How do we define whether a device is a noisy intermediate-scale quantum (NISQ) device with respect to number of qubits and their error rates? Does it make sense to do this? I believe I once saw a definition of a NISQ device as one with on the order of $n \le 10^5$ qubits and single qubit error rates on the order of $\epsilon \ge 10^{-4}$are there any references that attempt to define the 'intermediate-scale' part of NISQ? As far as I'm aware, John Preskill did not provide any explicit definitions (this seems to have been prudent at the time, perhaps it still is).

Attempting to unpack things a bit, it seems that a NISQ device must be one that is designed to leverage noisy, not logical (error-corrected), qubits. If we are to believe that, then the obvious question is how many logical qubits do we need to justify entering the fault tolerant quantum computing (FTQC) regime? Do we require a quantum supremacy experiment on a FTQC such as the one demonstrated by the Google-led collaboration or does a more appropriate alternative exist?

Let's say that the we answer the latter question in the affirmative and assume 53 qubits is sufficient to realize an equivalent quantum supremacy in the FTQC regime (i.e. 53 logical qubits), can we estimate the transition from NISQ to FTQC to meaningfully occur once a system capable of error correcting the $n$ number of noisy qubits with single qubit1 error rate $\epsilon$ to yield 53 logical qubits capable of completing a similar RCS task with XEB benchmarking? If so, what do we estimate the order of $n$ noisy qubits and $\epsilon$ error rate(s) to be?

Another way to think about this: What would an update to this 2018 graph from Google AI Blog look like?

2D conceptual chart showing the relationship between error rate and number of qubits. The intended research direction of the Quantum AI Lab is shown in red, where we hope to access near-term applications on the road to building an error corrected quantum computer.

Any answers which explain the current state of thinking with respect to the question would be fantastic.

1. Setting aside that single qubit errors are far from the only errors we're concerned with.

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    $\begingroup$ You seem to forget that the resource requirements for universal FTQC are usually dominated by the overhead needed for universality, e.g. magic state distillation or code switching. This overhead highly depends on the task at hand, for instance the frequency of $T$ gates. Moreover, the bar for FTQC should be higher than mere quantum supremacy ... To get an idea for the numbers see e.g. arxiv.org/abs/1905.09749 ... so the Google chart is not that wrong. Of course, this could (and hopefully will) change with further research. $\endgroup$ Apr 14 at 12:01
  • $\begingroup$ @MarkusHeinrich Yes I’ve read the Gidney and Ekerå paper, if I recall they discuss number of qubits and error rates with respect to a fault tolerant architecture leveraging magic state distillation. I have not forgotten — in fact, to your point, I see the overhead as being a consideration for answering to this question. Regardless, it sounds like you’re saying that the implication of the chart above, that the frontier of the NISQ era is in the neighborhood of ~10^5 qubits with an error rate of ~10^-3, is accurate? $\endgroup$
    – Greenstick
    Apr 15 at 0:43
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    $\begingroup$ I think this estimate is not totally unrealistic. IMO factoring seems like a good indicator to see what one is up against. However, I also think that the insights gained by separating the above graph into regimes is very limited. It suggests that one merely has to increase the number of qubits (and their quality) to achieve FTQC and in this aspect, having a 70 qubit processor over 50 qubits is an achievement. In the end, it could be more important to find a smart architecture, good codes and compilers etc to actually arrive at FTQC and not be stuck on the way. $\endgroup$ Apr 15 at 9:19

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