Quantum error correction is a beastly miracle: an astounding discovery which lets us dream of a future quantum computer but which quickly turns into a nightmare when we think of actually implementing it in the lab. Motivated in part by this, a large number of people have been thinking about ways to considerably simplify the problems of building a fault-tolerant quantum computer by designing physical systems where error correction is more a part of the natural dynamics of these systems. This approach is more in keeping with our classical computers where fault-tolerance is built into the physics of the devices we use to compute. There are now many different sketches of implementations of these ideas, ranging from very simple systems designed to energetically combat single qubit errors (quant-ph/0012018), to gapped systems which allow for topological quantum computation (quant-ph/9707021), to four dimensional systems which are fully fault-tolerant (quant-ph/0110143.) But these proposals are all just theory right now. What is interesting is to contemplate the actual physical implementation of these systems in the lab.
Along these lines, a lot of interesting ideas have been proposed. Some of these have been in condensed matter systems, like for example in engineered superconducting systems (cond-mat/0407663 and cond-mat/0111224). At the same time there has been a lot of interest in also engineering these systems in atomic and molecular systems. These proposals including using polar molecules in lattices (quant-ph/0612180, Nature Physics 2, 341 (2006)), engineering the appropriate interactions in optical lattices (cond-mat/0210564), systems of trapped ions (quant-ph/0509197, etc, etc. But care must be taken! Especially when you have to work to get the interaction you need in order to obtain the robustness offered by these energetic schemes.
Case in point is a paper which appeared today on the archive, “Energy protection arguments fail in the interaction picture” 0705.2370 by Ken Brown from Georgia Tech. The basic idea behind the energy protection schemes is as follows. Suppose you have a system whose ground state is a quantum error correcting code. Then you can encode quantum information into these energy levels. If you design your system Hamiltonian correctly then it is possible that single qubit errors, for instance, always take you out of this error correcting code space, i.e. that such single qubit errors all correspond to transitions from this ground state to higher energy excited states. More generally you can design systems, like Kitaev’s toric code, whose ground state can only be excited by errors on a large number of qubits. The basic idea is then that if your environment is cold enough, then the process which would normally cause decoherence are surpressed since EVERY single qubt error (or up to the number of errors the error correcting code can detect minus one) will take the system from the ground state to an excited state of higher energy. In contrast for most physical qubits it is possible to have errors (phase errors) which cause the phase of a quantum system to change without exchanging energy between the system and environment. [As an aside: decoherence when it is perturbative likes to preserve the separate energies of a system and its environment, thus decoherence can either be heating, cooling, or more insideously, these phase errors. Just making a system degenerate doesn’t help you protect against errors since then all errors are these insideous phase errors.] For an example of this type of argument see (quant-ph/0012018)]
Okay, so far so good, hopefully. Now suppose that you want to engineer such a system in an ion trap. Indeed, for example, in quant-ph/0703270, the authors consider engineering a system known as the long range quantum compass model in ion trap quantum systems (a long time ago the short range version of quantum compass model also appeared in the thesis of a lunatic thinking along similar lines.) The way that they achieve the necessary two qubit interactions to implement this model is to use what I like to call the Sorrenson-Molmer interactions (see quant-ph/0002024.) In particular it is possible to take ions in an ion trap, and apply lasers to the ions such that the internal energy levels of the ions couple to the motional state of the ions, and, if you do this just right, you can engineer two qubit effective interactions like, say Ising interactions along X and Y directions: [tex]$X otimes X$[/tex] or [tex]$Y otimes Y$[/tex] in qubit language. Now it turns out that the quantum compass model uses exactly these different competing Ising bonds. So the idea of quant-ph/0703270 is exactly to engineer the quantum compass model in these ion trap systems.
But wait! There was a bold word in that last paragraph. Effective. In particular in the Sorenson-Molmer scheme what you end up doing is showing that in the interaction picture of the ions the effective interaction is the requisite Ising terms along the appropriate directions. What does this mean, in the interaction picture? It means that there is a rotating frame for the ions (rotating both the internal degrees, i.e. your qubits, and the motional degrees of freedom, i.e. the bus) where the interaction looks like the Ising terms. But no system is isolated, so an important question to ask is what does the environment see in this frame of reference. In particular, since you are going to use these interactions in order to protect your system energetically, the arguments about protecting qubits by keeping the environment cold better work out.
And do they? Sadly, no and this is the content of 0705.2370. In particular if you make your effective interaction in the interaction picture, then you’d better look at you system environment coupling which you are tyring to protect against in this picture. Now when you do this what you find out is that the system environment coupling gets changed in an important way. In particular in this frame, the energetic arguments you make about decoherence being either cooling, heating, or phase-ish errors get modified (this argument is basically the rotating wave approximation.) Indeed what you discover is that there are interactions where you excite from the ground state in the interaction picture to a higher energy state and get a change in the zero temperature environment but still conserve energy. What is this process back in the non-interaction picture? Nothing more than spontaneous emission from your ion energy levels. Doh! [As an aside I should note that this effect occurs because the coupling you engineer is much much weaker than the bare energy levels of you ion.]
So the short of the long of this story is that because the interaction you engineer is created in the interaction picture, the arguments about how the system interacts with its environment gets modified. This in effect is a huge problem for proposals like quant-ph/0703270. Further it is a big problem for proposal which try to use Sorenson-Molmer type gates for doing things like simulation of many-body quantum systems (the system won’t thermalize to the thermal state of the interaction you are trying to create) and quantum adiabatic computation (same problem.) But to learn that you’d best check the details of 0705.2370.
