Recently I have been reading Quantum Field Theory Of Many-body Systems: From The Origin Of Sound To An Origin Of Light And Electrons
The place where I first learned about this sort of thing was some of the work I did in my thesis where I used the Jordan-Wigner transformation in one dimension (A good read: Michael Nielsen’s notes on the Jordan-Wigner transform.) Suppose you have a one dimensional lattice of fermions, where the fermions only interact between nearest neighbors. Let [tex]$a_i$[/tex] and [tex]$a_i^dagger$[/tex] be the annihilation and creation operators at the site [tex]$i$[/tex]. These being fermions, these operators satisfy [tex]${a_i,a_j^dagger}=delta_{i,j}$[/tex] and [tex]${a_i,a_j}=0$[/tex]. In the Jordan-Wigner transform, we replace each fermion site by a qubit, then we perform the map [tex]$a_irightarrow – prod_{j=1}^{i-1} Z_jfrac{1}{2} (X_i + i Y_i)$[/tex]. One can easily check that this mapping preserves the fermion commutation relations. Under this mapping, we can map our nearest neighbor fermion model to a nearest neighbor qubit model. It is exactly this kind of mapping, for more interesting systems, that Wen is excited about.
An interesting question to ask is how to perform the above mapping for lattices of dimension higher than one. To this end, you will notice that the mapping used above has a linear ordering and hence is not well adapted to such a task. In particular if you try to use the mapping in this manner, you will end up creating qubit Hamiltonians with very nonlocal interactions. In fact, many have tried to create higher dimensional Jordan-Wigner transforms, but in general, there were always limiations with these attempts. To this end, the recent paper cond-mat/0508353 by F. Verstraete and J.I. Cirac is very exciting. These two authors show that it is possible to convert any local fermion model into a local model with qubits (or qudits), i.e they effectively solve the problem of creating a Jordan-Wigner transform on higher dimensional lattices.
One of the points that Wen likes to raise from this work is the question of whether fermions are actually fundamental. From what I understand, while there are examples of fermions arising from these local interacting boson modes, it is not known how to do this with chiral fermions. Strangely I’ve always been more inamored with fermions than with bosons (holy cow am I a geek for writing that sentence.) But perhaps my love of bosons will have to start growing (oh, that’s even worse!)
Ticking Away the Moments That Make Up a Dull Day
Stephen Hawking and Leonard Mlodinow’s new book coming out this September: A Briefer History of Time
Beyond the Frinkahedron
An easy, well written discussion of the casual causal dynamical triangulations approach to quantum gravity: hep-th/0509010: “The Universe from Scratch” by J. Ambjørn, J. Jurkiewicz. and R. Loll. If I were young (wait a second), interested in quantum gravity, and this article didn’t get me interested in this technique, I think I would check my pulse (ducks shoe thrown by Lubos.)
physics.QP
The arxiv has announced a restructuring of its categories. Quant-ph will be known as physics.QP and its moderators are Daniel Gottesman and Lev Vaidman:
physics.QP Quantum Physics (Daniel Gottesman, Lev Vaidman)
quantum information and associated physical effects, quantum computation, experimental quantum devices, non-determinism experiments and interpretations
I have no idea what “non-determinism experiments” refers to. Apparently there are some really good experimentalists out there who run experiments with no noise.
Not Even Publishable?
Peter Woit has an interesting post today on his new book “Not Even Wrong.” In the post Peter describes his zigzag path towards publication of the book and the resistance he encountered. My favorite part is where Peter describes how during one round of review, he encountered a referee who
…dealt with the problem of not being able to find anything wrong with what I had written by claiming that arguing against string theory was like arguing against teaching evolution…
Not only are we going to stop teaching evolution in public high schools science classes, but now we aren’t going to teach string theory in these classes as well. On no! The horror! The horror!
Patterns? You Want Patterns?
I highly recommend this post over at Three-Toed Sloth about identifying coherent structures in spatiotemporal systems. What a pretty post! Oh, and the science is fascinating as well. It reminds me of a question I’ve always wondered about in cellular automata theory. It’s a bit long winded, so if I get the time I’ll write a post on it.
"An Introduction To Black Holes, Information And The String Theory Revolution: The Holographic Universe" by Leonard Susskind and James Lindesay
On the plane trip back from Washington DC I read the book An Introduction To Black Holes, Information And The String Theory Revolution: The Holographic Universe
This is a pretty cool book, I must say. First of all, however, we should address the word “Introduction” in the title of this book. On page three of this text (the first full page after the preface) the Schwarzchild metric is written down. Now this sounds like less than an introduction, but really it is a red herring. The book is actually easily readable by anyone who has taken an introduction to general relativity and a good course in quantum theory. I’d say an advanced undergrad could easily grok this book. I make this judgement from the fact that I was able to go through the book in one planeflight.
So what about the quality of the book? There are, basically, three parts to this book. I’m betting that it is based on lecture notes from the two authors: there seems to be major differences in the writing styles for the different sections. Part 1 is “Black holes and mechanics” , part 2 is “Entropy bounds and holography”, and part 3 is “Black holes and strings.” Of these, Part 1 is the largest and takes up most of the book. Which is good, because this is the most interested and best part of the book. In part 1, at a level lacking deep rigor, but a level comfortable for many physicists, the authors introduce the basics of black hole geometry, quantum field time in curved spacetimes, entropy calculations in such spacetimes, black hole thermodynamics and the information paradox for black holes. All in slightly less than a hundred pages. Hence the word “introductory”! This is not a book for those who want to become experts in this stuff, but none-the-less, this is a very beatiful introduction to some really cool ideas. My only major issue with this part of the book is the notation and “words” used to describe quantum theory, and in particular density matrices. If there is one thing quantum information scientists can be proud of, it is their clean and clear notation and exposition about quantum information. I guess everytime I see someone talk about information in quantum theory nowdays, it feels strange if they aren’t using the language of quantum information science. Oh, and for some reason they call the no-cloning principle the no-xerox principle.
Parts 2 and 3 of this book are interesting, but are not as tight as the first part. The entropy bounds deserve a lot more time than is devoted here, I think. But still one gets the basic ideas. Part 3 is very strange because it is so small (less than twenty pages.) It explains, in a very very rudimentary manner, the AdS-CFT correspondence (that supergravity in a certain anti-de Sitter universe can be mapped to a conformal field theory on the boundary of that space.) It’s nice to see this exponsition, but too many details were left out for me to really feel that I got any intuition about this important correspondence.
In total, this is a very nice book and I would definitely recommend it to nonexperts who know the general relativity and quantum theory necessary to understand the book. Part 1 is pretty smooth, I must say. There is only one thing which really bothered me about the book, and that was the lack of references. One of the purposes a book like this can serve is to point the reader to the more rigorous papers dealing with this subject. Unfortunately, the book has only ten references. This is a real shame.
Oh, and by the way, Susskind is, of course, famous for his belief in the anthropic principle. Fortunately it doesn’t make it’s appearance in this book (not that I have strong feelings about this subject 😉 )
Oh, and I especially liked the candor in this passage from the conclussion:
The theory of black hole entropy is incomplete. In each case a trick, specific to the particular kind of black object under study, is used to determine the relation between entropy and mass for the specific string-theoretic object that is believed to represent a particular black hole. Then classical general relativity is used to determine the area-mass relation and the Bekenstein-Hawking entropy. In no case do we use string theory directly to compare entropy and area. In this sense the complete universality of the area-entropy is still not fully understood.
The Incompleteness of Gravity
A new gravity theory, Intelligent Falling. Gotta love the Onion:
What the gravity-agenda scientists need to realize is that ‘gravity waves’ and ‘gravitons’ are just secular words for ‘God can do whatever He wants.’
Does this mean that LIGO can get faith-based funding now?
Bohring Fault-Tolerance
(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.
A Serious Version
A resubmission, quant-ph/0507189. Much, much, more serious:
Title: |0>|1>+|1>|0>
Authors: S.J. van Enk
Comments: A more serious version, almost 2.36 pages, but still an unnormalized title
Note: replaced with revised version Thu, 11 Aug 2005 17:17:52 GMT (6kb)