how such open wire exhibits gapless Majorana modes localized at its ends and describe their utility for a realization of a topological quantum qubit. [1,2]

Protected Order

Drawing from recent work by Huse et al., Bauer et al., and Kjall et al., we explore using the ideas of many body localization to protect quantum states and information in expanding the applications of quantum computing to non-zero energy density. [3-5] By introducing disorder into systems, it can exhibit a so-called random many-body localized state that has a potential for "quantum memory", which theoretically can protect quantum order, create long range order, and protect topological order that can be used for TQC.

© Matthew Radzihovsky. The author warrants that the work is the author's own and that Stanford University provided no input other than typesetting and referencing guidelines. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.

References

[1] J. Alicea, "New Directions in the Pursuit of Majorana Fermions in Solid State Systems," Rep. Prog. Phys. 75, 076501 (2012).

[2] R. M. Lutchyn, J. D. Sau, and S. Das Sarma, "Majorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures," Phys. Rev. Lett. 105, 077001 (2010).

[3] David A. Huse et al., "Localization-Protected Quantum Order," Phys. Rev. B 88 014206 (2013).

[4] B. Bauer and C. Nayak, "Area Laws in a Many-Body Localized State and its Implications for Topological Order," J. Stat. Mech. 2013, P09005 (2013).

[5] J. A. Kjäll, J. H. Bardarson, and F. Pollmann, "Many-Body Localization in a Disordered Quantum Ising Chain," Phys. Rev. Lett. 113, 107204 (2014).