- Quasi-particle, Majorana fermion : fermion which is its own antiparticle, creation operator = annihilation operator
- halfway between being an electron and not being an electron
- one ordinary Fermionic operator can always be described by 2 Majorana operators
- quasi-particles in condensed matter systems always come in pairs
→ in condensed matter the building blocks are always ordinary fermions
- encode qubits with fermionic states
- empty → 0
- filled → 1
- e.g. charge qubits
- problem - very sensitive to local perturbations
- 2 spatially separated Majoranas → can encode 1 fermionic degree of freedom in a very nonlocal way
→ topologically protected from any local perturbation by symmetry
- Topological qubit : 2 Majorana bound states form 1 topological qubit
- protected against almost all sorts of perturbations
- expected to have a very long coherence time
- usually impossible to form a superposition of an electron and a hole (opp. charge)
- good system for Majoranas - superconductors
- a sea of Cooper pairs
- Cooper pairs : states consisting of 2 electrons
- take out one Cooper pair → this pair with the hole looks like a single electron
→ distinction between electron and a hole effectively blurred
- a sea of Cooper pairs
- Majorana bound states
- particle-hole symmetry
- particle at energy E → antiparticle at energy -E
- zero energy → immediately get Majorana bound states
- One Majorana bound state : E = 0
- Multiple Majorana bound state : still E = 0
- N Majorana pairs
- 2^N-fold degenerate ground state
- allows topologically protected operations on Majorana bound states
- In a real system : not exactly E = 0, but exponentially close to 0
- Making a qubit - use 2 superpositions of an electron and a hole
- particle-hole symmetry
- just have to find states in superconductors with zero energy
- not easy
- superconducting gap - situation where there is a vortex in the superconductor
- vortex, magnetic flux penetrates the superconductor, locally suppresses the superconducting gap Delta
- bound state = a state at a finite energy
- due to the quantum mechanical zero-point motion
- to remove, need to consider unconventional superconductor
= P-wave superconductor
- to engineer a P-wave superconductor, need
- S-wave superconductor
- semiconducting nanowires with spin-orbit interaction
- tuning of magnetic field and chemical potential
- additional Berry phase of pi can cancel the zero-point motion
- to remove, need to consider unconventional superconductor
- due to the quantum mechanical zero-point motion
- superconducting gap - situation where there is a vortex in the superconductor
- instead of vortices that are hard to control, use 1D systems : Nanowires
- Majorana states at the ends of the wire
- engineer a nontrivial superconductor out of ordinary materials to find Majoranas
- e.g. semiconducting nanowire with spin-orbit interaction in proximity to a S-wave superconductor in a finite magnetic field can support Majoranas
- need to tune the magnetic field so that the Zeeman splitting exceeds the superconducting gap
- need to tune the chemical potential into the Zeeman gap
- e.g. semiconducting nanowire with spin-orbit interaction in proximity to a S-wave superconductor in a finite magnetic field can support Majoranas
- Majorana bound states always sit at zero energy and give rise to a resonant process at zero bias voltage
- not easy
- current experimental status
- measurement of Majoranas
- standard current-voltage measurement is enough (conductance spectroscopy)
- apply a voltage between a normal contact and the superconducting part
- measure the current across a tunnel barrier
- how?
- Andreev reflection → current flows in a normal-superconducting junction as an electron becomes a Cooper pair to enter the superconductor
- done in the presence of a tunnel barrier
- if bias voltage < superconducting gap
- need to enter as a Cooper pair
- current carried by Andreev reflection
- 2 tunnelling events; electron & hole → with a small tunnel probability T
→ total probability proportional to T^2
- if bias voltage > superconducting gap
- electron can just enter the superconductor as an electron
- 1 tunnelling event; electron through the barrier
→ total probability (conductance) proportional to T
→ higher than smaller bias
- if bias voltage < superconducting gap
- standard current-voltage measurement is enough (conductance spectroscopy)
- measurement of Majoranas
- Anyons
- if phase α = 0, particles are bosons (e.g. photon)
- if phase α = π, particles are fermions (e.g. electron)
- Anyon : a type of quasiparticle that occurs only in 2D, less restricted properties than fermions and bosons
- phase α can be any real numbers
- exchange operation may be a non-trivial unitary operation
- non-Abelian anyons : operations on the anyons generally do not commute
- has multiple quantum mechanical states with same, lowest energy
- has a degenerate ground state
- to perform coherent operations, stay in the ground state of the system
- qubit protected from noise or thermal fluctuations from its environment
- when energies of the fluctuations < gap between energy levels
- Quantum computation on anyons
- create anyons from actual electrons
- created pairwise
- quantum gates linked to the exchange of the anyons
- need non-Abelian property
- quantum operations on topological qubits = braiding
- Majorana bound state → form lsing anyons
- Z gate : exchanging a pair of anyons twice
- X gate : exchange of another pair
- Hadamard gate : sequential braiding operations on the anyons
- braiding operations are always discrete; happen or not happen
→ perfect quantum gates, fidelity = 100% - BUT discrete braiding operations cannot reach the entire Bloch sphere of a qubit
- supplemented with additional topologically not protected operations
→ fidelity < 100%
- supplemented with additional topologically not protected operations
- measurement : fusion
- lsing anyons result in
- a single electron
- no electron
- distinguish states with charge sensors (like spin qubits)
- lsing anyons result in
- create anyons from actual electrons
- Real-life implementation in nanowire networks
- problem - cannot exchange anyons (braiding) without colliding them
→ would cause fusion - solution - clever workaround (connect nanowire in Y-junction)
- interaction-based braiding
- valves to control
- degeneracy has to be broken by splitting off states needed for measurements
- remains to be seen which geometry will lead to the very first experimental demonstration of a topological qubit
- problem - cannot exchange anyons (braiding) without colliding them
NUS CS'25