This week’s Weakly Abstract is going to be highly controversial. You see, over the last month or so that I’ve been doing this I’ve followed a pretty tried-and-true pattern of picking either the most, or second most, scited paper on SciRate from any given week.

This week I’m going to dip waaayyyy down into the list (I think it’s currently the 4th most scited this week) to declare "Identifying phases of matter that are universal for quantum computation" by Andew ("Google Mouth") Doherty and Steve ("Old Man") Bartlett this week’s Weakly Abstract:

Identifying phases of matter that are universal for quantum computation

Andrew C. Doherty, Stephen D. BartlettA recent breakthrough in quantum computing has been the realization that quantum computation can proceed solely through single-qubit measurements on an appropriate quantum state – for example, the ground state of an interacting many-body system. It would be unfortunate, however, if the usefulness of a ground state for quantum computation was critically dependent on the details of the system’s Hamiltonian; a much more powerful result would be the existence of a robust ordered phase which is characterized by the ability to perform measurement-based quantum computation (MBQC). To identify such phases, we propose to use nonlocal correlation functions that quantify the fidelity of quantum gates performed between distant qubits. We investigate a simple spin-lattice system based on the cluster-state model for MBQC, and demonstrate that it possesses a zero temperature phase transition between a disordered phase and an ordered "cluster phase" in which it is possible to perform a universal set of quantum gates.

Now why is it that I "hate freedom" so much that I ignore the will of the intertubes and choose a paper with a measly 4 scites over papers like these:

- Austin G. Fowler, Ashley M. Stephens, Peter Groszkowski, High threshold universal quantum computation on the surface code. (7 scites)
- B. Dierckx, M. Fannes, C. Vandenplas, Additivity of the 2-Renyi entropy for PPT inducing channels. (6 scites)
- Grigori G. Amosov, Stefano Mancini, The decreasing property of relative entropy and the strong superadditivity of quantum channels. (5 scites)
- M. Cramer, A. Serafini, J. Eisert, Locality of dynamics in general harmonic quantum systems.

Is it because I support Barrack Obama and think that Ron Paul and is a loon?

No.

Is it because I’m an old drinking buddy of both of the authors?

No.

Is it because I’m just trying to shake things up a bit?

Oddly, no.

It’s because I’ve worked on the problem that Doherty and Bartlett are trying to solve and it is a thoroughly hard problem! You see, there’s this very odd little fact in our field that despite the measurement-based model of quantum computing being roughly 6 (or 7?) years old this year we don’t really have our heads around the problem of *which families of states are universal for measurement-based quantum computing*.

It sounds like it shouldn’t be that hard a problem, at least before thinking about it for about 30 seconds. In a more Hamiltonian controlish view of quantum computing we have a pretty good idea about which Hamiltonian evolutions, together with local control, are universal for quantum computation. Even when we don’t have complete local control we know how to map the problem to the theory of Lie Algebras in order to solve it (modulo some extra conditions).

I guess it could be argued that we also don’t understand which sets of unitary operations are universal for quantum computation either. That’s kinda true, it’s just that in Nature we are rarely given some fixed set unitary operations without any sort of control over the amount of time for which they are applied. So while it’s an interesting mathematical problem which I would dearly like to solve, it doesn’t really have a whole lot of connection to physics.

Where was I? Oh right, measurement-based quantum computing (MBQC) and identifying states which are universal for quantum computation. So, as it stands we know that 2D cluster states are universal for quantum computing, and we also know that all regular lattices graph states are universal for measurement-based quantum computing because there is a nice construction that shows which measurements we need to perform to turn a regular lattice state into a cluster-state.

We also know a fair bit about how the multiparty entanglement in a family of pure states should scale with the number of qubits in the state in order for that family to be used for measurement-based quantum computing (and given that we can’t change what our logical qubits are).

Unfortunately, it seems that when we are given a shiny new family of states with all the right entanglement characteristics we don’t have a good way of identifying *how to make logical gates via measurement and local unitary operations*. Gross and Eisert had a really good idea (link is to a longer paper with Schuch and Perez-Garcia as co-authors) a few years back when the realized that you can use a state’s description in terms of Matrix Product States (geez, those things keep cropping up don’t they?) to work out how to implement quantum gates. The problem with that is actually getting a description of your state in terms of MPS which is a pretty non-trivial thing to do.

Doherty and Bartlett have in this week’s Weakly Abstract demonstrated a whole class of states that can be used for measurement-based quantum computing while still using the same measurements that allow for MBQC on cluster states. The way that they do this is kinda far from obvious. Basically, they define a Hamiltonian that they dub the *transverse-field cluster model*, which is basically the Hamiltonian that has the cluster-state as its unique ground state with a transverse field thrown in. They demonstrate that as the transverse field strength is varied the ground state the system transitions from something that is universal for MBQC to something that isn’t. In short, they demonstrate that such a Hamiltonian undergoes a quantum phase transition between being universal for quantum computation and being something else.

Now, they don’t do this by actually finding out what the ground state is but rather they show that by studying the properties of certain long-range correlation functions you can work out the gate fidelity of a gate teleportation through the system. Essentially they are saying, among other things, that it is how these gate fidelities scale that determines your ability to do quantum computation. At least that’s my take on it anyway.

I should point out that I haven’t even begun to scratch the surface of the issues raised in this paper. I don’t fully understand a few parts of the paper and a lot of my confusion is a result of my lack of expertise in condensed matter physics. Yet it seems to me that this paper contains an abundance of compelling techniques. For instance, the way they determine which correlation functions are important for gate teleportation and the mapping between the transverse-field cluster model and the anisotropic quantum orbital compass model are both really interesting things to think about.

I guess you didn’t like the conclusion either.

Nah, I just really liked Steve and Andrew’s paper. I’ve worked on a similar problem and it gave me the chance to rant about it… As a blogger I never pass up the opportunity for a good rant.