You may have see this cool new paper on the ArXiv:

Observation of Co-tunneling in Pairs of Coupled Flux Qubits

(I believe there is something called a ‘paper dance’ that I am supposed to be doing)….

Anyway, here I’ll try and write a little review article describing what this paper is all about. I’m assuming some knowledge of elementary quantum mechanics here. You can read up about the background QM needed here., and here.

**First of all, what is macroscopic resonant tunneling (MRT)?**

I’ll start by introducing energy wells. These are very common in the analysis of quantum mechanical systems. When you solve the Schrodinger equation, you put into the equation an energy landscape (also known as a ‘potential’), and out pop the wavefunctions and their associated eigenvalues (the energies that the system is allowed to have). This is usually illustrated with a square well potential, or a harmonic oscillator (parabolic) potential, like this:

Well, the flux qubit (quantum bit), which is what we build, has an energy landscape that looks a bit like a double well. This is useful for quantum computation as you can call one of the wells ‘0’ and the other ‘1’. When you measure the system, you find that the state will be in one well or the other, and the value of your ‘bit’ will be 0 or 1. The double well potential as you might imagine also contains energy levels, and the neat thing is that these energy levels can see each other through the barrier, because the wavefunction ‘leaks’ a little bit from one well into the neighbouring one:

One can imagine tilting the two wells with respect to one another, so the system becomes asymmetric and the energy levels in each well move with respect to one another. In flux qubit-land, we ’tilt’ the wells by applying small magnetic fields to the superconducting loops which form the qubits. Very crudely, when energy levels ‘line up’ the two wells see each other, and you can get quantum tunneling between the two states.

This effect is known as macroscopic resonant tunneling. So how do you measure it? You start by initializing the system so that state is localised in just one well (for example, by biasing the potential very hard in one direction so that there is effectively only one well), like this:

and then tilt the well-system back a little bit. At each tilt value, you stochastically monitor which well the state ends up in, then return it to the initialisation state and repeat lots and lots of times for different levels of tilt. As mentioned before, when the energy levels line up, you can get some tunneling and you are more likely to find the system on the other side of the barrier:

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In this way you can build up a picture of when the system is tunneling and when it isn’t as a function of tilt. Classically, the particle would remain mostly in the state it started in, until the tilt gets so large that the particle can be thermally activated OVER the barrier. So classically the probability of the state being found on the right hand side ‘state 1’ as a function of tilt looks something like this:

Quantum mechanically, as the energy levels ‘line up’, the particle can tunnel through the barrier β and so you get a little resonance in the probability of finding it on the other side (hence the name MRT). There are lots of energy levels in the wells, so as you tilt the system more and more, you encounter many such resonances. So the probability as a function of tilt now looks something like this:

This is a really cool result as it demonstrates that your system is quantum mechanical. There’s just no way you can get these resonances classically, as there’s no way that particle can get through the barrier classically.

Note: This is slightly different from macroscopic quantum tunneling, when the state tunnels out of the well-system altogether, in the same way that an alpha particle ‘tunnels’ out of the nucleus during radioactive decay and flies off into the ether. But that is a topic for another post.

**So what’s all this co-tunneling stuff?**

It’s all very nice showing that a single qubit is behaving quantum mechanically. Big deal, that’s easy. But stacking them together like qubit lego and showing that the resulting structure is quantum mechanical is harder.

Anyway, that is what this paper is all about. Two flux qubits are locked together by magnetic coupling, and therefore the double well potential is now actually 4-dimensional. If you don’t like thinking in 4D, you can imagine two separate double-wells, which are locked together so that they mimic each other. Getting the double well potentials similar enough to be able to lock them together in the first place is also really hard with superconducting flux qubits. It’s actually easier with atoms or ions than superconducting loops, because nature gives you identical systems to start with. But flux qubits are more versatile for other reasons, so the effort that has to go into making them identical is worthwhile.

Once they are locked together, you can again start tilting the ‘two-qubit-potential’. The spacing of the energy levels will now be different (think about a mass on the end of the spring β if you glue another mass to it, the resonant frequencies of the system will change, and the energies levels of the system along with them. We have sort of made our qubit ‘heavier’ by adding another one to it.

But we still see the resonant peaks! Which means that two qubits locked together still behave as a nice quantum mechanical object. The peaks don’t look quite as obvious as the ones I have drawn in my cartoon above. If you want to see what they really look like check out Figure 3 of the preprint. (Note that the figure shows MRT ‘rate’ rather than ‘probability’, but the two are very closely linked)

From the little resonant peaks that you see, you can extract Delta β which is a measure of the energy level spacing in the wells. In this particular flux-qubit system, the energy level spacing (and therefore Delta) can be tuned finely by using another superconducting loop attached to the main qubit loop. So you can make your qubit mass-on-a-spring effectively heavier or lighter by this method too. When the second tuning loop is adjusted, the resulting change in the energy level separation agrees well with theoretical predictions.

As you add more and more qubits, it gets harder to measure Delta, as the energy levels get very close together, and the peaks start to become washed out by noise. You can use the ‘tuning’ loop to make Delta bigger, but it can only help so much, as the tuning also has a side effect: It lowers the overall ‘signal’ level of the resonant peaks that you measure.

**In summary:**

Looking at the quantum properties of coupled qubits is very important, as it helps us experimentally characterise quantum computing systems.

Coupling qubits together makes them ‘heavier’ and their quantum energy levels become harder to measure.

Here two coupled qubits are still behaving quantum mechanically, so this is promising. This means that in the quantum computation occurring on these chips involves at least 2-qubits interacting in a quantum mechanical way. Physicists calls these ‘2-qubit processes’. There may be processes of much higher order happening too.

This is pretty impressive considering that these qubits are surrounded by lots of other qubits, and connected to many, many other elements in the circuitry. (Most other quantum computing devices explored so far are much more isolated from other nearby elements).

Great summary Suz.

I thought it might be worth noting that Fig. 3 in the preprint is a plot of the tunneling rate vs. tilt; which is like plotting the slope of your Probability vs tilt graph above.

Can you throw up a graph showing what an MRT rate graph would look like in your example?