About universal quantum computation by “quantum wires”?

Question

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:

1. 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?

2. 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?

3. "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.)

4. 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?

5. 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?

6. 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?

Answer 1

I don't think I have the time to read the whole article so answering the rest of the questions is a little above me right now. But I can answer the first couple questions about the absract :)

1. Estimates of thresholds vary widely based on method of error correction and noise models considered. There is an entire field of science dedicated to this, but I've always found that reading this paper by Gottesman is a really good start if you've got just a little background in QEC and want to learn about fault tolerance.

2. Kitaev is proposing a method of protecting a single level system, not a 2-level system like a qubit, thus there are no meaningful quantum gates that can be performed. In other words, system only allows preservation of a state, not a the multitude of states required for meaningful computation. Thus fault-tolerant gates cannot be formulated in this way.

3. I imagine they are just talking about creating a state with a long coherence time. If you want to know more about figures of merit for preserving quantum states, this stackexhange thread has decent explanations of the most commonly used figures of merit: $T_1$ and $T_2$.