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Now I have understood that physical implementation of quantum computer need a universal quantum gate set like Clifford+T to realize any unitary quantum gate. However, I don't know if it is all the same gate sets for different physical implementation of quantum computer. For example, if the trapped ion and photonic implementation of quantum computer own the same universal quantum gate set. On the other hand, can the quantum computer of IBM and Google implement the same universal quantum gate?

I am interested in quantum simulator now, and I know that any unitary quantum gate will be decomposed into single qubit gate, and this process is done in simulator. After that, the simulation will send the after-decomposed single qubit gate into backend, which is a really physical implementation of quantum computer or emulator of classical computer and during this procedure the single qubit gates will be decomposed approximately into a sequence of gates within the universal quantum gates.

I don't know if my understanding of this issue is right.

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So any universal gate set can replicate any other, since both are universal, but different architectures generally have different physical gates. While Clifford+T is a universal gate set that is very nice to think about theoretically, it isn't generally close to the one used in the lab.

In most experimental setups, the physical level universal gate set used is composed of arbitrary angle Pauli rotations, along with a single entangling gate which is either always maximally entangling, or also variable angle.

For trapped ion systems, we use single qubit Paulis along with a gate known as the Mølmer-Sørenson gate. This gate is a rotation about the XX axis of two qubits which uses the shared motion of the ions in the trap to get distant entanglement.

Superconductors use different entangling gates, if I remember correctly IBM uses a gate called the 'Cross-Resonance gate' which I think is a ZX rotation gate, and on the Google Sycamore chip, they use a gate which ends up being a combination of CZ and iSWAP.

To understand how these are universal gate sets, lets use the ion trap gate set to build the pieces of the initial set you described, Clifford+T. First, lets condense that set into the three elements [H, CNOT, T]. A T gate is just a Z rotation, so by having the arbitrary Pauli rotations we are covered there. A Hadamard gate is an X rotation followed by a Y rotation, so we have that one as well. Through this we have all possible single qubit gates already. Now to get the CNOT, we can wrap the XX gate in 4 single qubit Paulis, as described in Fig. 1 of this paper. As a result we have a full universal set. The only difference for the SC gate sets would be a different decomposition of CNOT.

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  • $\begingroup$ Using {Clifford+T} can approximated simulate the unitary quantum gate, for example, Solovay-Kitaev algorithm can decompose any unitary quantum gate with a fixed accuracy. is there any specific accuracy in practical use. To be specific, what is the specific accuracy in IBM quantum system or Google Sycamore chip? $\endgroup$ – Henry_Fordham Jun 12 at 23:50
  • $\begingroup$ I don't think I understand your question. Any given universal gate set can approximate a unitary to arbitrary precision. I'm not sure if IBM/Google use a precise cutoff for their compilations. Practically speaking, there is a tradeoff between compilation error and the gate errors that a larger compilation would suffer from, so I don't think its a particularly one-size-fits-all problem. $\endgroup$ – Dripto Debroy Jun 13 at 1:20

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