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

2. 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

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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. Thank You