Yesterday we had a visit and seminar from Sonia Schirmer of the Cambridge Centre for Quantum Computation. The seminar was entitled ‘Quantum Engineering – Control Paradigms, Algorithms and Applications’
Whilst it was a little difficult for some of us experimentalists to follow, there’s definitely something interesting happens when experimentalists and theorists find a common discussion ground. I think the main problem is sometimes that theoretical work can be a bit too general for experimentalists to follow. Put it in terms of an example; a real-world system and it’s much easier. The loss of generality isn’t too much of a sacrifice from the experimentalist’s point of view. Conversely, sometimes it is difficult for theorists to realise what is easy (and what is hard) to implement in the lab.
So let’s talk!
We got into a rather intense discussion about superconducting flux qubit (SCFQ) realisations, and feedback control of such systems. For example, to manipulate SCFQ systems, you need a minimum of one control line. You can do some really simple but neat experiments with this system. (Disclaimer, when I say things are ‘simple’ I mean that they are possible in the best case scenario. The likelyhood of an actual experiment working is a probability of all its constituent parts working, which usually passes through my mind in the form of Psucess(experiment) = Psuccess(1)*Psuccess(2)*Psuccess(3)*…. See the end of this post for an example. But lets not get into the realm of half empty glasses; for now we will assume that everything works.)
For example, arbitrary rotations about the Bloch sphere are pretty simple to implement. Just apply a microwave flux pulse to your qubit at the frequency corresponding to the energy level splitting E(|1>)-E(|0>) and the qubit will swap between |0> and |1> at the Rabi frequency, going through a whole range of quantum superpositions α |0> + β |1> along the way. So usually experimentalists just apply a square shaped pulse modulated by a microwave frequency for a certain amount of time to bring the qubit into a particular superposition of states. For example, you might turn the microwaves on for 0.5ns to apply a π/2 pulse.
But you can also mess about with the shape of the pulse. Some people are already working on this (see for example THIS PAPER from MIT/NIST). This is where the control comes in. For the specific case of SCFQ, the control parameter would be the amplitude, phase, frequency, and pulse envelope shape of the microwave flux pulse applied to drive your qubit’s Rabi oscillations.
Open loop control – where you adjust the control parameters and check the output ‘figure of merit’ – this is usually something like the fidelity of the gate in question. (Like how often a CNOT gate gives the correct answer when you feed it known inputs).
Closed loop control – where you feed back the aforementioned figure of merit through some kind of algorithm to adjust the original control parameter.
SCFQ suffer from decoherence, which is believed to be mainly due to fluctuating electrons (fluctuators) in the junction barrier being excited into resonance at the same time as your qubit. If they are closely coupled to your qubit, they will interact (swap energy) and therefore act as a loss mechanism. By addressing the qubit with specially shaped pulses, it may be possible to ‘tiptoe’ around the fluctuators without waking them up.
Unfortunately (it seems) the systems are complex enough that you can’t predict what pulse shapes will work. But you can find algorithms which will converge onto good pulse-shape solutions if you have a simulation which gives you feedback. The best simulation of all is to use the real-world version. Hook up your control algorithm to a real experiment, and watch it go.
The other (more subtle but perhaps more interesting) thing is that it’s a two-way system. Your algorithm will also tweak the Hamiltonian which you are using to simulate the system in order to make it fit the data better. Your Hamiltonian will actually converge on one that accurately describes the physical system…. you win on both fronts! Your qubit gets treated better thus you get better data, and you get a more accurate theoretical picture of the mechanisms that were causing the issues in the first place.
The only two remaining problems are:
What algorithm to use to do this optimally?
What on earth does the complex mess of a Hamiltonian that you get out correspond to physically?
While I’m certainly not an expert in this field, (I’ve only really come across the idea of quantum control recently) it does seem like an interesting approach, especially when you can apply it to experimental realisations. Real-world data can keep simulations realistic and help optimise quantum control algorithms.
Wow, that even sounds like a conclusion.
 I chose the notation Psuccess for ease of calculation, although it is not really the way you end up thinking about scientific experiments after a while; Pfail is a better paradigm to adopt. Doubt and you’ll usually be right, but pleasantly surprised if you are wrong.
 I thought I’d give an actual example here with realistic values. All numbers are determined experimentally 😉
P(experimental success) = Psuccess(Helium leak doesn’t occur)*Psuccess(fridge cools to base correctly)*Psuccess(wiring doesn’t fail)*Psuccess(junction works)*Psuccess(noise level not too high) ~ 0.95*0.95*0.25*0.9*0.5
(Technically wiring doesn’t fail and noise level not too high are not independent events, but we’ll ignore this for now)
~ 0.101. So you’ll have to run the experiment 10 times to get a successful result. Each low temperature run takes about 2 weeks. And that’s why experiments take so long.