QCJC: Bilnov 2004

I received my introduction to quantum computing through the two “bibles” of ion trapping – a very careful study of Leibfried, Blatt, Monroe, and Wineland’s “Quantum dynamics of single trapped ions”, and a much less careful study of Wineland et al’s “Experimental Issues in Coherent Quantum_State Manipulation of Trapped Atomic Ions”. Those are not necessarily light and easy to read – they function more as giant review articles with detailed derivations. It doesn’t seem particularly practical to try to do any kind of journal club writeup of those papers, just because they are so large and unweildly. Instead, I think I’m going to use a series of articles from the “early days” of Ytterbium ion trapping to focus on individual topics, piece by piece. Also, I will point out that Jameson has already written about trapped ions here and here – I would especially recommend the Linke writeup for a wonderful comparison of trapped ion based qubits with superconducting qubits.

Bilnov’s 2004 review article features many of the same people from those other two tomes, but seems to be presented in a much more accessible way. The authors begin their discussion on hyperfine qubits, which are so named because of their usage of the hyperfine ground states of a single trapped ion. These separations, of order GHz, typically have incredibly stable line widths. This means that the spacing between these states are extremely well defined, to the point where the hyperfine splitting of the Cesium atom currently serves as the functional definition of the SI second. They also have very long radiative lifetimes, which mean that once you go to an excited state, it is less likely for the qubit to naturally decay back to the ground state, reducing decoherence. Some ions that have a non-zero nuclear spin include 9Be+, 25Mg+, 43Ca+, 87Sr+, 137Ba+, 173Yb+, and 199Hg+.

Before we can get anywhere with these trapped ions, we must first trap them. Earnshaw’s theorem limits us from using purely static (dc) forces to trap charged particles, so we need to use a (Paul) trap. This implementation uses a radio frequency (rf) potential, which is a very rapidly time-varying electromagnetic field creating a pseudopotential, along with dc electrodes to act as the end caps. This type of linear trap typically provides a reasonable harmonic well, so when the ions are cooled and trapped, they are confined to a single location. When multiple ions are loaded in, the Coulomb force forces the ions into a linear array.

At this point, the paper discusses some of the more fundamental limitations of an ion trap. For instance, it notes that each additional ion introduced to this chain will also bring along three more vibrational modes. These additional vibrational modes, which must be isolated to perform gates, create a denser “forest” of vibrational modes. Heisenberg uncertainty provides a fundamental limit on the tradeoff between frequency and time. Therefore, having small separations in frequency space require longer gates to achieve the same fidelities, making it difficult to scale up ion trapping. However, this is not an impossible problem. For instance, most of the time a quantum algorithm only needs to act on a few qubits at a time. Therefore, if there was a way to split of “computational” qubits from “storage” qubits, and merge them back together later, then we might be able to perform large algorithms with a smaller maximum ion chain length.

After ions are trapped, the experimenters are able to perform optical pumping in order to bring the ions to either the excited or the ground spin state, which would then be able to represent the 1 or 0 in the qubit. Afterwards, there is a circularly polarized laser that is resonant with one of the transitions of the Cadmium ion that they use. This transition will only allow for photons to be scattered when they are in the excited spin state; however, it is possible for the camera or photomultiplier tube used to see dark counts because of randomly scattered light in the chamber, or because of poor quantum efficiency from the detector. Therefore, it is useful to both have a very good imaging objective to collect the maximum amount of light, as well as to integrate the collection time for some extended amount of time (in this case, 0.2ms). Therefore, by defining some photon “cutoff” between the 0 and 1 states, the authors are able to achieve detection efficiency of >99.7%, as shown in Figure 3.

One thing to note is that the wavelength of the imaging transition seems to be around 214.5nm, which is incredibly far into the ultraviolet regime. That might be one of the issues in using Cadmium ions, as that wavelength seems like it would be both difficult to source high power lasers, as well as difficult to find optical coatings that would not have issues handing that type of power needed to control long ion chains.

I think I will bump discussion about gate control to another paper, but it is useful to note that the paper demonstrates their success using the Cirac-Zoller gate, which is a CNOT gate that is a predecessor of the more modern Molmer-Sorensen gate that is more commonly used today. I am interested in bettern understanding the difference between these two gates, and especially why it seems to have so strongly shifted towards using MS gates today.

Reference: Bilnov, B. B., Leibfried, D., Monroe, C., Wineland, D. J., Quantum Computing with Trapped Ion Hyperfine Qubits Quantum Information Processing 3, 45-59 (2004)

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