I have a technical question on the "measurement-only"-proposal for topological quantum computation on anyons. First some background:

Background. While it has become a common idea that topological quantum gates could be implemented by braiding operations on anyons, it is rather unclear, both experimentally but also conceptually, how to go about actually moving anyons around in physical position space, in a useful way. This is particularly the case for defect-like anyons such as most Majorana realizations under discussion, which tend to be unmovable.

In order to address this key obstacle to realizing TQC, there is a proposal, apparently embraced by the leading TQC laboratory, that the desired braiding operations on the anyon wave functions can be implemented, alternatively, by a suitable sequence of measurements ("forced measurements") of topological charges of the system. Essentially, the idea is to use quantum teleportation protocols to teleport the otherwise immobile anyons around, and the claim is that this can be done in a way such that the end result is the same braiding operation on the anyon wavefunction that would have been realized had the anyons actually be moved around in an analogous fashion.

The references. This proposal/claim is originally due to:

  • P. Bonderson, M. Freedman, C. Nayak: "Measurement-Only Topological Quantum Computation", Phys. Rev. Lett. 101 010501 (2008) (arXiv:0802.0279)

  • P. Bonderson, M. Freedman, C. Nayak: "Measurement-Only Topological Quantum Computation via Anyonic Interferometry", Annals Phys. 324 (2009) 787-826 (arXiv:0808.1933)

Somewhat more streamlined discussion is in:

  • [Bon13] P. Bonderson: "Measurement-Only Topological Quantum Computation via Tunable Interactions", Phys. Rev. B 87 (2013) 035113 (arXiv:1210.7929)

That this claim is quite remarkable is nicely brought out by referring to it as "braiding without braiding":

  • S. Vijay, L. Fu: "Braiding without Braiding: Teleportation-Based Quantum Information Processing with Majorana Zero Modes", Phys. Rev. B 94 235446 (2016) (arXiv:1609.00950)

But is this actually true? Are we really braiding without explicitly invoking braiding? That's my question.

The protocol. Concretely, the proposed protocol considers (Bon13, (18)) four anyons, ordered 1-4, and is advertised as consisting of nothing but the succession of forced measurements:

  • first on the pair (2,3),

  • then on the pair (1,2)

  • then on the pair (2,4)

  • finally again on the pair (2,3).

protocol for braiding-by-measurement

For the present purpose it does not matter what exactly this forced measurement on a pair of anyons does, it is only important that it is some (repeated) operation "$b_{i j}$" that is represented by a string diagram on a pair of adjacent anyons.

The issue. That's the crux: One of the above four steps acts on non-adjacent anyons, namely the operation "$b_{2 4}$" acting on the pair (2,4). In order to actually apply the measurement in this case, all the above authors implicitly (namely in their diagrams, but not in the accompanying text) insert "by hand" a braiding of anyon 2 past anyon 3 to make anyon 2 become adjacent to 4 (and finally to move it back to where it was). This is evident by a glance at the protocol diagram Bon13, (18), at the positions highlighted in yellow above.

The next step in the argument is to re-arrange this allegedly "measurement-only" string diagram and to conclude that, under suitable conditions, it is equivalent to the desired braiding of anyon number 1 with anyon number 4 (Bon13, (22)).

The question. But since a braiding $2 \leftrightarrow 3$ is used to draw this conclusion, it would seem plainly counterfactual to say that the braiding $1 \leftrightarrow 4$ has been obtained "without braiding", just by measurement. In actuality, it has been obtained by (measurements and) braiding $2 \leftrightarrow 3$. No?

I guess the point is that, in the given situation, anyons 2 & 3 serve as "auxiliary" anyons, and that the anyons 1 & 4 are the actual anyons of interest in the process. Still, a question seems to remain: How to operationally implement the braiding $2 \leftrightarrow 3$ that is manifest in the protocol Bon13, (18)? Is it clear that this is something that can be done not just on paper but in the laboratory? And if this can be done on the pair (2,3) why can't it be done on the pair (1,4) right away, omitting the auxiliary anyons 2,3 altogether?

Clearly, I must be missing something. I apologize if I am missing the obvious. Probably I am missing something in the translation from string diagrams on paper (which I understand well) to actual operations on anyons in the laboratory (about which, indeed, I know little). In any case, I'd be grateful for a hint as to whether and how there really is "braiding without braiding" of anyons in the lab.

  • $\begingroup$ As a matter of fact, the protocol involves no physical transportation of anyons. The string diagram is not supposed to represent physical motion of anyons, rather it is a convenient way to express calculations in the braided tensor category. So I think one has to be a bit careful in translating the diagrams. Regarding whether this could be done in lab, at moment we do not have non-Abelian anyons, at least none that can be controlled to this degree. But for Majorana zero modes, it is an elementary quantum-mechanical calculation to verify the "braiding by tuning interaction" protocol. Cont... $\endgroup$
    – Meng Cheng
    Commented Mar 24, 2022 at 14:19
  • $\begingroup$ ...proposals for how to do this in actual physical realizations of Majoranas, like nanowires, have been put forward and carefully analyzed. For one example, see iopscience.iop.org/article/10.1088/1367-2630/14/3/035019 There are more refined ones from Microsoft Station Q group, but the basic idea has not changed much. $\endgroup$
    – Meng Cheng
    Commented Mar 24, 2022 at 14:22
  • $\begingroup$ Thanks for the comment. I understand that there is no motion and that the string diagram is not reflecting a motion in space. Instead, the string diagram is meant to reflect a protocol of operations to be done on the system. But if we agree that the $b_{i j}$ reflect plain measurement operations, then there is clearly, in addition, also a braiding operation involved in this protocol. I am guessing that its implementation is somehow implicit in how we translate $b_{i j}$ to an actual operation in the lab. But I am wondering how to make this implicit step explicit. $\endgroup$ Commented Mar 24, 2022 at 14:30
  • $\begingroup$ One way to see that there seems to be some hidden assumption is to ask the question: Does the "measurement-only"-protocol produce over- or under-braiding?? And how is that determined by the measurement process? $\endgroup$ Commented Mar 24, 2022 at 14:31
  • $\begingroup$ All these protocols assume that one has the ability to pairwise measure the charge/tune the interaction between anyons, at least for the four that are involved in this protocol. Strictly speaking, it means that we need to do something non-local since these anyons are supposedly far away from each other. This is not discussed in the papers cited in the question, and is clearly going to be very system-dependent. The paper I cited above discussed how to implement $b_{ij}$ for Majoranas in lab, and indeed they essentially make use of a non-local interaction in nature --the Coulomb interaction. $\endgroup$
    – Meng Cheng
    Commented Mar 24, 2022 at 14:37


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