How to add qubits to an array one by one.
by Anne-Marije Zwerver
So it is winter and it is cold. Cold? It is freezing! But the air is nice and dry outside, so you decide to take a wintery walk in the forest. If you’re in a part of the world where you can currently fry an egg on the street, just wander along in your head – this is a small gedanken experiment. The walk is nice, yet cold and by the time you arrive home, the only thing you want, is to take a nice and warm shower. You turn on the tap and you feel the water running, splashing on your arms and shoulders, slowly defrosting your fingers. But then, for goodness sake, your roommate turns on her (cold) tap and your water temperature rises instantly. In a reflex, you jump out of the water jet, your skin already showing red stains. Luckily it was just an instant and soon you can go back into the shower. But then, of course, your other roommate needs some hot water and with a scream you, again, jump out of the now ice-cold shower. Time for a cup of tea…
Over your cup of tea you muse. How about a system in which, when your roommate turns on the tap, your water heating system gets a signal, such that it can respond and mix in some extra hot/cold water accordingly such that your water temperature doesn’t change. Of course, in most modern houses, this water temperature problem is not much of an issue these days. . But as a matter of fact, our qubits are getting a cold shower all the time. And, in order to keep our qubits happy and be able to work with them, we have to prevent that.
To see how this works, we first have to take a look at how we make qubits; we have to look at the quantum hardware. And with ‘we’, I mean the people in my research group (actually, QuTech studies four or five different ways of quantum hardware, depending on how you count). In the Vandersypen lab in QuTech we employ so-called spin qubits. In this blogpost from last April, Stephan nicely explains why spin qubits are very promising candidates for larger scale quantum computers. Here, I would like to explain a bit further how we make these spin qubits and scale them to larger arrays.
Our spin qubit is encoded in the spin momentum of an electron. Spin is a quantum mechanical property of many particles, that you can imagine as an intrinsic magnetic moment. For electrons, this spin can be either in a lower energy state (‘spin up’ ), or a higher energy state (‘spin down’ ) (or in a superposition of up and down). But, to be able to make such a qubit and fully control it without any unwanted interactions, we first have to catch a single electron and isolate it from its surroundings. We catch the electrons on tiny islands; tiny droplets of electrons. Each droplet can contain from many hundreds of electrons down to a single one, or even zero. Many of these droplets next to each other, each containing only one electron, form an array of electron-spin qubits .
So how do we make these islands to catch the electrons? We do this by confining electrons in 3 dimensions. We start with a material that contains a 2-dimensional electron gas (2DEG), which is basically a 2-dimensional sea of electrons. You can compare it to a layer of billiard balls on the deck of a billiard table; the electrons can extend very far on length and broadness, but in height they contain about one layer. On top of the material (and a bit above the 2DEG), we fabricate metallic plates; gates. When we apply a negative voltage to these gates, these gates start to repel the also negatively-charged electrons (the wave function is confined to a certain area and thus becomes more localized). When we deposit the gates in a certain way, they form an energy landscape in which some places are energetically more favorable for the electron to reside than others. On our billiard table, the balls under the gates are pushed away, whereas the other balls can stay, such that tiny islands are formed separated from the rest of the billiard balls (the non-island part of the 2DEG is called the Fermi reservoir). These islands – called quantum dots – are so small, that electrons that want to jump on the island are to a certain value repelled by the electrons that already reside on the island (Coulomb repulsion). Thus, they have to pay an energy price to jump on the island. This makes the energy levels of the quantum dot discretized; each energy level can contain only a single electron. Since electrons prefer to live at places with the lowest energy, only energy levels below the reservoir energy can be occupied by an electron (otherwise an electron has to gain energy to live on the island). This allows us to accurately control the number of electrons in the quantum dot.
So far so good, but what does the whole shower story have to do with this? Well, the electrons that we caught in our quantum dots are very sensitive; even the slightest difference in energy landscape can make them jump on or off an island. To compare: just as you want your shower water to be the exact right temperature, the electrons want the quantum dot to be at the exact right potential . As said, we can adjust this potential with the metallic gates. But because the metallic gates are capacitively coupled to their quantum dots, the gates do not only influence the potential landscape of the quantum dot directly underneath them, but also the potential landscape of the quantum dots around them; this effect is called cross capacitance and the amount of this cross capacitance from a gate to a dot can be measured. So when we change a single gate, it changes the landscape for a lot of islands. This means that, when we form an island with exactly one electron on it and, thereafter, with a second gate try and form a second island with one electron on it, the first electron will feel this. You could say that its roommate in the room next door turned on a tap and, if the energy around him changes significantly, our first electron will ‘jump out of his droplet’. With two electrons, we work in a two-dimensional parameter space and we’ll be able to keep track and compensate. But by adding more and more electrons, we have to, simultaneously, keep track of a higher and higher dimensional parameter space. This makes it very hard to form arrays of more than approximately three droplets.
Since we encode one qubit per island, but, keeping cross capacitance in mind, we cannot make more than approximately three islands simultaneously, our quantum computer will be out of business rather quickly. Therefore, we have to protect our electrons from cross capacitance. We do this by first forming a quantum dot loaded with one electron and then measuring the cross capacitance of every gate in our sample to this quantum dot. In this way, we know how much each gate influences the energy landscape of our dot and so we know, if we use a particular gate, by how much we have to change the gate of the quantum dot itself to compensate for this change and make sure the energy landscape around our dot stays the same. Using this principle, we can change neighbouring gates of our first quantum dots to add a second quantum dot, while compensating with the gate of the first, such that its potential landscape is not influenced. Then, again, we measure the cross capacitance from all gates to the second quantum dot. If we now change a random gate, again, we know exactly how much the energy landscape around the second dot changes and by how much we have to compensate the gate of the second. To keep track of all the cross capacitances, we store them in a matrix, with the quantum dots we formed on the rows and all the gates of the sample on the columns. Inverting this matrix gives our compensation factors for the quantum dots. How well our compensation matrix works, can be measured by means of a so-called charge stability diagram.
Now we can see that our quantum dots form indeed a perfect two-dimensional orthogonal space, we can add a third quantum dot – a third dimension if you want – by forming the third quantum dot and, again, measuring the cross capacitance from all gates to this third quantum dot. We fill the adequate row in the matrix, to engineer a 3D orthogonal space. This method can be repeated until all our eight quantum dots are formed and we can exactly determine the number of electrons on each of the quantum dots, without influencing the number of electrons on the other dots . So, in principle, we engineered a system that knows exactly when your roommate turns on the tap and how much it should change the temperature of your water to compensate for the change that all your seven roommates induced. Such that we made sure that our qubits never get a cold shower anymore! 
Footnotes and References
 It seems a bit weird that the lower energy state is called ‘up’. This does not need to be the case. Actually, it depends on the material you use which state is higher in energy. The g-factor of a material determines the splitting of the energy levels due to a magnetic field. Since the g-factor in Gallium Arsenide, the material we use in this project, is negative, the spin up state becomes lower in energy.
 In general these energy states are degenerate, but we can split them by means of a magnetic field.
 You can see this when you look at the energy discretization: quantum dot levels that are lower than the Fermi energy can be filled. When the energy of the quantum dot changes, the number of energy levels that are below the Fermi level changes and hence the number of electrons on the dot.
 Volk, Zwerver et al. Loading a quantum-dot based “Qubyte” register. npj Quantum Information. 5, 29 (2019).
Ever since she met Schrodinger’s cat, Anne-Marije has been fascinated by quantum mechanical phenomena. This brought her to QuTech where she is now preventing qubits from getting a cold shower. When not in the lab, she loves speed skating, running and wind surfing.