The Quantum Pontiff

And now, coming to an arXiv site near you:

Simulating Hamiltonian dynamics using many-qudit Hamiltonians and local unitary control
Michael J. Bremner, Dave Bacon, and Michael A. Nielsen
When can a quantum system of finite dimension be used to simulate another quantum system of finite dimension? What restricts the capacity of one system to simulate another? In this paper we complete the program of studying what simulations can be done with entangling many-qudit Hamiltonians and local unitary control. By entangling we mean that every qudit is coupled to every other qudit, at least indirectly. We demonstrate that the only class of finite-dimensional entangling Hamiltonians that aren’t universal for simulation is the class of entangling Hamiltonians on qubits whose Pauli operator expansion contains only terms coupling an odd number of systems, as identified by Bremner et. al. [Phys. Rev. A, 69, 012313 (2004)]. We show that in all other cases entangling many-qudit Hamiltonians are universal for simulation.

Often when I am thinking about the foundations of quantum theory, I am struck by the universality of the theory. Quantum theory (or its related cousin, quantum field theory) applies generically to all physical systems (disregarding the transition to some “classical” theory and of course, difficulties with both QCD and gravity.) Thus we apply quantum theory to our basic theories of physics, electromagnetism, the weak force, the strong force, but we also apply quantum theory to simple atoms and complex molecules, to single electrons and electron gases in metals, etc. Quantum theory is the universal language we use to describe any physical process. If we are thinking about ways to explain quantum theory, then this universality is a bit mysterious: the explanation had better apply to all of these different physical systems and that seems like a lot of work! Of course, this reasoning is flawed: it seems the universality is an illusion. The reason we can describe a complex molecule by quantum theory is that the fundamental constituants of that molecule obey quantum theory. Separation of different energy scales (and other scales, like localibility) allow us to ignore some of the constituants details, and the complex system behaves like a quantum system. So really any explanation of quantum theory need only apply to some basic level of physics (where this level is I refuse to speculate.) While quantum theory appears mysteriously universal, this is an illusion for those persuing understanding the mystery of the quantum.

Last week I attended a workshop sponsered by DARPA on “Scalable Quantum Information Processing via Error Control.” The idea behind the workshop was to bring together theorists who know something about error control with experimentalists who know all about different proposed quantum computing implementations and examine the feasibility of each of the different implementations in light of the requirements which arise from error control. It’s been a while sense I attended a workshop with “real physicists” (a.k.a. experimental physicists and their physics calculating theorist brethren.) The implementations covered were “Ions and Neutrals”, “Superconducting”, “Spins in GaAs”, “Spins in Si, Si/Ge”, “NMR”, “Linear Optics”, and “Electrons on Helium.” DARPA has recently said that a new program “FoQuS” will be starting and will be narrowing down the field of DARPA funding for the different implementations. Needless to say this has caused a lot of stress on those currently receiving DARPA funding.
Here are some of my observations from the workshop.
Ion traps rock. If were a starting graduate student who wanted to do quantum computing and do some really rocking quantum computing experiments during my graduate career I would make a dash straight towards an ion trap quantum computing group. The era of NMR is over and a new era of Ion trap quantum computing has begun! Why do I say this at the expense of the other possible implementations? First of all, the ion trap people have already successfully show the basics of coupling and manipulating qubits and they can do these with good to great fidelity. One great advantage they have (and share with other AMO proposals) is that they can do high fidelity, fairly fast (approx 10x two qubit speed) measurements. Second, they have really nailed down exactly what is going on in their system: what are the decoherence times and mechanisms, what are the heating rates when you move ions, etc. Third, ion traps have always been questioned due to their scalability and thus there has been a lot of thought in the ion trap community about how to fabricate traps for which will realize concrete quantum computing architectures. The recent demonstration of teleportation in ion traps, I think, is the beginning of a long line of beautiful protocols for multiqubit quantum protocols.
Locality has been too often ignored. The threshold for fault-tolerance is hard to analyze when you restrict yourself to particular geometric architectures (with the important exception of calculations done with toric (surface) codes which have a local quantum structure and a nonlocal classical structure. Here the thing we would like to get around is the ancilla state preparation factories which require the ancilla states to be swapped into the surface codes: this is just the old locality issue again, but in a much more tame setting. Also the surface code setup is very nice for three dimensions, but difficult to imagine in two dimensions.) If you are going to use a concatenated structure for error correction, then you really need to sit down and think about flying qubits. If not, it seems you might be in deep trouble when trying to construct your architecture.
Fitting it all together. Many implementations will have serious technical dificulties when you try to lay the control circuitry on a realistic architecture: for some spin implementations this may cause serious problems.
Superconducing qubits visibility Superconducting qubits have this (not understood?) property that they don’t get a high visibility when they do single qubit Rabi flopping. Of course there are two reasons for this to be occuring: one is that the state was not properly prepared and the other is that the measurement is not high fidelity. Until this visibility problem is well understood, superconducting qubits may be in trouble.
When things take off? In the next few years, implementations will be working on implementing quantum error correcting techniques. So, suppose you implement the five qubit quantum error correcting code. What should we expect to see? Well since these experiments will most probably be below the memory threshold, the effect of the circuit in terms of real fidelity will be to make things worse! So we will have some years where we start measuring how badly we are making things worse. What I can imagine happening is that there is a group of people who will work on these small codes on their systems and constantly improve them until they pass the threshold. Simulatenously I expect a number of people will work on architectural scalings for the implementations. And when these two meet their goals, I’m a relative optimist that all hell will break loose and we will see quantum computing being scaling in a remarkable fashion. When, then, will breakeven be reached? And how do I say this without invoking images of fusion?
State of the implementations. My impression of the state of implementations is as follows. NMR quantum computing in liquid state has reached its terminus. Ion traps will be taking the lead where NMR quantum computing has left off. I expect the ion traps to press the next four or five years of the quantum Moore’s law. Quantum dot and superconducting qubits are at the stage where they need to pen down the characteristics of their systems. It was impressive to hear that first attempts at single shot measurements in quantum dots have achieved 64% measurement fidelity. I’m a bit worried about the Kane implementations, and I’m sure there not at the ion trap level. Thus I’d say most solid state implementations are a few years (like 3 to 5 years) behind Ion traps. Linear optics quantum computing is the wild card in the whole picture. I worry most about the requirements of state preparation in linear optics quantum computing. I worry also about the device complexity, but this doesn’t seem an insurmountable barrier because I’m not an engineer or an experimentalist. Is mode matching a killer? I know the least about electrons on Helium, but I get the impression that they are all on the cusp of demonstrating two qubit interactions. They will then have to begin the quantificiation process a la ion traps. Implementations with neutrals also fit in somewhere, but I’m not quite sure where. I have always been shocked by the lack of experimental progress in neutrals: the number of quantum optics people should have lead to some nice results by now, but I haven’t seen this (but I claim no authoritative status as I’m just a lousy uneducated theorist!)
In all it was a fun workshop. The talks were super short, but the conversations after the presentations were at times very interesting. What was amazing to see was to watch an expert in ion traps talk to an expert in superconducting qubits and other such cross disciplinary conversations. Normally these two wouldn’t give the time of day to each other, but through quantum computing there is a common language. And not just a common language, but also a common set of problems with many common solutions. Quantum computing is so multidisciplinary it is scary. But it’s also the reason it is such a beautiful and exciting field.