This post is technical

Today I released a new paper on the arXiv with my co-author Sean Barrett. In this paper we look at the problem of preparing cluster states using non-deterministic enangling gates. We introduce a numerical approach for simulating preparation procedures and use this to estimate how many gate operations are required when preparing large cluster states. We examine a few different types of entangling gates, as well as different preparation ‘strategies’ (i.e. algorithms for constructing large clusters from single qubits). For different gates and strategies we consider how the required number of gate operations scales with the success probability of the gate.

You can read our paper here.

For those of you planning on attending QIP in Brisbane next year, DON’T FORGET TO REGISTER. The earlybird registration deadline is this Friday. Additionally, the conference is limited to 200 attendees and is filling up very quickly. So, register early to avoid disappointment. Don’t say I didn’t warn you.

I’ve been at the Australian Institute of Physics (AIP) Congress here in Brisbane all this week. Yesterday I gave a talk on “Error tolerance and tradeoffs in loss- and failure-tolerant quantum computing schemes”.

In this talk I discussed observations we previously made in quant-ph/0603130, where we noted that many recent loss-tolerant protocols for performing quantum computing in the presence of qubit loss and gate failure have the undesired side-effect that they magnify the effects of depolarizing noise.

You can download my talk here.

Just a reminder that Quantum Information Processing 2007 is being held here in sunny Brisbane from 30th January to 3rd February next year. The abstract submission deadline for contributed talks is November 4. See the conference site to register or submit abstracts.

This post is slightly technical.

I submitted a couple of new papers to the arXiv today (probably inaccessible until Monday), joint work with Christine Silberhorn and Wolfgang Mauerer that we have done together during my visit to the Max-Planck Institute here in Erlangen.

The first paper, “Spectral structure and decompositions of optical states, and their applications”, considers the description of optical states in the spectral degree of freedom, as opposed to the number degree of freedom, which is how such states are usually described. We look at general properties of such spectral characterizations, and discuss a decomposition technique that simplifies many types of calculations. Most notably, processes like spectral filtering, mode mismatch, and various interference effects can be very easily described using our techniques.

The second paper, “Quantum state tomography of single photons in the spectral degree of freedom”, describes an approach for performing Quantum State Tomography (QST) of single photons in the spectral degree of freedom. Normally QST of optical states is performed in the photon number degree of freedom, using techniques like Optical Homodyne Tomography (OHT). However, for many quantum information processing applications we employ single photons and we need to ensure that they have both a high degree of purity and indistinguishability. Given a single photon, tomography in the photon number degree of freedom cannot quantify this. Instead, we suggest characterizing the state in the spectral degree of freedom, in which case spectral purity and indistinguishability can be readily calculated.

Note: This article was originally posted on July 25, 2006, but is being reposted now that embedded LaTeX is working.

This post is on quantum mechanics, but is intended to be understandable by anyone. The small amount of mathematical notation that I introduce should be fairly digestible to anyone without a physics background.

A couple of nights ago I had a dinner table conversation with a fellow physicist, who raised the highly emotive, almost religious issue of the interpretation of quantum mechanics. For the non-physicists, quantum mechanics has some very unusual properties that we are not used to encountering in the world around us (see my previous post on quantum mechanics). The most well known of these unusual properties is the notion of a superposition, whereby something can assume multiple states simulatenously. A state is simply a property of a system. For example, the position of a particle is represented by a state. A superposition in this context means that the particle is in multiple positions simultaneously. Obviously this is very counterintuitive indeed. This, as well as numerous other counterintuitive properties of quantum mechanics give rise to the argument over the interpretation of quantum mechanics. That is, how do we reconcile such strange behavior with our inherent human desire for cartoon descriptions of everything.

Mathematical background (for non-mathematicians and non-physicists)

Let me now expand on this. I’ll do this by introducing the mathematical tools that physicists actually use, but in a way that can hopefully be understood to anyone without such a background.

Suppose we have a system, which we will label A. This could be anything – a particle, a cow, a person, a banana. Using quantum mechanical notation we will represent the state of a system as , where s represents the actual state, and the subscript A denotes the system we are referring to. For example, suppose we have a two-level cow, which can be in one of two states – it can be eating, or not eating. The mathematical descriptions of these two states are and . In quantum mechanics we denote a superposition using a ‘+’ symbol. So a superposition of two states a and b is just denoted .

This completes the quantum mechanical notations we use for representing the states of systems. Next we’ll turn our attention to the evolution (i.e. the dynamics) of a quantum system. In quantum mechanics there are precisely two (or possibly just one, as we will discuss later) types of dynamics. The first is so-called unitary evolution. The technical definition of this is irrelevant here. Just think of it as an operation that maps one state to another. For example the unitary evolution describing a two-level cow that is initially standing around and then decides to have dinner, is described by , where denotes the unitary evolution operation.

The second type of evolution that is possible is the so-called projective measurement. As the name suggests, this type of evolution takes place when we measure a system. As an example, suppose we start with a system in a superposition of two states, , and we measure it to establish whether the system is in state a or state b. When we perform this measurement we always get either one answer or the other. Our measurement device never says ‘you have a superposition of a and b‘. This is why we don’t observe quantum mechanical effects in the world around us. So the question now is, if we have a superposition, which measurement result do we get? The answer is that our result is random. Truly random, in the sense that it is impossible, even in principle, to predict it. So 50% of the time our measurement device will say ‘you have a‘, and the other 50% of the time it will say ‘you have b‘. After the measurement, the state of the system collapses (according to the standard interpretations of quantum mechanics) into either or . Notice that after a projective measurement, the system is no longer in a superposition. In other words, the strange quantum mechanical properties dissapear. To describe this mathematically we’ll introduce corresponding measurement operations. If we perform a measurement and get outcome x, we denote the measurement operation as . So, referring to the discussed example we have

It turns out that this kind of evolution cannot be described by the unitary evolutions I mentioned before.

The measurement problem

The previous discussion gives rise to an obvious question. What constitutes a measurement? We know from numerous experiments that if we have an isolated system of say, atoms interacting with one another, then the system evolves unitarily. It is only when we look at it that this projective evolution takes place. In general, isolated small systems evolve unitarily, but as soon as they interact with a macroscopic system they undergo projective dynamics. The distinction seems very arbitrary. What is it that actually causes this transition from one type of dynamics to another? This is the so-called ‘measurement problem’, and is one of central points of contention in the interpretation of quantum mechanics.

Many worlds

One interpretation of quantum mechanics, which resolves the measurement problem, is the so-called ‘many worlds’ interpretation, which I will now describe. Suppose we begin with two systems, an observer and an atom. We assume the atom is in a superposition of two states, 0 and 1. The observer could be you or I, and can assume one of three possible states: ‘doing nothing’, ‘I think I measured 0’ and ‘I think I measued 1’. We’ll assume the observer is initially in the ‘doing nothing’ state. So, the state of the entire system can be expressed

Expanding the brackets this is equal to

Now we are going to apply a unitary evolution that performs the following mapping

In other words, this evolution maps an idle observer to the state where he believes he has measured whatever the respective state of the atom is. So what happens when we apply this evolution to the initial state?

We now have a state which is a superposition of the atom being in state 0 and the observer thinking that the atom is in state 0, and the atom being in state 1 and the observer think the atom is in state 1. This is a so-called entangled state, a superposition where the states of distinct systems are correlated within the superposition. Entanglement is another one of the quirky features of quantum mechanics, which gives rise to all sorts of strange phenomena. But I won’t go into that now.

With a little bit of algebraic rearrangement, it can easily be seen that this state is equal to

In this expression we have a superposition of an observer who thinks he measured a 0 and 1 respectively, and the state of the atom is described by the respective projection. Note that we have not inserted projection operators explicitly at any stage, it arises naturally as a result of the entanglement between the observer and the system he is measuring. This is how the ‘many worlds’ interpretation of quantum mechanics arises. There is no such thing as a projective measurement. Instead, whenever we (an observer) become entangled with a system we enter a superposition with it. Note that from the view-point of a particular term in the superposition, projective measurement has taken place. Thus, according to this interpretation, the reason we observe projective dynamics in the real world is because we are simply one of the terms in the superposition.

This interpretation resolves several issues. First, there is no arbitrary scale at which interactions become projective. Instead it arises very naturally as a result of our entanglement with that which we are observing. Second, the laws of physics are once again completely deterministic, since these non-deterministic projections no longer exist. Unfortunately, this interpretation also raises a somewhat depressing and overwhelming possibility. That is, we are simply one term in an almost infinitely large superposition of parallel worlds. Every time you look at an atom and observe a 0, there’s another you in a parallel world that observed a 1. For some reason I find this quite distressing.

For the physics readers, let me draw your attention to the upcoming QIP’07 conference, which will be held from Jan. 30 – Feb. 3, 2007, in Brisbane, Australia. Details are at www.qipworkshop.org. For students and postdocs, there may be a limited amount of travel support available.

Nature recently launched a trial of a new peer reviewing system for manuscripts, where pre-prints are made available on the web and anyone, including the authors, can post comments in an open forum in a blog-like fashion. I think this is an excellent idea and developments like this are long overdue. I’ve always been a bit uneasy with the usual blind review system. The reasons for this are twofold. First, if the acceptance/rejection of a paper is determined entirely by one or two blind reviewers, it’s a bit of a lucky-dip whether you get good ones who make fair comments, understand the work and are actually interested. Second, science is all about the open discussion of ideas, and its only logical that this should be extended to peer review. Deciding the quality of a piece of work is something I think could benefit a lot from community feedback. I would have given anything to have been able to comment on this paper that I commented on in a previous post.

This is a technical post

I’ve just posted my latest paper to the arXiv . In my previous paper we discussed the fact that several recent qubit loss and gate failure tolerant quantum error correction schemes come at the expense of a blow-out in depolarizing noise rates. In this paper I’ve generalized this notion and derived an upper bound on the tradeoff between qubit loss and depolarizing error rates for the class of non-degenerate quantum error correcting codes. I’ve approached this by modifying the well-known quantum Hamming bound, which enforces an upper bound on the number of depolarizing errors a non-degerate code of given length can correct for.

The standard quantum Hamming bound is

where a k qubit logical state is encoded into an n qubit codeword, and t depolarizing errors occur. The generalized bound I derive is

where t_l located (i.e. qubit loss) and t_u unlocated depolarizing errors occur. Roughly speaking, this result implies that an optimal non-degenerate code can correct against twice as many located as unlocated errors with a linear tradeoff between the two. It would be interesting to establish whether a class of codes exist which are able to saturate this bound. If so, I think such codes would be quite useful, and have broader applicability than codes which are tailored specifically to either located or unlocated errors.