I understand that a quantum computer is universal if it can compute anything that a quantum Turing machine can. Another way to think about universality is that any unitary transformation on, e.g., a registry of qubits or their continuous-variable counterparts, qumodes, can be approximated. One may also consider ways to decompose unitary transformations into circuits of quantum gates and then introduce the concept of a universal gate set, that is to say a set of gates that can be combined into a finite circuit that can approximate any given unitary transformation as accurately as one desires. These topics have already been discussed in this SE for example here and here.

Is it the case that the universalities of a discrete-variable quantum computer and a continuous-variable quantum computer are in fact different? Different in the sense that what we consider to be the set of all unitary transformations appear at least to me to be different in the two cases? Or am I missing something and they both can be shown to be equivalent to the same quantum Turing machine?

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    $\begingroup$ you can do universal quantum computation with both, if that's what you're asking $\endgroup$ – glS Oct 17 '20 at 20:20
  • $\begingroup$ @glS The sets of unitary operations that a universal DV quantum computer and a unitary CV quantum computer can realize are the same then? For example, both can do a C-NOT gate? I know it must be some very elementary misunderstanding I have but I would be happy if it could be clarified. $\endgroup$ – Kiro Oct 19 '20 at 7:11

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