# QM/QC: Frustrated bosons on a two-leg ladder : a chat with Simon Trebst

Simon Trebst is a computational physicist working at Microsoft Station Q , a Microsoft research group at the University of California in Santa Barbara working on topological quantum computing.

Topological quantum computing is a theoretical approach to the problem of error correction in quantum computing. If we can't always keep transistors honest, each of whose 1/0 value is formed of tens of millions of electrons, how are we going to guarantee accuracy when a qubit might be a small agglomeration on the atomic scale?

TQC's concept is to make qubits of condensed matter whose quantum state can be easily read from quasiparticles excited above the ground state yet whose quantum state is partly encoded in the ground state which cannot be so easily altered. Readers may wish to read a previous blog posting Topological quantum computation : a chat with Michael Hermele for a background on the topic of topological quantum computing.

Our title today was taken from a 2008 paper Trebst co-authored, Strong-coupling phases of frustrated bosons on a two-leg ladder with ring exchange, which conjures up the image of a shotgun wedding between two leptons caught in a compromising position and rivals The Effect of Gamma Rays on Man-in-the-Moon Marigolds as a wonderful title for a play.

I spoke by phone with Trebst at his office in Santa Barbara.

JW: I know it's not immediately relevant to topological quantum computing, but you must tell me what *Strong-coupling phases of frustrated bosons on a two-leg ladder with ring exchange* is about!

ST: It's a quantum mechanical effect. It's work that we have done to understand another collective phenomenon in two-dimensional systems. These things can occur in heterostructures. They can also appear in the solid state of a crystal, where two layers are not adjacent to each other, there are other atoms in between, so a reduction to two spatial dimensions can occur. Copper oxide superconductors are an example where such layers are separated enough you can think of the layer as primarily two-dimensional. There's always some interaction between layers, but it's weaker than the interaction in the plane.

JW: What is Station Q doing and what is your work?

ST: Station Q is a research group composed of persons of different scientific backgrounds, headed by a mathematician, Michael Freedman, a topologist. Beside those with a background in math, there are also two more topologists, amd there are those with backgrounds in physics, especially condensed matter physics. We are working on topological quantum computing, characterizing and understanding topological states of matter with an eye to their use in storing and manipulating quantum information.

JW: I recently spoke with Prof. Michael Hermele who explained non-Abelian statistics to me. How does Station Q's work relate to his research?

ST: What makes topological quantum information distinct from other approaches is that it is stored in a system with many degrees of freedom. A conventional qubit could be stored as a spin or in an ion with one degree of freedom. Now we are moving to a many-body system. One example would be fractional quantum Hall liquid, a liquid of electrons confined to two dimensions at really low temperature and in a strong magnetic field. These conditions can allow the formation of a different state of matter with all these properties we are looking for.

JW: Including endurance so it doesn't change as easily.

ST: Perturbing it locally in a single degree of freedom doesn't disturb the overall topological state. You're much more shielded from the enviroment with decoherence protection as oppposed trying to shield a single spin in a more conventional approach.

JW: Prof. Hermele said that experimentally proving non-Abelian statistics may await laboratory confirmation for a decade.

ST: There are two things one should distinguish: One thing is that topological states of matter have established experimentally through the observation of the fractional quantum Hall states. This work, done around 1982, garnered the Nobel prize in physics for 1998. There are many different quantum Hall liquids at different filling fractions. Most are Abelian in their topological characteristics. The second class, non-Abelian, still needs to be characterized experimentally. There is very strong evidence established experimentally for these different fractional states. But it's still difficult to prove that these quantum Hall liquids have the non-Abelian properties they are thought to possess.

JW: Typically, an American corporation doing research expects a payoff in 5 years or less. Is Microsoft's Station Q that sort of effort?

ST: What you will *not* see in 5 years is a working quantum computer you can buy in the store. It's a much longer-term strategy of exploring a unique approach to quantum computing. This idea was originally spearheaded by two people, Alexei Kitaev, Caltech and Michael Freedman**. **Michael has been with Microsoft a long time, probably 1997. In this time he developed these ideas for topological quantum computing further and further, eventually leading to the establishment of Station Q in 2005 in Santa Barbara.

It's a fundamental research endeavor with visionary mathematicians and physicists working on fundamentals of topological quantum computing . It is interesting from scientific perspective in that it is aiming at characterizing a particular quantum state of matter. From an engineering perspective, it might open the route to a very stable quantum computer.

JW: I'm looking for hardware on my desk!

ST: You'll need to be patient. Hardware on the desk, no. Hardware in some labs, yes. In the next 5 years we hope for the quantum equivalent of a transistor in the lab. That would be an essential breakthrough.

JW: What about the borderline between quantum computing and conventional using Josephson junctions ?

ST: There are various thoughts about how you can use such a device in building a quantum computer. There is the thought that you can use a bunch of Josephson junctions to constitute a topological quantum computer. It's a different approach from quantum Hall liquids, but similar in that you use it to preserve quantum information through topological protection. There are other ways to use Josephson junctions. You can also use a quantum mechanical system to, for example, generate random numbers, but that's a very different stage to play on, a merger of classical computing and quantum computing.

JW: So what are *you* actually doing?

ST: My personal research is this:

Assuming you have non-Abelian statistics, you have these quasiparticles, anyons , and the idea of performing computation is that you manipulate quasiparticles in a topological liquid. Each anyon helps you store information. The way that you manipulate them is that you move them around each other, braiding them. However, if you have such a set of multiple anyons, they still see the presence of each other if you push them close together and they start to interact.

There's an exponentional number of states that you can generate. More and more anyons produce states exponential to the number of anyons. It is in these states that you produce your quantum computations. In comparison, when we consider "spins", then that allows really two different states. Now, if you take anyons, there are different quantum numbers similar to spin that quantify which quantum state these anyons are in, and if you take more of these anyons, the number of states also grows exponentionally. The subtlety, however, is that the expansion of the number of quantum states is not based on an integer, but rather on a number larger than 1 and less than 2, and that's what being exponentiated.

JW: What is that number? Is it like natural log *e*?

ST: The "quantum dimension", e.g. the characteristic number with which the quantum mechanical state space grows for non-Abelian theories that can be labelled by an integer k, where k goes between 2 for the simplest example and k infinitely large which corresponds to conventional quantum mechanical spin-1/2's, can be calculated as

d = 2*cos ( pi / (k+2) )

Here are some examples. If k=2, and this is the case which is relevant to the most promising candidate fractional quantum Hall liquids, then d = sqrt(2) = 1.4142.... which implies that a system with 2 non-Abelian anyons (which in this case are called Ising anyons) can form 2 distinct states, with 4 non-Abelian anyons 4 states, and with 6 non-Abelian anyons 8 states and so on...

The next case would be an anyonic theory with k=3, where the quantum dimension becomes d = (sqrt(5)+1)/2 = 1.6180... which is the golden mean . Because for these anyons the number of states (asymptotically) grows like the Fibonacci numbers (which are well-known to grow as an exponential of the golden mean), these anyons are also called "Fibonacci anyons".

If k becomes an infinitely large number, then the expressions to "2" which is effectively a spin-1/2 particle.

We have a variety of candidate systems wherein we have the simplest theory, that this quantum dimension is sqrt(2). There are many fewer proposals for candidate systems where that number is the golden mean. As k is increased, the systems become more rare, except k=infinity which brings us back to spin-1/2 particles.

So now we have these degenerate states that are possible with these anyons. One might think that if we take a topological liquid and we excite all these anyons, these different non-Abelian states are all equivalent, that is, all the states have the same energy. What we are asking is, what are the effects of the interactions between these anyons? The answer is that these interactions start to split the energy of the various non-Abelian states. It is then interesting to ask which one becomes the lowest? That is the one we call the ground state which is the new collective state of all the anyons. I am working on how to characterize and identify this lowest energy state.

In comparison, if in classical computing you had a bunch of transistors with ones and zeros, any patterns are equally valid. With 4 classical bits, 1111 is something similar to 1010 ... I'm asking whether anyons representing these two states come with the same energy level. It turns out interactions can have the effect to make certain superpositions of patterns lower-energy patterns than others.

JW: What are the engineering implications?

ST: Number one, it's good to know what happens in the presence of such interactions. Two, given that this might happen, you might want try to avoid these interaction effects, that is, to make sure in your topological liquid that the anyons are well separted from one another at all times and make sure that in the course of a computation, which is the braiding of anyons, they don't come closer than a particular distance one can calculate. This way we could make sure that you don't have this splitting and change in energy states.

Another idea is to positively make use of the splitting effect. That's a whole other problem, mathematically, using these effects.

So I am attacking three problems, the physics of characterization of non-Abelian states, the engineering of avoidance of certain interaction effects, and the mathematics of potentially using this splitting for computation. It's a rough and tough area.

JW: Not for closeted specialists.

ST: It's a great opportunity for a reasearcher to work at the intersection of several fields.