# What physically limits qubit connectivity in superconducting chips?

All the superconducting based quantum computers I'm familiar with have a maximum of 4 nearest neighbors per qubit. Trapped ion architectures seem to be able to drive entangling gates between all pairs of qubits in devices containing up to something like 50 qubits. What is it that is limiting building superconducting chips where each qubit is connected to say, 10 others? Is it a control limitation, geometric, frequency crowding, perhaps something we can do but the hit in fidelity or coherence times makes it infeasible?

From my understanding, there is no fundamental reason why a given superconducting qubit cannot be connected to many others.

In practice, however, you are facing the following limitations that are clearly technology-dependent (they depend on how you perform your two-qubit gates). Those limitations are typically either related to a frequency issue (there is a limitation in the total number of frequencies you can allocate to your qubits: it will give a limitation to how much qubits one given qubit can communicate to), to very "practical" issues (roughly speaking you would need a much more complex architecture if you want a high connectivity), or noise issues (the less interactions can be turned on, the less noise can enter in the system).

## Method 1 to perform two-qubit gates: intermediate coupler scheme

One way of performing two qubit gates is based on a coupler scheme. Basically between qubit Q1 and qubit Q2, you use a coupler "C" (that is usually a resonator, sometimes also a qubit, but which does not contain any information from the algorithm). By tuning the frequency of this coupler you can make $$Q1$$ and $$Q2$$ interact in some "effective frame" (and hence perform a two qubit gate). One ref for this.

The issue with this scheme is that you would need a coupler per pair of qubit that need to interact. It can quickly be a lot (if you have $$N$$ qubits in your computer and you want to perform an arbitrary two qubit gate between each pair of qubits, you would need $$\binom{N}{2}$$ couplers). I let you imagine the nightmare in wiring and overall scalability.

## Method 2: you couple different qubits with a bus

In this method, you put many qubits in contact with a bus. This bus will mediate the interactions and can allow "in principle" to make any pair of qubits connected to this bus to interact to make a two qubit gate. This is much easier to scale up with that.

One example is the cross-resonance scheme. The principle is to send a microwave pulse on $$Q1$$ at the frequency of $$Q2$$. Doing so you can perform two-qubit gates between $$Q1$$ and $$Q2$$. In this scheme, one natural limitation is the number of qubits that can be connected to the bus. You need to be sure to have enough frequencies to avoid "accidentally" performing a three body interaction. For instance, if you only have two frequencies available but that you put three qubits on the same bus, you will not have enough space in frequency to make the thing work.

This question leads to another one: how do I know that I lack frequencies? An easy criteria for that is to consider that transmons qubits are typically at frequencies in the range $$6 GHz \pm 1 GHz$$. Then the total bandwith you can have is $$\Delta f \approx 2 GHz$$. The spectral width $$\Delta s$$ of a signal is typically given by the inverse of its duration (shorter signals are more spread in frequencies). Then, knowing the duration of your gates you can deduce how much frequencies you can reasonably put. Typically, you can have $$\Delta f / \Delta s$$ different frequencies for your qubits. What limits to make $$\Delta s$$ as small possible is because you want to manipulate your qubit in a "fast enough" manner (otherwise they will decohere).

As a last comment: there might exist other methods than the one I have put here. But they are the two main ways of performing two-qubit gates I am thinking of. The first method has been used in the Sycamore processor from Google. I know that the second method (in particular cross-resonance) is used in IBM quantum processors.

• Concerning the method one, it sounds like tyranny of numbers known from classical mainframe computers before integrated circuits mass deployment is back. So 3D structures can help. Right? Commented Feb 19, 2022 at 8:01
• @MartinVesely My guess would be that it would help, yes. But there are also other issues by connecting one qubit to many others which is that if you do that you also open new potential noise channels for your qubits. This is true for the two methods. Commented Feb 19, 2022 at 11:10

If you increase the number of transmon qubits (for example) connected to one transmon qubit, the charging energy will increase. If this happens, qubit frequency and anhamornicity will change and you will be out of the transmon regime you need to operate the qubit.

I would add that there are also physical limitations. Transmon qubits are realized in planar semiconductor structures, so you are limited to only two dimensions. Of course, you can make multilayer chips like in classical semiconductor instruments, however, there is a issue with "induction" from lines in one layer to another etc. Moreover, you have to plug qubits to reading lines and input lines which makes the multilayer design even more difficult.

Notwithstanding that even single layer chips have currently issues with connecting many lines. This could be solved with multi-chips design, i.e. more quantum processors connected each other.

Still you will face issues mentioned by Manuel García.