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In the theory of universal quantum gates,I have known a common universal gate set is the Clifford + T gate set, which is composed of the CNOT, H, S and T gates. Then there is a concept called "accuracy", which means use the composition of this set of gates can achieve any accuracy of the gate we want to approximately simulate.

However, in real quantum computer hardware, for example, ion-trapped quantum computers, the two parameterized gates in this paper can be regareded as universal quantum gates to realize any quantum gates (https://arxiv.org/pdf/1603.07678.pdf), and the concept of "accuracy" is also no longer mentioned.

It mentioned a concept of "fidelity" instead. So does that means, the process of simulation will be 100% right if we don't take the influence of hardware realization into account? If so, why do we still need to learn so much about "universal quantum gates"?

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The author in that paper allows himself arbitrary angle single and two qubit gates. With this set it is generally pretty easy to exactly match a given unitary, since the two qubit gates can give you the backbone of the correct entanglement structure, and the single qubit gates can conjugate the gate into the exactly correct basis.

The reason that accuracy is a concern for the CNOT/H/T gate set is that it is a discrete set, so for a rotation of some irrational angle (in multiples of $\pi$), I believe you would generically need infinite gates to exactly match it. The reason this gate set is still considered is that due to the Eastin-Knill theorem, Quantum Error Correcting Codes can only allow a discrete logical gate set, so in Fault-Tolerant Quantum Computers, a discrete set might be the best that we can do.

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Considering a fixed set of universal gates has several advantages. As a broad principle, standardization has proven extremely useful in computer science. Choosing a fixed set of primitive operations - even if they aren't always the best ones for every possible specific application - allows everyone to be on the same page, which makes it easier to focus on optimizing the physical realizations of those particular operations. Knowing that you only need to engineer a finite number of quantum gates in order to approximate an arbitrary unitary gives a much more achievable target than needing to be able to generate an infinite number of different primitive unitaries. Also, it makes it easier to compare performance across different algorithms, etc. if everyone's working from the same toolkit.

It's too earlier to tell whether actual useful implementations of quantum computers will only use a finite set of primitive unitaries, or whether they'll be able to engineer arbitrary unitaries out of a continuous set. But either way, it's useful to think about the possibilities enabled within both paradigms.

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