(Update: Sean Barret points out that his comment when I first posted this was exactly the point I talk about in this post. Somehow my brain didn’t register this when I read his comment. Doh.)
Back from a workshop in Arlington, VA.
One of the most interesting events in this workshop was that Daniel Lidar talked (all to briefly) about his (along with Alicki and Zanardi’s) objections to the theory of fault-tolerant quantum computation. I’ve talked about this before here, where the resulting discussion in the comments was very interesting. At the workshop, Hideo Mabuchi brought up something about the paper which I had totally missed. In particular, the paper says that (almost all) constructions of fault-tolerant quantum computation are based upon three assumptions. The first of these is that the time to execute a gate times the Bohr frequency of the system should be on the order of unity. The second assumption is a constant supply of fresh ancillas. The third is that the error correlations decay exponentially in time and in space.
What Hideo pointed out was that this first assumption is actually too strong and is not assumed in the demonstrations/proofs of the theory of fault-tolerant quantum computation. The Bohr frequency of a system is the frequency which comes from the energy spacings of the system doing the quantum computing. The Bohr frequency is (usually) related to the upper limit on the speed of computation (see my post here), but is not the speed which is relevant for the theory of fault-tolerance. In fault-tolerance, one needs gates which are fast in comparison to the decoherence/error rate of your quantum system. Typically one works with gate speeds in implementations of quantum computers which are much slower than the Bohr frequency. For example, in the this implementation of a controlled-NOT gate in ion traps at NIST, the relevant Bohr frequency is in the gigahertz range, while the gate speeds are in the hundreds of kilohertz range. What is important for fault-tolerance is not that this first number, the Bohr frequency, is faster than your decoherence rates/error rates (which it is), but instead that the gate speed (roughly the Rabi frequency) is faster than your decoherence rates/error rates (which is also true.) In short, the first assumption used to question the theory of fault-tolerance doesn’t appear to me to be the right assumption.
So does this mean that we don’t need to worry about non-Markovian noise? I don’t think so. I think in many solid state implementations of quantum computers, there is a strong possibility of non-Markovian noise. But I don’t now see how the objection raised by Alicki, Lidar, and Zanardi applies to many of the quantum computing proposed systems. Quantifying the non-Markovian noise, if it exists, in different physical implementations is certainly an interesting problem, and an important task for the experimentalists (and something they are making great progess on, I might add.) Along these lines it is also important to note that there are now fault-tolerant constructions for non-Markovain noise models (Terhal and Burkard, quant-ph/0402104 and Aliferis, Gottesman, and Preskill, quant-ph/0504218.) Interestingly, these models postulate non-Markovian models which are extremely strong in the sense that the memory correlations are possibly infinitely long. However, it is likely that any non-Markovian noise in solid state systems isn’t of this severly adversarial form. So understanding how the “amount” of non-Markovian dynamics effects the threshold for fault-tolerance is an interesting question.