In this paper: A. Y. Kitaev, "Unpaired Majorana fermions in quantum wires", *Phys.-Usp.* **44** 131 (2001), arXiv:cond-mat/0010440, it says:

Unlimited quantum computation is possible if errors in the implementation of each
gate are below certain threshold. Unfortunately, for conventional
fault-tolerance schemes the threshold appears to be about $10^{−4}$, which is
beyond the reach of current technologies. It has been also suggested that
fault-tolerance can be achieved at the physical level (instead of using quantum error-correcting codes). The first proposal of these kind was based
on non-Abelian anyons in two-dimensional systems. In this paper we describe
another (theoretically, much simpler) way to construct decoherence-protected
degrees of freedom in one-dimensional systems (“quantum wires”). Although
it does not automatically provide fault-tolerance for quantum gates, it should
allow, when implemented, to build a reliable quantum memory.

My questions:

why "conventional
fault-tolerance schemes the threshold appears to be about $10^{−4}$, which is
beyond the reach of current technologies" Why is it $10^{−4}$? and what is this ratio?

the “quantum wire” does not automatically provide fault-tolerance for quantum gates? Why is that? and what correspond to the **quantum operations for quantum gates in “quantum wire”**?
are these quadratic Majorana operators or higher order Majorana operators?

"Although it does not automatically provide fault-tolerance for quantum gates, it should
allow, when implemented, to build a reliable quantum memory."--> What are the criteria for "reliable quantum memory?" (does it make difference in 1d or other dimensions.)

Later there is a claim "Even without **actual inelastic processes**, this will produce the same effect as decoherence" (this here means "different electron configurations will have different energies and thus will pick up different phases" due to the fermion number $a^\dagger a$ term on a local site). What are the **actual inelastic processes** mean in quantum sense?

In p.4, "if a single Majorana operator can be localized, symmetry transformation S should not mix it with other operators." What does it mean to have S not mixed with other operators? Is that S commutative to other operators? Namely other operators respect the symmetry S?

In a footnote "3-dimensional substrate can effectively induce the desired pairing between electrons with the same spin direction — at least, this is true in the absence of spin-orbit interaction" --> What does (with or without) the spin-orbit interaction affect then?

This post imported from StackExchange Physics at 2020-12-03 17:30 (UTC), posted by SE-user annie marie heart