Finite-Time Braiding Dynamics within Topological Nanowire Qubits

Abstract

Topological Quantum Computing has largely evolved towards a paradigm of manipulating edge localized Majorana within $p$-wave topological superconducting nanowires. To bridge the gap between physical qubit systems and quantum algorithms, we perform a dynamical analysis to extend what is known in the adiabatic regime, providing time-dependent gate elements for further qubit and algorithm modeling efforts. Our analysis covers dynamical considerations for two methods of shuttling domain edge bound Majoranas in a single nanowire system which both function by applying spatiotemporally dependent onsite and hopping parameters within the system's Hamiltonian. We then complicate this model by converting it into the T-qubit to calculate the finite-time gate representation of the shuttling techniques used in a more practical setting. These contributions provide insight for realistic experimental setups in the next-generation of qubit implementation and will hopefully facilitate fault tolerant scalable systems and universal gate design.

Figures (6)

• Figure 1: Simple System Finite-Time Gates. Left plot depicts $U(t)$ produced by the P-H symmetry breaking perturbation for a single nanowire. The right plot depicts similar information for the time-dependent coupling of two nanowires. For both scenarios, the plots of $E(t)$ imply that these perturbations change the content of the ground state manifold in such a way to spoil fault tolerance.
• Figure 2: Adiabatic Shuttling. Left plot depicts the spectrum for adiabatic $\mu-$shuttling by shifting a domain wall sigmoid potential profile along the chain. Right plot depicts identical calculation for $\phi-$shuttling for a travelling phase discontinuity. We use the associated eigenstates as a moving basis in our time-dependent calculation.
• Figure 3: Finite-Time $\mu-$Shuttling. Left column depicts the time dependence of a MBS being $\mu-$shuttled towards one another, and the right column shows comparable information for MBS moving away from one another. To visualize this trajectory, we plot $\rho(t)$ in row one. We calculate $E(t)$ in row two to compare how the energy gap of the initial configuration affects energetics in future states. $U(t)$ is nearly identity for all time in the far-out scenario, while the close-in scenario leads to a complex rotation as well as significant scattering into the bulk as indicated by $q(t)$.
• Figure 4: Finite-Time $\phi-$Shuttling. Left column depicts the relevant dynamic quantities for the 1D braid with a step-like profile and the right depicts a softened $\phi-$sigmoid system. Here we see the gapless nature of the sharp profile system spoils the protected status of the ground state manifold as indicated by the rising $E(t)$ and sharp increase in $q(t)$. The softened system maintains its ground state manifold for all values of $t$.
• Figure 5: $\mu-$protocol. (a) Diagram of the braiding action used in this calculation. The grey (black) sites are topologically (non)trivial, and MBS sit at the red sites. While the right MBS moves across the junction at $t/T = 3/7$, we rephase the lower branch of the system to keep the bulk states from approaching the ground state manifold. (b) We plot the adiabatic energy spectrum for the unphased and phased system in light grey and green respectively. Notice, the bulk state which enters the gap after $t/T=4/7$. We also plot the dynamic $E(t)$ in solid black and red. (c) Computing $q(t)$ demonstrates the effect that the junction bound state has on dynamics. (d) $U(t)$ is approximately diagonal with a nontrivial complex rotation on the computational manifold.