Quantum Computing: the Majorana Fermion Solution By: Ryan Sinclair Physics 642 5/5/2016

Outline Majorana Fermions Qubit construction Errors and Noise Effects What are they? Are they found in nature? How do you make them? Qubit construction Errors and Noise Effects Surface Code example Topology Logical gates

References C.W.J. Beenakker, Annu. Rev. Condens. Matter Phys. (4), 113-136 (2013) L.S. Ricco, et al., arXiv: 1512.02725v1 (2015). Y. Li, arXiv: 1512.05089v1 (2015). S. Vijay, et al., arXiv: 1504.01724v3 (2015). A. G. Fowler, et al., Phys. Rev. A 86, 032324 (2012).

Majorana Fermions MFs are particles which are their own antiparticle Best chance for a natural MF is the neutrino/antineutrino MFs have been produced artificially by exploiting natural processes The hole and electrons in superconductors form the MF Requires available states at E = 0, the Fermi surface Break spin degeneracy using strong spin-orbit coupling associated with a topological insulator

Qubit Construction MF bound to a topological defect creates an Ising anyon Obey non-Abelian statistics Evolution is described by a unitary transformation Binary states are formed by two zero modes coupled by tunneling States can be distinguished by presence/absence of an unpaired quasiparticle

Errors and Noise Effects Classical error resistance due to parity conservation Quantum errors require coupling between MFs Relies on nonzero gap for quasiparticle excitations Joint measurement required for parity determination Only ~100 MFs needed to construct logical qubit Errors can be corrected by classical software

Surface Code Surface code relies on a series of measurements that move and braid the logical qubits Logical qubits are composed of physical qubits defined by two energy levels. Logical qubits are created by ceasing certain measurements in order to create holes with different possible anyon charges Universal quantum computing for bosonic surface codes requires CNOT, Hadamard, S- gate, and T-gate

Gate Construction CNOT gate Hadamard gate Constructed by braiding logical qubits Simplest example involves a single braiding that produces an overall sign change Hadamard gate In general, requires a series of Hadamard gates acting on physical qubits MF code only requires transferring a qubit between distinct sublattices S- and T-gate can be composed from CNOT and Hadamard gates

Conclusions Majorana fermions are their own antiparticles MFs may not occur in nature, but have been observed experimentally Logical qubits constructed from MFs are theoretically resistant to both errors and noise effects Surface Codes can be implemented in order to construct the necessary logical gates CNOT Hadamard S- and T-gates

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