Topological Quantum Computing

For some time, I’ve heard theorists give talks about topological quantum computing. Every one of them jumps right in and never discusses what the hell this is. And it has never made sense. They talk about braiding of particles and somehow that’s equivalent to quantum gates. What the hell is braiding? Nothing ever made sense.

But as of yesterday, it finally does. Or at least some small part of it does. I am not a solid-state/condensed-matter physicist. So I understand absolutely nothing about proposed solid-state quantum computing, and the proposed architecture for topological quantum computing either. But at least, I now have some vague idea of this. Sankar Das Sarma gave a talk yesterday, and, well I think things are starting to make sense.

Either that, or I’ve heard these terms far too many times.

Here’s the main idea:

Local geometry is a redundant way of encoding topological information. Slight denting or stretching of a torus does not change its genus.

What does that mean? Provided I understood this correctly, if we can find a physical quantum system that is insensitive to slight perturbations in its topology, then the quantum states are, in a sense, immune to decoherence. This insensitivity cannot be due to symmetry, as we can break the symmetry. Somehow, this magically leads to “non-Abeliean quasiparticle braiding statistics.”

Briefly on quantum statistics and identical particles: (some lecture notes here) If we have two identical particles (for example, electrons), we can describe the state of both of them by the wavefunction , where are the positions of the particles. Since these particles are identical, then the dynamics of the system is the same if I swap the positions of the two particles. So, the dynamics of particle A at and particle B at is exactly the same as the dynamics of particle A at and particle B at .

If I interchange the particles twice, I should end up with the same function. That is interchange once: . Interchange a second time: . Because the dynamics are the same, then in three spatial dimensions, we have two types of wavefunctions:

This leads to bosons and fermions, which obey different statistics, and all the wonderful things that we know (BEC, fermi-degenerate gasses, etc.). Fermions obey Pauli’s Exclusion Principle, while bosons do not.

However, strange things happen in two dimensions. It turns out that you can have another type of particle, an anyon. Here, an interchange (?) picks up an additional phase. You can view the two particles as points on a sheet, and the interchange as winding one around the other.

Here’s where the topology comes in. I believe that because the fundamental group of is trivial, and thus you get these bosons and fermions. However, the fundamental group of is non-trivial.

What this means is that any loop in three dimensions (with some points removed) can be pulled together into a point. But a loop restricted to a plane (minus some points) cannot. If I had a loop around one of those points on the plane, I cannot pull it past the hole.

This is what’s meant by braiding. By moving particles around each other, these lines get braided together and knotted up. Somehow (magically) this is equivalent to gates and such, although, it makes no sense as to how that is done.

So, it seems like I’m starting to get the idea, but I don’t have a clue as to how the actual ‘computing’ comes into play here.

The quantum–classical crossover of a field mode

On the ArXiv today, there’s a paper that answers my earlier question about the classical limit of the Jaynes–Cummings model. Apparently, if I just tried a little harder, I could have had another paper. Such is life.

Basically, in the Jaynes–Cummings model, the dynamics of a qubit and a single photon is given by

That is, the energy of the qubit, the energy of the photon, and the interaction between the two.  This predicts energy exchange between the photon and qubit.  The qubit will oscillate between zero and one.  But the theory predicts a decay of this oscillation as well as revivals.

A classical field interacting is described by .  This predicts oscillation of the qubit state between zero and one, but no decay.

It seems that the timescale for the oscillation in both cases is inversely proportional to the electric field strength.  The decay time is independent of the field strength.  So, in the large photon limit, we expect there to still be a decay of the oscillation, but we should still see many oscillations before.

The paper also has another correspondence, that I haven’t quite finished yet.  But it involves the Lindblad equation, which should make it fun.

Quantum Entanglement at a Distance

We have a new paper in the latest issue of Nature, about remote entanglement of two ions. Here’s a couple press releases: Michigan press release, and Maryland press release. We were also on Slashdot. You can read the paper here (pdf). Unfortunately, I’m not on this paper—this was the other project in the lab.

So, what’s this all about? Well, suppose we had two computers, and we wanted to transfer information from one to another. Typically, this would be done by sending bits across the internet (or wireless network, etc.). Now, what if we wanted to do this with a quantum computer? Well, now we have two types of information: classical and quantum. We could just read out the information of one computer, and send it to the other. The read-out process collapses the quantum information into regular classical bits. We could easily send this to the other computer easily and work with it (The classical bits could trigger the control logic of the other quantum computer).

Now what if what the other computer needed was the quantum information? We need a way to transmit this quantum information. It turns out one way of transferring this information is through quantum teleportation. This requires quantum entanglement.

Suppose I had three qubits, q0 q1 q2, where q1 and q2 are entangled. I give you qubit q2, and want to transfer the information in q0 to you. I perform a joint measurment on both qubits that I have (q0 and q1). When I tell you the result of this measurement, then you can perform a quantum operation on q2 to recover the information in q0. The two entangled qubits are like a quantum information bus. But we still need classical information to determine how to recover the information.

What we have acheived in the lab is a means of entangling these two qubits without needing them to be right next to each other. In my description above, I initially put q1 and q2 into an entangled state, and gave you one of them. Now, you could have your own qubit and live in Michigan, and I have my own in Maryland. We can then ‘open up’ the quantum channel by entangling the two qubits, allowing transfer of quantum information.

How is this done physically? First, we have a single ytterbium ion in an ion trap, and excite it to an excited state. The ion has two decay channels. If it decays to the qubit state |0>, it emits a photon of one color (say ‘blue’). If it decays to qubit state |1>, it emits a photon of a different color (’red’). In this method, we have generated a single photon entangled with a single ion. So, we have both ions do this, giving us a photon entangled with the first ion (q1), and a photon entangled with the other ion (q2).

We then interfere these photons on a beamsplitter. If a photon hits a beamsplitter, it can go one of two ways. We have detectors at each output port to detect the photon. Now, if the two photons interfere on the beamsplitter, we will only coincidence detection if the photons are of a different color. So we detect a ‘red’ photon and a ‘blue’ photon. But we don’t know which ion emitted which photon.

What we do know is that one ion is in |0> and the other is in |1>. That’s the key. These two ions are now entangled. And the quantum communication channel is open.

Random Quantum Optics Question

Consdier a coherent photon state |alpha> interacting with a two-level system.  Now, in the large photon limit, we ought to be able to recover the semi-classical interaction: i.e. a classical EM field interacting with a two-level system.  Conceptually, this should be the case.  Has anyone shown this?  From what I can tell, the interaction with a coherent state gives rise to Rabi flopping that decays, and has revivals.

Light traveling faster than light

There has been a small splash in the past couple days about a couple Germans who have purportedly violated special relativity (see here, and here). So what’s going on here? Is this another superluminal propagation of light in a region with strong anomalous dispersion? For starters, here’s another article with a bit more detail, as well as a comment from Aephraim Steinberg (a quantum opticist at University of Toronto). Additionally, here’s the paper on the ArXiv: arXiv:0708.0681v1.

Well, what they’ve done is worked with Frustrated Total Internal Reflection (FTIR). First of all, total internal reflection is when all the light is reflected at the interface. This only happens when moving from a denser optical material to a more rarefied material. Past a certain angle, all the light is reflected. You’ve seen this if you look up while underwater. Typically, you can see the sky above you, only in a small disc. Outside that, all you see is the water. This is the same principle by which fiber optics works—the light is confined in a small glass rod.

Now, here’s a slightly weird part. Even though all the light is reflected, there is a small portion of it that hangs out at the interface, and extends into the rarer material. This portion is called the evanescent wave. Now, it turns out if another dense material is placed close enough to this evanescent wave, the light can actually be transmitted to the second material. This is very similar to Quantum Tunneling, where a particle can be transmitted through a potential barrier. This process is called frustrated total internal reflection.

Back to the paper and experiment. They conducted a FTIR experiment with microwaves, and measured the arrival times of both the reflected and transmitted signals. They claim that the signals arrived at the same time. This means that since the transmitted signal travels a longer distance, it must be moving faster than the speed of light. The paper is slightly dubious, as there is no data present in the paper.

The crux of the matter is the tunneling time. Now, the notion of time is slightly problematic in quantum mechanics—there is no ‘time’ operator to give us the arrival time of particles. In fact there has been much research on this. If I had decided I wanted to be a theorist, I probably would have been working with Herbert Winful on the theory of this.

The two questions are: Is there superluminal propagation in tunneling? How do we interpret what is going on?

Now we’re getting somewhere interesting. In their paper, they have an interesting take on this. They identify the evanescent wave with virtual photons:

All three properties - the violation of the Einstein energy
relation, the zero time spreading, and the non observability of
evanescent modes - can be explained by identifying evanescent
modes with virtual photons as predicted by several authors, see
for instance references 6, 7, 8, 9, 10. The corresponding Feynman
diagram is sketched in Fig. 2. Tunneling and evanescent modes
are properly described by quantum mechanics.

That is, a photon that doesn’t really exist, but can have an effect on the system. Virtual particles are abound in particle physics, where we typically detect the decay products of a virtual particle. Does this interpretation hold water? Perhaps. The theorist in me wishes for there to be a thorough, rigorous QED calculation showing this.

Steinberg offers a different interpretation:

Steinberg explains Nimtz and Stahlhofen’s observations by way of analogy with a 20-car bullet train departing Chicago for New York. The stopwatch starts when the centre of the train leaves the station, but the train leaves cars behind at each stop. So when the train arrives in New York, now comprising only two cars, its centre has moved ahead, although the train itself hasn’t exceeded its reported speed.

That is, given a pulse of light, the back end of it is eaten up, giving the impression that the center of the pulse has moved faster. I’m not sure if this holds water in this particular case. Perhaps when dealing with superluminal propagation in an region of anomalous dispersion, sure. But for evanescent waves, I don’t know.

Winful has another explanation of this effect (a long paper on this is available in the New Journal of Physics). From PRL 90, 023901 (2003), the final paragraph is:

In conclusion, we have shown that the apparent superluminal
tunneling of pulses is a quasistatic phenomenon
in which the output envelope adiabatically follows the
input. The incident peak does not actually propagate to
the exit which means that the notion of a transit time is
meaningless. The input field merely modulates the amplitude
of a standing wave created through the interference
between forward and backward waves. When properly
interpreted in this context, no superluminal transport is
seen

So what does this mean? It’s sort of like saying that the input merely increases and decreases the level of an existing wave inside the barrier. The pulse that comes out is not the pulse that goes in.

Suffice to say, I don’t think that this issue will be resolved any time in the near future.

In case you were wondering…

I have an article in latest issue of the IEEE Spectrum, explaining what we do. The IEEE Spectrum is the magazine for the Institute of Electrical and Electronics Engineers. This obviously justifies my work as an EE in a physics lab. Check it out, I hope it is readable for everyone. Thoughts? (Looking at Chris and Mike in particular)

I should do this more often.

Monday I was at Ashley’s, celebrating the end of classes. I have finished all my coursework, so I will now be spending, well, pretty much the same amount of time in the lab (since I hardly went to class). Now that I don’t have classes, I can spend free time studying what I want. Over several beers, I read/skimmed most of Atomic Physics by Christopher Foot. As well as rambling in my journal about what a quantum state physically means.

I found that I have about 6 pages of random notes ranging from what it is that we actually measure, whether entangled states and mixed states corresponds to some actual reality outside of their mathematical construct, a touch on the Copenhagen interpretation, addition of angular momenta (two spin 1/2 systems), and the EPR paradox.

I need to go through these notes, and write something coherent about them.

My last problem set

I’ve been cooped up this weekend, working on the last problem set that I’ll ever have to do (for a grade).  So, that’s good.  Unfortunately, it’s in particle physics.   That’s bad.  But I think particle physics is interesting, and an elegant theory.  That’s good.  But I don’t understand a lick of it.  That’s bad.

Can I go now?

Algorithms! I need algorithms!

Often, I am asked something along the lines of: “Why Quantum Computing? What does it offer that is better than regular computers?” To be perfectly honest, I don’t know. For me, I just think that it is really cool to be able to control and engineer the quantum states of individual ions. And working with lasers. But as an engineer, I feel like I need to offer some sort of practical and useful application of Quantum Computing (or Quantum State Engineering).

Are there certain problems that are nearly impossible on a regular computer, but are a snap on a quantum computer? This implies some sort of complexity of the problem. But since I’m no computer scientist, I don’t know anything about P vs. NP & complexity classes. But I do know this: a quantum computer won’t be any faster than a regular computer, unless it is running a quantum algorithm (an algorithm that exploits some aspect of quantum mechanics). Because, otherwise, it could be done on a regular classical computer. And to be honest, with all the infrastructure we need (electronics, lasers, vacuum chambers, etc.), it’d be faster to just use a regular computer.

So, what quantum algorithms are there? Again, I’m no computer scientist, and only know of three: the Deutsch-Jozsa algorithm, Grover’s search algorithm, and Shor’s factoring algorithm. Wikipedia gives a good description of the Deutsch-Jozsa algorithm:

In the Deutsch-Jozsa problem, we are given a black box quantum computer that implements a single function f(x1, x2, …, xn) that takes n binary bits x1, x2, …, xn and returns the binary value f(x1, x2, …, xn). We know that the function is either constant (0 on all inputs or 1 on all inputs) or balanced (returns 1 for half of the input domain and 0 for the other half); the task then is to determine which it is (constant or balanced) by applying inputs to the black box and observing its output.

Grover’s search Algorithm searches an unsorted database. Suppose you find just a phone number in your pocket, and you can’t remember if it is the number of the hot girl you met last night, or the the ugly one. To save face, you decide not to call to find out. Rather, you want to look them up in the phone book. Unfortunately, the phone book is ordered by name, not number. So how would you search? If you had a quantum computer, Grover’s algorithm could help you.

Shor’s factoring algorithm is just that. It factors a number into its primes. Why is this interesting? The difficulty of factoring has it’s uses in encryption. An example of the use of this is in RSA Encryption. Of course, this has spy stuff written all over it—breaking codes, secret messages. It also offers the best speed-up, and uses the Quantum Fourier Transform. And I’m quite partial to any sort of Fourier Transform.

So, obviously, I’ll mention Shor’s algorithm.

Perhaps it is time that I learn some Computer Science & some other quantum algorithms. If you’re interested in how Shor’s factoring algorithms works, I highly recommend this post by Scott Aaronson. Incedentally, he’s got another post about the recent breakthrough in quantum algorithms: the NAND tree.

Under the radar

D-Wave supposedly has unveiled it’s 16-qubit Quantum Computer today, according to their blog. What does this all mean? Well, if it truly is a Quantum Computer, then good for them. I’m quite skeptical, mainly because this thing would have gone completely under our radar. Their method is using superconducting Quantum Interference Devices (aka “SQUIDs”), which are large solid-state devices. One reason ions hold so much promis in this field is that their qubits can have long coherence times (i.e. it keeps a |0> or |1> longer), because we choose atomic states that do not interact strongly with the environment. Solid-state implementations of Quantum Computing generally have short coherence times. So, given that I’ve not seen many papers demonstrating rapid progress in solid-state quantum computation, I think I’m right in being rather skeptical about this announcement. My advisor Chris is skeptical too.

I don’t know anything about adiabatic quantum computing, which is what this system supposedly is. From what I can gather, it looks like they’re just simulating (solving?) a 2-D ising model in a magnetic field. Correct me if I’m wrong, but isn’t there an analytic solution?

I have this sneaky suspicion that what we have is a 2-D array of qubits—a qubit consisting of a large domain of a single state—interacting via tunneling from one domain to another. I don’t think that we have actual entanglement, nor coherence. This tunneling interaction can be written as a 2-D Ising model. Hence, all I can see this ‘quantum computer’ doing is simulating the 2-D ising model.

Note: The Quantum Pontiff, Dave Bacon, had a roundup of this. We’ll see what goes on with this. If this is bunk, hopefully it won’t be like Cold Fusion, which discredits the whole field.