Category: Quantum

(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.

On a couple of blogs (Not Even Wrong and Luboš Motl’s reference frame) a question has creeped up which is what role studies of the foundations of quantum theory have in a future theory of quantum gravity. At the Strings2005 conference in Toronto recently, this question was raised during a panel discussion. When someone claimed that foundations hasn’t contributed anything to physics, Lee Smolin apparently said something to the effect that study of foundations of quantum theory has given us quantum computing.
It is true that the original thinkers about quantum computers, and in particular I’m thinking about David Deutsch, where inspired by interpretation issues in quantum theory. But I think the relationship between foundational studies of quantum theory and what is now quantum information science is a bit different. I think that foundational studies have played the role of a “clarifier” in quantum information science. Instead of many results being motivated by foundational issues directly, studies in foundations have lead those in quantum computing to a better understanding of what quantum theory is and what it is not. I certainly think that banging my head up against the different interpretations of quantum theory has given me a very good grasp of the limits of what quantum theory can say and what it cannot say. Thus I think that quantum information science has benefited immensely by listening to the foundation crowd and learning just exactly what is so different about quantum theory. So, while the interpretations themselves don’t have any gigantic successes (one could argue for smaller ones!), I think they are an essential base which betters the field of quantum computing.
Now back to quantum gravity. Whether or not the foundations of quantum theory has anything to say about quantum gravity, I think, is a question we can debate until the cows come in. There are certainly very good philosophical points of view that the strain between gravity and quantum theory must be broken in some nontrivial manner, and whether we break quantum theory or gravity is, I think, an intriguing question. But if we take quantum computing as an example, the lesson that may be learned is that by careful study of quantum theory you can gain an appreciation for it which might allow you to make progress in forming a quantum theory of gravity. I must say that I am often shocked by the lack of understanding of the details of quantum theory among high energy physicists. Sure they use it every day. But do they really deeply get the theory and what it can and can not do? Of course this is a very prejudiced statement coming from a very prejudiced quantum computing dude. We hold quantum theory fairly sacred and hate to see it abused. I’m also sure that high energy physicists are greatly pained by quantum information scientists lack of understanding of “real” physics!
My person views on the relationship between foundations and quantum gravity are about as close as I get to pseudoscientific gobldy-gook. I gave a talk on those views once. It was supposed to be one hour and it ran to two. Sometimes I contemplate writing a book about these views… Penrose did it, so why can’t I? 😉