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My understanding is that the magnetic fields needed to hold the ions in place in ion trap quantum computers are very complex, and for that reason, currently, only 1-D computers are possible, therefore reducing the ease of communication between qubits. There does seem to be a proposition for a 2-d system using a Paul trap in this preprint but I can't seem to find if this has actually been tested.

Does the scalability of ion trap quantum computers depend upon this alone (whether or not the ions can be arranged in configurations other than a straight line) or are other factors entailed? If the former, what progress has been made? If the latter, what are the other factors?

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Ion trap quantum computers hold ions in empty space using electric not magnetic fields. That is impossible using static fields (Earnshaw's theorem) so an alternating field is used. The effect is that charged particles such as ions seek a field minimum; this type of ion trap is also called a quadrupole trap because the simplest (lowest order) field having a minimum in space is a quadrupole field. It is simple to arrange fields that confine ions either to a point or to a line and ion trap quantumcomputers use the latter. Yet this does not scale because computations involve motional modes of the ions which become harder to distinguish when there are more ions.

There are two approaches to make this approach scalable: Couple strings of ions either using light (photons) or by shuttling ions from one to another such linear ion trap section. Using photons is particularly difficult and far from currently workable for a quantum computer that meets an error correction threshold, so let's focus on shuttling ions.

Mathematically true quadrupole traps cannot be built to have intersections but that hasn't stopped physicists from making them anyways. The trick is that, although one cannot arrange to have a quadrupole field at the center of the intersection, one can still have confinement. And by slightly driving ions into the confining (alternating) field using a static field, one can get sufficiently strong confinement. It has even been shown that such shuttling across an intersection is possible without significantly heating the ion (changing its motional state).

With such intersections, ion traps are scalable.

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  • $\begingroup$ There's a section on loopholes in that article on earnshaws theorem...would any of them apply? $\endgroup$ – snulty Apr 15 '18 at 22:29
  • $\begingroup$ @snulty No, unfortunately, none apply here. $\endgroup$ – pyramids Apr 16 '18 at 11:15
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You may want to check out this Schaetz et al, Reports on Progress in Physics of 2012 "Experimental quantum simulations of many-body physics with trapped ions" (alternate link in semanticscholar). In sum: yes, the arrangement of the ions is one key limitation to scalability, but no, configurations are not currently limited to a single line of atoms. On that paper, check Figure 3 for experimental fluorescence images of laser-cooled ions in a common confining potential of a linear RF trap, including a single ion, a single line, a zig-zag chain and a three-dimensional construct.

From Figure 3 in the paper above by Schaetz et al: "Structural phase transitions can be induced between one-, two- and three-dimensional crystals, for example by reducing the ratio of radial to axial trapping frequencies." I am sure more recent review papers should exist, but this is the first one I found that was satisfactory. Admittedly, current results are more about direct simulation rather than universal computation, e.g. from figure 13 in the same paper: "Changing the experimental parameters non-adiabatically during a structural phase transition from a linear chain of ions to a zigzag structure, the order within the crystal breaks up in domains, framed by topologically protected defects that are suited to simulate solitons."

On the same topic, and also from 2012, another paper worth checking out would be Engineered two-dimensional Ising interactions in a trapped-ion quantum simulator with hundreds of spins (arXiv version) (Nature version. You have the experimental picture as Figure 1; it is a Penning trap in this case rather than a Paul trap. Indeed, it is not universal quantum computing but rather the specialized application of quantum simulation, but still it is undeniably experimental progress towards holding ions in place in a 2-D trap and thus advancing towards scalability.

I am myself no expert in traps, but this is what I got on scalability in a recent (2017) conference:

  • Experimentalists play around with the potentials and achieve interesting combinations, with central zones that are quasi-crystalline (chains, ladders, ribbons etc) and exotic tips (e.g. ribbons or ladders that finish in a single atom).
  • The majority of the popular ions have a configuration of the type [noble-gas]$s^1$ (like Ca$^+$), preferredly with no nuclear spin but this is for convenience and simplicity. Accessing hyperfine states and/or a more complex spin level structure (like Yb$^+$=[Xe]f$^{14}$s$^2$) opens the door to a richer Hilbert space per ion.
  • Collective vibrations are used as the basis of interqubit communication. As in the previous point, the breathing mode is uniquely stable and thus convenient to use, but other vibrations are also accessible and would allow more interesting interqubit communication schemes.
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While I’m not an experimentalist, and have not studied these systems in any great depth, my (crude) understanding is the following:

In ion traps you (more or less) have to trap the ions in lines. However, this isn’t a limitation in terms of the ease of communication because what you’re probably thinking about is when a linear system has nearest neighbour interactions, I.e. each qubit can only interact with its immediate neighbours. In ion traps, this isn’t really true because you can access a common vibrating mode of all the ions in order to make arbitrary pairs interact directly. So actually, that’s really good.

The problem is that number of qubits that you can store. The more atoms you put in the trap, the closer together their energy levels are, and the harder they become to individually address in order to control them and implement gates. This tends to limit the number of qubits you have in a single trapping area. To get around this (and with the added bonus of parallelism, necessary for error correction), people want to make multiple distinct trapping regions interact, either with flying qubits, or by shuttling the atoms between different trapping regions. This second approach seems to be very much in progress. This is the theory proposal, but I have certainly seen papers that have demonstrated the basic components.

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