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  About universal quantum computation by “quantum wires”?

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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?

This post imported from StackExchange Physics at 2020-12-03 17:30 (UTC), posted by SE-user annie marie heart
asked Sep 28, 2020 in Theoretical Physics by annie marie heart (1,205 points) [ no revision ]

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