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The Solovay-Kitaev theorem (and more recent improvements) explains how to efficiently compile any 2-qubit unitary into any universal (dense) finite set of gates. My question is if this theorem is relevant for modern hardware?

Take, for example, superconducting qubits. There, modulo noise and errors, arbitrary single-qubit gates can be executed. Having the capacity to perform arbitrary single-qubit gates allows to compile any 2-qubit gate with at most three CNOTs.

I am not sure, but I'm guessing that other hardware platforms also allow for the continuous single-qubit gates. Does this make the Solovay-Kitaev theorem obsolete for practical purposes, or am I missing something?

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  • $\begingroup$ Note that you cannot perform any rotation. There is still a minimal angle you can use. The situation is similar to representation of real numbers on a classical computer. Althought they are called real, we only work with finite set of numbers as a memory is also finite. $\endgroup$
    – Martin Vesely
    Commented Dec 8, 2021 at 8:23
  • $\begingroup$ @MartinVesely the minimal angles are due to errors, right? Say it's 1$^\circ$ error. Is it possible to get a smaller error, say 0.5$^\circ$, by using a combination of gates with $1^{\circ}$ eror? $\endgroup$
    – Nikita Nemkov
    Commented Dec 8, 2021 at 8:27
  • $\begingroup$ No, but I meant that probably you could use combination of H and T gates (if they are native and not implemented with rotation gates) to reach a better accuracy, but this is only idea...maybe there is a flaw. $\endgroup$
    – Martin Vesely
    Commented Dec 8, 2021 at 15:19

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I think you'll find that most hardware, at the hardware level, gives you arbitrary single qubit rotations. So, in that sense, it is true that Solovay-Kitaev is not directly applicable to current systems. However, obsolete is certainly not the right word. It's rather the opposite - it's ahead of its time.

The importance of the Solovay-Kitaev algorithm is really once we start to do error correction (which requires bigger devices). When you implement an error correcting code, even if you have arbitrary single-qubit rotations at the physical level, you (generally) only have a finite gate set at the level of logical qubits. So to get an arbitrary rotation, you need Solovay-Kitaev or similar.

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    $\begingroup$ An analogous comment applies to topological quantum gates, some references on that point are listed here: ncatlab.org/nlab/show/… $\endgroup$
    – Urs Schreiber
    Commented Dec 24, 2022 at 9:49

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