###### 20.04.2018difficulty level - QQ

## Making quantum computers with spin qubits

In one of the previous blog posts, David DiVincenzo reviewed his criteria. Here we will follow this theme and look how these criteria translate onto a physical system. Currently, there are a few qubit implementations that look quite promising. The most prominent examples are superconducting qubits, ion traps and spin qubits. We will focus on the latter one, since that’s the one I’m working on. All the platforms mentioned above fulfill the so called DiVincenzo criteria. These criteria, defined in 2000 by David DiVincenzo, need to be fulfilled for any physical implementation of a quantum computer:

- A scalable physical system with well characterized qubits.
- The ability to initialize the states of the qubits to a simple state, such as |000⟩.
- Long relevant coherence times, much longer than the gate operation time.
- A “universal” set of quantum gates.
- A qubit-specific measurement capability.

In this article we will go through all these criteria and show why spin qubits fulfill these criteria, but before doing that, let’s first introduce spin qubits.

Spin qubits are qubits where the information is stored in the spin momentum of an electron. A spin of a single electron in a magnetic field can either be in the spin down (low energy) or in the spin up (high energy) state. Comparing to a classical bit, the spin down state will be the analogue to a zero and spin up to a one.

One of the first steps in a spin qubit experiment is to obtain a single electron that is isolated from its environment. The isolation from the environment is needed to make sure that the electron does not undergo unwanted interactions that will affect its quantum state in an uncontrollable manner.

Isolated single electrons are obtained by shaping a two dimensional electron gas (2DEG).

When stacking different materials on top, one can get, under certain conditions, a 2DEG. A two dimensional electron gas can be seen as a plane where electrons are free to move wherever they want. One could compare the 2DEG with a very thin metal layer.

Once we have a plane of electrons, the trick is actually to push away from or attract electrons to this plane by using gate electrodes placed on top of the 2DEG. This will allow you to make a row of electrons (= isolated electrons). Here each electron will form one spin qubit on the chip.

Now we covered the basic concepts of spin qubits, we can return to the discussion why spin qubits can form a good platform for quantum computation. In the following I will go through the DiVincenzo criteria, one by one, and show why and how spin qubits satisfy these criteria.

*A scalable physical system with well characterized qubits.*

This criteria consists of two parts, a scalable and characterized system. The characterized part means that you know how to simulate the qubit (e.g. you know the Hamiltonian). This knowledge is needed to design the required gate operations. In case of spin qubits, the system is described well by the Fermi-Hubbard model.

The second requirement for this criterion was the argument of scalability. Where scalable means that you can extend your qubit control to millions of qubits.

A first thing to look at could be the size of one qubit. For a spin qubit this will correspond to an area of 70nm × 70nm. This means one could easily put a billion of qubits on a chip of 1cm × 1 cm.

However, this is not the big challenge. Problems arise when you start thinking about the control of the qubits. In the lab we observe that each qubit has its own personality. Examples are, different gate voltages for every qubit, the different amount of power required to operate a qubit, different energy differences between spin up and spin down…

The reason that every qubit has its own character has to do with the fact that we work with imperfect materials and an imperfect fabrication process. This is one of the big reasons we are collaborating with industrial chip manufacturers (e.g. Intel). Aside from the fabrication issues, there are also very practical problems, for instance how to connect all the control electronics to the qubits. For example, to have full control of a 5 qubit system, you need 40+ connections. 14 of these connections are high bandwidth connections. This means that very fast signals need to be sent through these lines to the qubits. Therefore they need to be connected by dedicated coaxial cables (which are relatively large). The picture below shows what a chip carrier for a 5 qubit chip looks like and where the coaxial cables need to be connected. Note also that such a chip needs to operate at extremely low temperatures, which also means you have a limited thermal ‘budget’ (e.g. μW range). A typical operating point for spin qubits is 10mK, which is close to absolute zero.

So are spin qubits scalable? The answer is, we don’t know. Until now, people have only focused on small systems (a few qubits). The same is true for superconducting qubits and ion traps.

*The ability to initialize the states of the qubits to a simple**state, such as |000⟩.*

When working with a quantum computer, one could imagine that, before running any algorithm, the user should be able to start from a well known quantum state. Typically, the lowest energy state is taken (e.g. 3 qubits in 0, |000⟩). For spin qubits one can achieve this using the Elzerman method. The idea is to load one electron with spin down from the reservoir in the quantum dot. The reservoir is an area next to the qubits filled with a lot of electrons. Note that when you put a lot of electrons together, there will be no difference anymore in the energy between spin up and down (electrons strongly interact with each other).

Now the idea is to align the energy level of the spin qubit (yes, you can control those!) with the reservoir in such a way that only an electron with spin down can tunnel in.

*Long relevant coherence times, much longer than the gate operation time.*

In any quantum system there is a characteristic time that tells you how long a quantum state will be maintained if left alone. This time is called the coherence time. It is important that the coherence time is much longer than a typical gate time (e.g. changing a qubit from 0 to 1 and vice versa). When this is not the case, all the quantum information will be lost before the algorithm is done. The loss of quantum information is mainly due to the interaction of the qubits with the environment.

The coherence time of spin qubits in mainly limited by the nuclear spins in silicon. When considering a spin qubit, one can imagine the electron as a charge cloud that is sitting at the Si/SiGe interface. This charge cloud overlaps with the nuclei of the silicon atoms. These nuclei can also possess a spin. When one of the spins of the nuclei flips (even at low temperatures there is enough energy to make this happen at random), the energy of the spin qubit will change a bit. This causes random energy fluctuations and will affect the coherence time of the qubit. In practice, we try to minimize this effect by using isotropically purified Silicon (e.g. ^{28}Si has no nuclear spin).

The coherence time of spin qubits can be quite long (e.g. 100μs, with ^{28}Si). The time for a single quantum operation will take about 100ns or less. This means there is a factor 1000 difference between gate time and coherence time, meaning that you can run 1000 operations before your information is lost.

*A “universal” set of quantum gates.*

A quantum algorithm typically consists of a set of operations, let’s call them U1, U2, … .These operations represent changes to the state of the qubits. One can think of this as the quantum mechanical equivalent of classical AND, NAND, OR, … gates. In the quantum mechanical case, you have two kinds of gates, single qubit gates and two qubit gates.

Single qubit gates, as the name suggest are used to manipulate a single qubit. For spin qubits, this is done using microwave photons. The energy of these photons corresponds to the energy difference of the spin up and down state. When both energies are exactly matched, one can controllably change the spin from up to down or any combination in between.

Two qubit gates are gates that entangle two qubits. This means that both qubits get a kind of awareness of each other. In practice, making two qubit gates is quite straightforward. This can be done by pulsing the barrier gate which is in between the two qubits. One could visualize this interaction as pushing the electrons close together, so that they interact with each other. One of the topics I’m actively working on, is making this kind of gates very reliable. At the moment these gates are often plagued by a lot of noise. In practice, this means that your gate operation is not reliable anymore. To solve this, we are exploring different regimes where you can make a two qubit gate. The interaction between the qubits is, for example, highly dependent on the energy difference between the two qubits you want to entangle.

A gate set is universal when you can perform single qubit gates and two qubit gates on all the qubits. As just described, this is the case for spin qubits.

*A qubit-specific measurement capability.*

When a quantum algorithm is done, you want to know what the outcome is. To know the outcome, you need to measure the state of the system (e.g. all or a part of the qubits that were used for the experiment). When performing a quantum measurement, the wavefunction of the qubit is collapsed. A wavefunction of one qubit could look like:

$latex |Psi rangle = 0.1 |0rangle + 0.9|1rangle $

This function describes the probabilities of the electron being in 0 or 1. In this case there would be 10% chance of a 0 and 90% chance of a 1. The collapse of the wavefunction means that, after a measurement, it will become 1 or 0 depending on which state you measured. A quantum algorithm typically starts by making a full superposition. When you have a lot of qubits, this would mean that you are in all the possible states at the same time (e.g. for 2 qubits, $latex |Psi rangle = |00rangle + |01rangle + |01rangle + |11rangle$). Then some computation is done, which should yield one result. The power of quantum computation is that you can evaluate all the possibilities at the same time. The hard thing is to describe your problem in such a way that you have a high likelihood of measuring the right outcome and a low probability of measuring bad outcomes.

Measurements for spin qubits can be done using the Elzerman method. It is quite similar to the initialization. When manipulating your qubits there is a big barrier between the reservoir and the spin qubit (single electron). When an electron is in spin up, it will be able to tunnel out into the reservoir. When this happens, there is for a short while no net charge in the place where the qubit was. After this, a spin down will tunnel back in, as we saw before. Presence or absence of charge is a property we can measure. A charge detector is placed close by the place where the electron can tunnel out. This sensor will give a change in signal whenever you have a spin up. In case of spin down, this charge sensor will not respond since no electron jumps out. Note that this readout method also directly initializes the qubit in spin down.

In conclusion, going through these 5 criteria shows that spin qubits have good potential for quantum computation. However, spin qubits are currently not the front runner in the field. That honour belongs to superconducting qubits, which are being pushed hard by IBM and Google. Superconducting qubits are running ahead because they are easier to make, tune and interconnect. The fact that superconducting qubits can be “easily” interconnected, allows for convenient schemes for quantum error correction, which is essential for bigger quantum computers (e.g. >> 10 qubits). But as for spin qubits, the real challenge for superconducting qubits is also the scale up. For superconducting qubits, the control, wiring and cooling of bigger devices becomes much harder, which might make it hard to go beyond a few thousands of qubits.

I suspect in the coming year(s), that spin qubits will reach a quality (e.g. fidelity of operation) similar to one demonstrated for the transmons, today. This will be partially due to the availability of better devices (e.g. Intel made) and improvements in the way we do gate operations.

In most experimental realisations of spin qubits, the qubits are placed in a row. This is not ideal for large scale operation, as it is not convenient to make two qubits interact at the ends of the spin chain. One of the next big steps for the spin qubit community would be to investigate the feasibility of fabrication and operation of 2D arrays of spin qubits.

Stephan is an experimentalist in the Vandersypen group working towards the first logical qubit in silicon. In his free time he is passionate about eating/making food and bouldering.