Where is “the two level system” of a topological qubit encoded in the Majorana-fermion-based platform of quantum computing? If the Hamiltonian in a topological quantum field theory is absent (H=0), then what dictates the evolution of a topological system in the circuit model of quantum computation?


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There are many proposals for using Majorana zero modes to encode a qubit; one of the most popular is called the hexon, discussed in this paper: https://arxiv.org/pdf/1610.05289.pdf.

The main idea here is that a one-dimensional topological superconductor hosts Majorana zero modes at its ends. Two Majorana modes combine into a Dirac fermion, so the ground state is doubly degenerate; either the fermion built from the two Majoranas at the ends is occupied or unoccupied. Having $2N$ Majorana zero modes gives a Hilbert space of dimension $2^N$. The architectures discussed in many of these papers are variants on this concept designed to protect the system from unwanted effects like quasiparticle poisoning.

So the qubits in the Majorana models consist of occupying or deoccupying the fermions created by these Majorana zero modes. However, the Hamiltonian is not $0$ -- we're not dealing with a TQFT, we're dealing with a condensed matter system. To implement a quantum circuit we perform an adiabatic evolution of the Hamiltonian, taking it along some closed path in parameter space. While the Hamiltonian returns to its original form, the states evolve into one another. If the zero modes are kept far apart during this evolution, this evolution is topologically protected; it's a quantum gate.

If we did want to use TQFT language, you would consider your surface punctured with several holes; these holes are defects which bind the zero modes. Protected operations then generate a representation of the braid group.


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