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It seems to me that a good geometry for an ion trap quantum computer would be to constrain the ions to lie on the surface of a sphere. Their mutual repulsion would space them out evenly. If the ions were positively charged, a negative charge at the center of the sphere could put them all in a “spherical well potential”. The force on each ion would be the outward radial force of the sphere substrate on the ion plus the inward radial force produced by the Coulomb force between the ion and the central charge. One could achieve full connectivity between the qubits by putting them all in the same energy level of that spherical well potential. For each ion, one could place a microlaser at the same polar angles but at a slightly larger radius than the ion. This way, one could address each ion separately.

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Visually this looks cool, but there are two big reasons this kind of geometry isn't feasible. These reasons hold true even if you could somehow figure out an ion trap that gives you a perfect spherical potential that locks all ions in place.

  1. Despite appearances, this geometry doesn't offer higher qubit connectivity than a simple linear chain.

  2. The spherical geometry is probably the worst possible geometry to do qubit readout.

Now in more detail. The way trapped ions get their connectivity is by carefully exciting vibrational normal modes (phonons) so that only the two ions you care move and undergo a two qubit gate. This applies equally to both ions in a linear chain or in a sphere, so there is no connectivity benefit for a sphere, since both the sphere and linear chain have the exact same number of vibrational modes.

More detrimentally, the sphere does not allow you to read out your qubits properly. The way readout is done via shining resonant laser light so that only ions in the bright state fluoresce, and the image of the ions is collected on a camera. However, in the spherical case you would have a blurry photo since the sphere would be larger than the depth of field of the optics/laser. Moreover, when one qubit is read out in this spherical geometry, its emitted fluorescent light has a very high probability of hitting another ion and accidentally reading it out as well (i.e. large crosstalk). So single ion readout (and quantum error correction!) are near impossible in this geometry without crosstalk. For comparison, the linear geometry has none of these problems and is very easy to read out with low crosstalk.

Nonetheless you are on the right track, in principle circular ion traps (not spherical) are a good idea and there are people in the field exploring that.

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Very interesting concept! I'm not a trapped ion expert, but few concerns that immediately come to my mind to think through are two-qubit gates, small spatial perturbations cascading, and practicality.

Two-qubit gates on trapped ion systems are typically implemented by exciting a vibrational mode between the two qubits. Isolated two qubits from the others vibrationally seems difficult, especially if they're all supposed to "press" on each other electromagnetically to maintain the spherical arrangement.

Second, I'd be worried that one ion shifting slightly due to laser pressure or some other small perturbation would bump its neighbor and cause the entire trapped ion structure to shift slightly as that vibration is absorbed.

Third, aligning all the microlasers with the ion sphere could be difficult if they like to shift. Additionally, there's the problem of how stable the configuration is. If you're trying to keep it all floating in vacuum (which is desirable for your qubit coherence) and only use E/M fields to constrain the atom positions, designing and implementing those fields in 3 spatial dimensions would be practically challenging.

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