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I have heard about the diamond norm, and from what I understood it is a "nice" tool to quantify quality of quantum gates in the NISQ era. I would like to know a little more before going in detail in the litterature.

My question are the following:

Why is it considered as a good tool to quantify quality of quantum gates? Why this tool and not another one?

What are the nice properties this tool has? I have seen in Is the diamond norm subadditive under composition? that it is subbaditive which make the tool useful to bound the diamond norm of a full algorithm given the diamond norm of each gate. Are there other some useful properties? I guess this question will be related somehow to the previous one. I tried to find this information in literature, but unfortunately all papers I read just give the definition and the diamond norm and a way to compute it numerically.

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    $\begingroup$ Mark Wilde's textbook might be a good resource to look up these properties. There is a chapter called 'Distance Measures' which discusses this norm. You can access a free pdf at this link : arxiv.org/abs/1106.1445 $\endgroup$
    – Purva Thakre
    Commented Dec 27, 2019 at 20:17
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    $\begingroup$ It is natural in the same way you want CP maps, not just positive maps: You want your map also to work - or your gate quality also be good - if you act on part of a larger quantum system. $\endgroup$
    – Norbert Schuch
    Commented Dec 28, 2019 at 0:39
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    $\begingroup$ @PurvaThakre thank you for the link, it is really a nice reference ! $\endgroup$
    – Marco Fellous-Asiani
    Commented Jan 1, 2020 at 19:55
  • $\begingroup$ @NorbertSchuch allright, I see what you mean. Thanks. $\endgroup$
    – Marco Fellous-Asiani
    Commented Jan 1, 2020 at 19:56

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