Particle physicists have always considered themselves the kings of physics. Murray Gell-Mann famously called solid state physics by the moniker “squalid state physics.” In the ivory towers where scientists picture themselves as selfless serfs in the service of knowledge, particle theorists have long occupied the attic. At the same time, there is another community of the mathematically inclined who claim that they do their work for the greater good of knowledge: programmers. In particular the open source spirit of programming, that good code is in some way eternal and should be shared and contributed to the greater cause, gives good coders an air of superiority not dissimilar to that found in particle theorist.
And when I think about these two fields, I begin to think that perhaps quantum computing is today’s version of the selfless king in search of knowledge. Not only are we learning about the fundamental ways in which quantum information and computation differs from classical information and computation, I think many of us in the quantum computing community also feel that our work will have some greater consequence once a quantum computer is eventually built. We are, therefore, I think a rather smug community not very dissimilar to particle theory or the ethic of the eternally beautiful algorithm. Whether this smugness will be our undoing, our triumph, or our own psychosis with which we will beat ourselves over the head is another question.
Physicists Are Crazy
Best title of the year on quant-ph? Or Worst title of the year on quant-ph?
Is Quantum Suicide Painless? On an Apparent Violation of The Principal Principle
Perspective
If you were a quantum computer, would the mystery you be working on not be the mystery of quantum theory but the mystery of the classical world?
Beyond Quantum Theory
Nothing is more mysterious in quantum theory than the fact that states are rays in a Hilbert space and that the probability law comes from the modulus squared of overlap between the input and output states. I like to phrase this question as “Why Hilbert space?” Of course there may be no “why”! To quote Feynman: “Do not ask yourself, if you can possibly avoid that, ‘how can it be like that?’ because you will lead yourself down a blind alley in which no one has ever escaped.” But let’s assume that there is something “beyond quantum theory.” What could such a structure look like? There are many paths we can imagine for what such a structure could look like. But all of these structures must in some limit or even exactly given an explanation for the Hilbert space structure and measurement postulate for quantum theory. So here it makes a certain sense to begin thinking about what exactly quantum theory is and what exactly quantum theory is not before we embark on exploring what is beyond quantum theory. But I think today, thanks in large part to years of foundational people yelling and screaming as well as the comfort developed with quantum theory from practicing quantum information science, we understand intimately what quantum theory is and what quantum theory is not. Perhaps it is time to move on!
After going through many phases of thinking about where quantum theory comes from, I’ve now entered a new phase. My earliest phases in thinking about quantum theory stressed the information theoretic notions of quantum theory. Thinking like a computer scientist, statistician, or information theorist leads one to a much cleaner idea of what quantum theory is and what quantum theory is not. The quantum state should never, for example, be mixed up with a realistic description of a system. Noncontextuality and the nonlocal nature of quantum correlations are best understood as telling us how we can and can’t think about the information in quantum systems. And, while these points of view are certainly enlightening, this point of view can be taken too far. For example, I have spent a considerable amount of time trying to understand if the correlations produced by measuring entangled quantum states can be seen to arise because these correlations are best for, say, winning some information theoretic game. The best success of this type of reasoning, I think, is the result of William Wootters (two ohs two tees), who showed in his Ph.D. thesis that for real quantum theory the quantum measurement postulate follows from the question of how to best send distinguishable signals through a channel with angular symmetries. But it may be, and this is where my change of heart has occured, that quantum theory does not arise because it is “best at some game” or “natural under information constraints.” This does not mean that we don’t listen to what quantum theory is and isn’t saying from an information theory perspective, but it does mean that we need to move on and look for a deeper structure behind quantum theory.
How might we do this? Well my new phase is based on a philosophical argument I have discussed here before: the nonlocal nature of quantum correlations implies that any deeper theory which explains quantum theory must take seriously that our notions of spacetime topology are wrong. If all our descriptions of quantum theory must have parts which explain nonlocality, then what is the difference in such a description between having nonlocal quantities and saying that our notion of spacetime topology is wrong. In fact I might go so far as to suggest that the failure to quantize gravity (shut up string theorists…just kidding) is evidence that this is the correct approach. Since general realtivity is our theory of spacetime structure, the reason, in this view, for why we can’t quantize general relativity is that general relativity, or some deeper theory of spacetime, is what gives rise the quantum theory. So now, in my new phase, instead of looking for the game quantum theory is best at playing, I think about the geometric constructions which might give birth to Hilbert space and the quantum probability law. I think the most inspiring connection to date of this idea are results in topological field theories, where the topology of the manifold is a dynamic quantity. And there are many who argue that gravity might be a similar such theory where we have a topological field theory with the extra structure of local degress of freedom. A beautiful paper along these lines (but not far enough along these lines) is Quantum Quandaries: a Category-Theoretic Perspective by John Baez.
Old Bacon
From Scott Aaronson’s upcoming thesis:
For better or worse, my conception of what a thesis should be was influence by Dave Bacon, quantum computing’s elder clown, who entitled the first chapter of his own 451-page behemoth “Philosonomicon.”
Isn’t it great to be in a field where at age 29 you can be considered “elder”? In fact I was just looking at the schedule for the upcoming QIP conference at MIT and was a bit taken back by the youthfulness of the invited speaker list.
Quantum Dollars
At the workshop here a claim was made that the world has now spent over one billion dollars in the field of quantum information science since the discovery of Shor’s algorithm in 1994. How many billions more before a useful quantum computer is built?
Ion Trap Milestone
The slow steady advance in ion traps! A milestone: Realization of quantum error correction,” J. Chiavernini, D. Leibfried, T. Schaetz, M. D. Barrett, R. B. Blakestad, J. Britton, W. M. Itano, J. D. Jost, E. Knill, C. Langer, R. Ozeri & D. J. Wineland, Nature 432, 602–605 (2004)
Scalable quantum computation and communication require error control to protect quantum information against unavoidable noise. Quantum error correction protects information stored in two-level quantum systems (qubits) by rectifying errors with operations conditioned on the measurement outcomes. Error-correction protocols have been implemented in nuclear magnetic resonance experiments, but the inherent limitations of this technique prevent its application to quantum information processing. Here we experimentally demonstrate quantum error correction using three beryllium atomic-ion qubits confined to a linear, multi-zone trap. An encoded one-qubit state is protected against spin-flip errors by means of a three-qubit quantum error-correcting code. A primary ion qubit is prepared in an initial state, which is then encoded into an entangled state of three physical qubits (the primary and two ancilla qubits). Errors are induced simultaneously in all qubits at various rates. The encoded state is decoded back to the primary ion one-qubit state, making error information available on the ancilla ions, which are separated from the primary ion and measured. Finally, the primary qubit state is corrected on the basis of the ancillae measurement outcome. We verify error correction by comparing the corrected final state to the uncorrected state and to the initial state. In principle, the approach enables a quantum state to be maintained by means of repeated error correction, an important step towards scalable fault-tolerant quantum computation using trapped ions.
Quantum Gravity 2+1
Talking about different physical theories in different dimensions is one of physicists favorite pastimes. Thus, for instance, we often move the theory to the infinite dimensional case or to the case of dimension slightly greater or less than some integer dimension. Often we work with low dimensional theories because of constraints–for example we do this when we talk about an electron gas trapped in a two surface or a one dimensional line or even a zero dimensional dot. But sometimes we work in different dimensions to see if we can gain insight into the theory in a dimension where we can’t seem to make much progress. Such is the case for a quantum theory of gravity. Our success in understanding quantum gravity in two spatial dimensions plus one time dimension (2+1) has been far better than our success in understanding quantum gravity in three spatial dimensions plus one time dimension (3+1). Why?
Well there is an easy way to see why quantum gravity in 2+1 dimensions is quite a different beast than quantum gravity in 3+1 dimensions. Let’s look at the 3+1 dimensional case first. At a fixed time, the spatial metric has six degrees of freedom (it’s a real tensor and it’s a symmetric tensor and it’s in dimension three.) But the laws of general relativity are invariant under general coordinate changes. This means that there are four gauge degrees of freedom which correspond to the choice of four spacetime coordinates. Thus there are two physical degrees of freedom in 3+1 dimensional gravity.
But what happens in 2+1 dimensional gravity? At a fixed time, the spatial metric now has three degrees of freedom (it’s a real tensor and it’s a symmetric tensor and it’s in dimension two.) But now there are three gauge degrees of freedom. Thus in 2+1 dimensional gravity there are no physical degrees of freedom!
In fact what happens in 2+1 is that the curvature tensor vanishes! Now recall that if we parallel transport a vector around an area where the curvature tensor vanishes, then the vector doesn’t change. Thus if we work in a spacetime where there all loops encose areas, then parallel transport will be path independent and there will be a global notion of parallelism. Quite a boring theory right? Well yes, if you only consider simply connected spacetimes (i.e. spacetimes where all loops are continuously contractible to a point.) But if you consider spacetimes which have noncontractible loops (think of a torus and the circles which form circumferences of this torus) then the parallel transport around one of these noncontractible loops doesn’t enclose an area. Now the geometry of this flat spacetime is characterized by the results of parallel transport around noncontratible loops (holonomies.) So right away we see that gravity in 2+1 dimensions will be an interesting theory when we allow topologically nontrivial spacetimes. In fact, when we construct the solutions and quantize gravity in 2+1 dimensions we are led to a topological quantum field theory! Actually things get quite interesting in 2+1 dimensions when we try to quantize the theory. In fact there are many different approaches to this quantizations, and, strangely, not all of these are consistent (this is why you have to pay attention when all these mathematical physicists go on and on about all these different methods to quantize classical theories!)
Well enough quantum gravity for today. Just remember, theories of physics are never as complicated as most theoretical physicists would like you to believe.
Where is SETI, Where is SETI, Here I am, Here I am
If I were an extraterrestrial, I might be very cautious about communicated my location to the rest of the universe since the universe might be full of other hostile extraterrestrials. This is an argument which must scare the bejebus out of those working on the search for extraterrestrial intelligence (SETI). And this begs the question: is there a way to transmit a signal such that the location of the transmitter cannot be discovered? In this 2003 article by Walter Simmons and Sandip Pakvasa, the authors claim that it is possible to design such a protocol by using entangled photons. Now I haven’t fully understood their protocol, but I do worry that it requires the transmitters to bounce their entangled photons off of relay stations which make a large angle on the reciever’s sky. And if you can create a large angle on the reciever’s sky, why don’t you just send a signal from somewhere where you don’t have any of your cute little alien colonies?
CEPI Seminar 10/10/04 Wim van Dam
Complexity, Entropy, and the Physics of Information Lecture Series
Wednesday, November 10, 2004, 5:00 PM. Refreshments 4:15 PM.
Robert N. Noyce Conference Room, Santa Fe Insitute
Wim van Dam
Computer Science Dept., University of California, Santa Barbara
Quantum Computing, Zeroes of Zeta Functions & Approximate Counting
Abstract:
In this talk I describe a possible connection between quantum computing and Zeta functions of finite field equations that is inspired by the ‘spectral approach’ to the Riemann conjecture. The assumption is that the zeroes of such Zeta functions correspond to the eigenvalues of finite dimensional unitary operators of natural quantum mechanical systems. To model the desired quantum systems I use the notion of universal, efficient quantum computation.
Using eigenvalue estimation, such quantum systems are able to approximately count the number of solutions of the specific finite field equations with an accuracy that does not appear to be feasible classically. For certain equations (Fermat hypersurfaces) I show that one can indeed model their Zeta functions with efficient quantum algorithms, which gives some evidence in favour of the proposal.