From Majorana fermions to topological order

Barbara M. Terhal, Fabian Hassler, and David P. DiVincenzo

Phys. Rev. Lett. 108, 260504 – Published 28 June 2012

Abstract

We consider a system consisting of a 2D network of links between Majorana fermions on superconducting islands. We show that the fermionic Hamiltonian modeling this system is topologically ordered in a region of parameter space: we show that Kitaev’s toric code emerges in fourth-order perturbation theory. By using a Jordan—Wigner transformation we can map the model onto a family of signed 2D Ising models in a transverse field where the signs, ferromagnetic or antiferromagnetic, are determined by additional gauge bits. Our mapping allows an understanding of the nonperturbative regime and the phase transition to a nontopological phase. We discuss the physics behind a possible implementation of this model and argue how it can be used for topological quantum computation by adiabatic changes in the Hamiltonian.

Received 23 January 2012

DOI:https://doi.org/10.1103/PhysRevLett.108.260504

Authors & Affiliations

Barbara M. Terhal¹, Fabian Hassler¹, and David P. DiVincenzo^1,2

¹Institute for Quantum Information, RWTH Aachen University, 52056 Aachen, Germany
²Department of Theoretical Nanoelectronics, PGI, Forschungszentrum Jülich, 52425 Jülich, Germany

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Vol. 108, Iss. 26 — 29 June 2012

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Images

Figure 1
(a) Fermionic model studied in this Letter. Each island (light gray square) has four Majorana fermions (yellow dots) labeled as $a$ , $b$ , $c$ , $d$ . Fifty percent occupancy is favored for these two fermionic modes, as expressed by the parity constraint in Eq. (1). A weaker quadratic interaction exists between Majorana fermions on islands $i$ and $j$ along diagonal links $V_{i = μ \pm \hat{z}, j = μ \pm \hat{x}}$ , Eq. (2). Periodic boundary conditions are assumed. (b) Zoomed view of a single island. The Majorana wire has a $C$ shape (black line) in order to be able to tune the overlap between the $a$ - and $c$ -Majorana and thus implement an $X$ gate. A $Z$ gate/measurement is implemented by increasing the ratio of $E_{C}^{'} / E_{J}^{'}$ , see main text.Reuse & Permissions
Figure 2
(a) Toric code on a $L \times L$ lattice with $2 L^{2}$ qubits on vertices and periodic boundary conditions in both directions. The Hamiltonian is a sum over white and gray plaquette operators $A_{μ} = Z_{μ + \hat{z}} Z_{μ - \hat{z}} X_{μ + \hat{x}} X_{μ - \hat{x}}$ . (b) The Hamiltonian in which the hatched plaquettes are omitted has ground space degeneracy of 8 in the topological phase, hence encodes an additional qubit. The logical operators of this qubit are (blue and red) loops encircling the hole. We can call this a white hole qubit as it is obtained by omitting plaquettes in the Hamiltonian, i.e., making a hole in the lattice, which are centered around a white plaquette.Reuse & Permissions
Figure 3
The gauge bits $σ$ set the Ising interactions to ferromagnetic (FM, black edges) except for an AFM (red bold edges) loop around the torus. This AFM boundary will be felt in the FM phase, but not in the paramagnetic (PM) phase of the model, leading to the topological degeneracy. A loop operator $C_{γ}$ in the fermionic model becomes a product of Ising edges which winds around the torus; note that for the depicted sign pattern the loop operator which crosses this domain wall will have $- 1$ eigenvalue.Reuse & Permissions
Figure 4
Sketch of the spectrum of system as a function of $λ / Δ$ . For small $λ$ , the system is in a topological state with a fourfold ground state degeneracy on the torus. The first excited states for small $λ$ are Ising models with frustration as determined by the gauge bits. All these models are degenerate for $λ = 0$ ; and the degeneracy lifts in fourth-order perturbation theory in $λ$ , see Eq. (5). The gap of the frustrated model increases monotonically for increasing $λ$ . The phase transition to a state without topological order happens at the transition point $(λ / Δ)_{c}$ of the unfrustrated Ising model. At this transition the gap of the Ising model closes and the degeneracy of the topological states vanishes.Reuse & Permissions

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