I just thought I’d try to get in there to be the first to denounce LP for being an influential blog.

Damn you LP, you tall poppy you!

/sarcasm off

Below is a YouTube video of a speech that Obama gave at his campaign headquarters after winning the Democratic domination. It’s one of the best speeches I’ve seen in a long time. Note, there are no autocues and he isn’t simply reciting his stump speech (although he is clearly riffing off parts of it). It takes a while to get going but from about half-way through it really picks up.

Well, it’s been a really long while since I posted. I guess a lot of that has to do with travel. I don’t know what it’s like for others, but doing the whole travel thing throws my rhythm totally off. This is both a blessing, especially when it comes to cranking out the theories, and a curse, as in the case of regular blogage goodness.

Anyway, seeing as travel is on my mind but mostly because it’s 5:30 AM and my girlfriend is stuck at the airport due to a canceled flight I thought I’d point people to a VERY USEFUL little EU regulation, EC 261/2004 (wiki version). Basically, EC 261/2004 is a kick-ass little regulation that was designed to discourage airlines from screwing over passengers. Some of the handy things in this regulation include:

• If your flight is cancelled for a reason that has been caused by the airline then they owe you a flight.
• On top of that, if they haven’t given you enough notice (14 days) or your travel plans get screwed about they owe you €. What’s more the amount they must pay you, in some circumstances, be much more than the cost of the ticket.
• This holds for all carriers operating in the EU. That is, even if the airline is some bodgy budget thing they still have to abide by the regulation.
• The airline is supposed to offer the compensation to you and make you aware of your rights.
• They can’t opt out by offering you flight vouchers etc. If they owe you, they owe you money in € as this is compensation and it is not a gift from the airline (as some airlines will try to tell you).
• Oh, and if you are delayed by more than a few hours they owe you food and accommodation on top of the € and the flight that you’ve already paid for.

Essentially, if you are traveling in an EU country and your flight is unexpectedly delayed (by more than a couple of hours), cancelled, or overbooked because of something within the airline’s control then you should always enquire about compensation under EC 261/2004!

Geez, sorry everyone I really dropped the ball on the Weakly Abstract this week. Frankly with Easter and everything I just didn’t get the time to crank out a post.

I should mention that there probably won’t be a Weakly Abstract next Monday either as I’ll be on the road. Hopefully I’ll manage to squeeze in a post later in the week as there have been some very good papers in the last couple of weeks.

Anyway, from today onwards I’ll be on the road for about the next 6 weeks!

From Monday next week I’ll be visiting the Quantum Information and Technology group at the National University of Singapore for a couple of weeks. After that I’m going back home to Brisbane for a week, then I’ll be in Sydney for a couple of weeks visting the Quantum Information Science and Security group at Macquarie Uni. Finally I’ll be heading to Brissie for another week before my journey back to Brizzle.

Now, back to getting my packing finished…

As I mentioned in the Weakly Abstract post yesterday I’m currently attending the Quantum Information Processing Spring School here in Bristol.

One of the cool things about this meeting has been the venue. See that photo in the header of my blog? Well, the school is being hosted at the Burwall’s Centre for Continuing Education which is on the left hand side of the suspension bridge in the photo. The Burwall’s Centre is basically an old mansion overlooking the Avon Gorge which Bristol University has converted into a conference centre.

The view from up here is pretty spectacular especially today as Spring has finally decided to happen. Unfortunately I’ve left my camera at home but hopefully I can harass one of the students here to take a photo today so I can post it later…

Sorry everyone, I dropped the ball and didn’t manage to get out the Weakly Abstract on time this week. My excuse is kinda lame, yesterday I was giving a talk at the Quantum Information Processing Spring School and with one thing and another I didn’t get time to crank out a blog post. What’s doubly lame is that I’m still at the Spring School so this post is gonna be a quick-un.

There’s a whole host of papers that caught my eye in the last week and unfortunately they can’t all be the Weakly Abstract. So, in the finest tradition of the interwebs I’m gonna simply choose a paper by a friend of mine in order to give it a plug. Well, that and this paper fits into the whole Spring Schoolish vibe that I’ve got going on. This week’s Weakly Abstract is Optical Quantum Computing by Jeremy O’Brien:

Optical Quantum Computing
Jeremy L. O’Brien

In 2001 all-optical quantum computing became feasible with the discovery that scalable quantum computing is possible using only single photon sources, linear optical elements, and single photon detectors. Although it was in principle scalable, the massive resource overhead made the scheme practically daunting. However, several simplifications were followed by proof-of-principle demonstrations, and recent approaches based on cluster states or error encoding have dramatically reduced this worrying resource overhead, making an all-optical architecture a serious contender for the ultimate goal of a large-scale quantum computer. Key challenges will be the realization of high-efficiency sources of indistinguishable single photons, low-loss, scalable optical circuits, high efficiency single photon detectors, and low-loss interfacing of these components.

I gather that this is the arXivish version of a review paper that Jeremy published in Science late last year. I don’t have time to get into the details of the paper, but basically it’s a bit of a snapshot of where the field of optics based quantum computation is at right now. It’s easy to read so, um, get to it I guess.

I’d just like to go on the record and say that I’m very pro this whole quick review paper thing and I wish that people would do it more. So, people, get out there and start summarizing stuff!!

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&quot Doherty and Steve ("Old Man&quot Bartlett this week’s Weakly Abstract:

Identifying phases of matter that are universal for quantum computation
Andrew C. Doherty, Stephen D. Bartlett

A 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:

1. Austin G. Fowler, Ashley M. Stephens, Peter Groszkowski, High threshold universal quantum computation on the surface code. (7 scites)
2. B. Dierckx, M. Fannes, C. Vandenplas, Additivity of the 2-Renyi entropy for PPT inducing channels. (6 scites)
3. Grigori G. Amosov, Stefano Mancini, The decreasing property of relative entropy and the strong superadditivity of quantum channels. (5 scites)
4. 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.

For years now I’ve been a big fan of Terry Pratchett’s Discworld series. A year or so ago I discovered a fantastic book that Terry Pratchett co-wrote with another author, Neil Gaiman. That book was "Good Omens" and was one of the best books I’d read in years.

I guess "Good Omens" was a bit of a gateway book that gave me a bit of a taste for Gaiman but somehow softened by Pratchett’s familiar themes. A few months ago I picked up Gaiman’s "American Gods" and, well, now I’m hooked. To push the lame drug metaphoralizing even further it seems that Gaiman and his peeps know the value of a free sample and have made "American Gods" available for free online till the end of the month (I think).

I’d also like to give a plug to Gaiman’s blog which he updates pretty much daily with stuff that is actually interesting to read!

Checking out SciRate there’s been a fair bit of good quant-ph action over the last week. For instance you might want to check out the following papers:

1. Guillaume Aubrun, A remark on the paper “Randomizing quantum states: Constructions and applications”. (10 scites)
2. Huw Price, Toy Models for Retrocausality. (7 scites)
3. Keisuke Fujii, Katsuji Yamamoto, Fault-tolerant quantum computation in concatenation of verified cluster states. (5 scites)
4. Shesha Raghunathan, Todd Brun, Continuous monitoring can improve indistinguishability of a single-photon source. (5 scites)
5. Paulo E. M. F. Mendonca, Alexei Gilchrist, Andrew C. Doherty, Optimal tracking for single qubit states. (5 scites)

Unfortunately the world is cruel and the law of the land clearly states that there can only ever be one Weakly Abstract (that is until I decide to do things differently of course). The paper that caught my eye this week was The computational difficulty of finding MPS ground states by Schuch, Cirac, and Verstraete:

The computational difficulty of finding MPS ground states
Authors: Norbert Schuch, Ignacio Cirac, Frank Verstraete

We determine the computational difficulty of finding ground states of one-dimensional (1D) Hamiltonians which are known to be Matrix Product States (MPS). Therefore, we construct a class of 1D frustration free Hamiltonians with unique MPS ground states and a polynomial gap above, for which finding the ground state is at least as hard as factoring. By lifting the requirement of a unique ground state, we obtain a class for which finding the ground state solves an NP-complete problem. Therefore, for these Hamiltonians it is not even possible to certify that the ground state has been found. Our results thus imply that in order to prove convergence of variational methods over MPS, as the Density Matrix Renormalization Group, one has to put more requirements than just MPS ground states and a polynomial spectral gap.

Where to begin, where to begin. Well, once upone a time there was a classical algorithm called the Density Matrix Renormalization Group (DMRG). Now DMRG is an algorithm that varies over Matrix Product States (MPS) in order to determine which MPS describe the low energy structure of some Hamiltonian. Typically, DMRG works very well in 1D systems. When I say "works well" I mean that the DMRG algorithm converges on a MPS that is a good approximation to the ground state of the Hamiltonian. Indeed, Hastings proved that if a 1D Hamiltonian has a local structure and is "gapped" then it’s ground state can always be approximated by an MPS.

The key question is how hard is it to find the right MPS to describe the ground state of a given Hamiltonian. Jens Eisert demonstrated in 2006 that performing DMRG on 1D frustrated systems while trying to optimize multiple local sites at once is an NP hard problem. Eisert pointed out that if one optimizes DMRG locally then it is possible to get stuck in local minima, whereas if a more global optimization is performed the problem becomes computationally intractable. Eisert’s results suggest that it can be difficult to make DMRG converge to the correct answer even in somewhat simple systems. Unfortunately his approach doesn’t say much about the typical behaviour of DMRG as it depended on arguments about specific systems (those which are frustrated) and also it is restricted to particular optimization methods.

In this week’s Weakly Abstract paper Schuch, Cirac, and Verstraete have taken a more general approach to this problem. In particular they have leveraged recent results of Kitaev and Aharonov et al on the QMA completeness of the LOCAL HAMILTONIAN problem to the problem of finding MPS descriptions of the ground states of 1D systems.

Importantly, they use techniques from the study of the LOCAL HAMILTONIAN problem to construct classes of Hamiltonians with the following properties:

1. There is a class of 1D local frustration-free Hamiltonians of length L, a gap of order 1/poly(L), with a unique MPS of size poly(L) describing the ground state and MPS describing the low-energy states where finding the MPS corresponding to the ground state is in NPcoNP. Notably, this class of problems includes factoring which is thought to be hard.
2. There is a class of 1D local Hamiltonians of length L, a gap of order 1/poly(L), the ground state is an eigenstate of every local term and where the ground state manifold is spanned by MPS for which finding the ground state is NP-complete. (Note that finding the ground state of some classical 2D systems has long been known to be NP-complete)

They argue (quite well!) that existence of the above two classes demonstrates that it is very unlikely that one can find a certifiable version of DMRG. That is, a version of DMRG where one can efficiently check whether a particular solution is indeed the correct solution.

It should be emphasized that there are lots of classes of Hamiltonians for which DMRG will converge to the correct answer, as is often the case with these sort of NPish results. It’s just that we can’t say really which ones converge and we can’t write an algorithm that can generally check this quickly for 1D systems.

One of the more interesting aspects of this paper, which I’d like to think a lot more about, is how they used the techniques developed for QMA-completeness proofs of the LOCAL HAMILTONIAN problem of Kitaev and co to find NP-hard ground states. They suggest in the paper that there may be other classes of Hamiltonians with more NP trickery happening in their low-energy eigenstates waiting to be found by using these techniques…

Another Monday, another Weakly Abstract. This week’s Weakly Abstract is "Creation, manipulation, and detection of anyons in optical lattices" by Aguado et al:

Creation, manipulation, and detection of anyons in optical lattices
Authors: M. Aguado, G. K. Brennen, F. Verstraete, J. I. Cirac

Anyons are particle-like excitations of strongly correlated phases of matter with very exotic physical properties. Unlike bosons and fermions, they have so-called fractional statistics, which are characterized by non-trivial changes in the quantum wavefunction when two of them interchange their positions. Those changes can, in turn, be used to perform quantum computations [A. Yu. Kitaev, Annals Phys. 303, 2 (2003), arXiv:quant-ph/9707021v1], something which has renewed the interest in the investigation of physical systems where anyons may be present, manipulated and detected. In this work we show how this can be accomplished in the context of optical lattices. Our proposal just requires one (or several) ancilla particle(s) which can: (i) undergo single particle gates; (ii) be moved close to each constituents of the lattice and undergo a simple quantum gate; (iii) be detected. Recent experimental progress with atoms in optical lattices makes our proposal feasible with present technology.

Aguado et al propose a method for generating and controlling anyonic excitations in optical lattice systems via coupling atoms in the lattice to well-controlled ancilla particles. The amount of stuff that I don’t know about topological quantum computing could fill a few textbooks, so if anyone would like to elaborate further on the results of this paper they should feel free to do so in the comments section.

I think that Gavin Brennen (the second author) talked about these results at QEC07 but I can’t verify this as his talk doesn’t seem to be available on the QEC website (btw, there are some fantastic talks available there). Does anyone else remember Gavin’s talk?