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Current quantum processors have constrained connectivity among qubits. For example Starmon-5 processor by Quantum Inspire has only one qubit connected to others, effectively it looks like a star. The situation is similar in case of IBM Q processors (the most obvious case is 15 qubits Melbourne processor).

I understand that connectivity among qubits is constrained by used technology. In case of semiconductor processors, probably the fact that structures are planar is the biggest obstacle. To tackle this, it would be perhaps needed to have three dimensional structures. Moreover, it seems that having full connectivity would increase a noise in NISQ processors.

My question: Is there any proposal how to build a quantum processor with full connectivity?

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Small ion trap quantum computers have all-to-all connectivity. Two-qubit gates can be executed between any arbitrary pair of ions in the trap. This has been demonstrated on up to 11 qubits (see, for example, https://www.nature.com/articles/s41467-019-13534-2).

But it becomes difficult to maintain this control when the number of ions gets too large. A linear chain of somewhere around 50 ions is the limit of feasibility to maintain good two-qubit gate fidelities. There are proposals for allowing multiple chains of ions to interact, but you no longer have direct all-to-all connectivity the way you would in a single chain.

So if by "full connectivity" you really mean all-to-all connectivity, such that you can execute gates directly between any arbitrary pair of qubits, this requires $n(n-1)/2$ physical connections between qubits. It really seems that this becomes infeasible for large $n$ beyond, say, a few hundred. Even going to three-dimensional structures wouldn't get you too far.

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