Measuring Topological Charge: Ising Qubit
Date -- 30.07.23
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Broadly, there seem to be two approaches to measuring anyonic charge (see Figure 1):
A1. Direct measurement. Bring a neighbouring pair of anyons sufficiently close together that their fusion channels may be distinguished by some local observable (e.g. energy, force).
A2. Probing / interferometry. One or more probe anyons of known charge are fired towards a region of unknown charge. As a probe charge approaches said region, it is split into two waves which travel around opposite sides and then recombine. We might then be able to infer the value of the unknown charge from the resulting interference.
These approaches are briefly summarised in Chapter 11.2.2 of Steven H. Simon's upcoming book, 'Topological Quantum' [1]. I've been using the proto-book as a reference since I was a PhD student, and can't recommend it enough! You can find the latest draft copy here. The final product is available for pre-orders here.
The theory for A2 appears to have first been laid down by Overbosch and Bais in 2001 [2]. It was then further developed by Bonderson, Kitaev, Shtengel and Slingerland during the mid 2000s [3,4,5,6].
0. Figures
Figure 1. Two approaches to measurement of topological charge.
Figure 3. Graphical depiction of exchange / R-matrices via worldlines.
Figure 2. Mach-Zehnder setup for Ising qubit. Purple trajectories show the path of the probe Ising particle/wave; BS1 and BS2 are balanced beam-splitters; the gold-shaded region encloses two Ising anyons whose state is given by the Ising qubit; D1 and D2 are detectors, only one of which will click for each probe charge sent into the interferometer.
1. Pros and Cons
There are some obvious pros and cons to these approaches.
A1 (Pros): Measurement of fusion channel is direct.
A1 (Cons): Requires control on microscopic length scales (observable highly localised).
A2 (Pros): Anyons whose fusion channel is to be determined can be kept in situ.
A2 (Cons): Measurement of fusion channel indirect, and host system should support interferometry.
The details of A1 probably depend on the system. If fusion outcomes could successfully be measured this way, it sounds straightforward and ideal! Recall that part of the appeal of using anyons for quantum computing lies in the nonlocal encoding of information in their fusion states; so long as the constituent particles are kept apart, noise operators (which are typically local) cannot 'see' the information. It follows that the information is safer if the operational anyons are kept far apart -- and of course, gate implementation (i.e. braiding) can be done over large length scales. Operating at larger length scales also has the advantage of not requiring control over anyon trajectories on smaller length scales: herein lies the appeal of A2.
The measurement approach of A1 involves reducing these length scales for pairwise measurements, and so the risk of corruption is higher. In order to safely realise A1, one would need very fine control over the patch of the system where the measurement occurs.
Of course, A2 is not without drawbacks. First of all, the underlying measurement theory appears to be somewhat involved (at least for the most general case) [4,6]. Secondly, performing interferometry (in an already complex system) sounds challenging. It seems that in practice (e.g. in fractional quantum Hall systems), the interferometers of choice go by the name Fabry-Pérot, or two-point contact. While the details of how such interferometers are applied in these systems are likely complicated, the underlying theory is illustrated by a Mach-Zehnder interferometer (as is done by Bonderson et al. in [4,5,6]). We follow suit in the simple example below. We note that in a Mach-Zehnder interferometer, accurate measurement also relies on having precisely tuned beam-splitters!
2. Example: Mach-Zehnder Interferometer for an Ising qubit
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