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In this paper (or more pedagogically here) it is given a way to compute the Quantum Volume of a quantum computer (qc), specifically aimed at the IBM qcs.

In all this process I found some dark spots that I would like to understand. First of all we need the heavy outputs $H_U$ which can be obtained from a "classical" computation. Obviously in the implementation the transpiler will do some optimizations so from an original $U$ ($m$ qubits and depth $d$) we will get the transpiled $U'$. The probability of sampling a heavy output with this $U'$ is strangely

$$ h_U = \sum_{x\in H_U} q_U(x). $$

  1. Ok I buy that this quantity works, but why to sum it over all possibles $x\in H_U$ is still unclear to me.

  2. The probability of observing a heavy output is then $h_d = \int h_UdU$. In the paper mentioned the Algorithm 1 computes $h_d$, right?

  3. As far as I understand, this algorithm simply constructs the frequency plot between a realization of $U$ vs the number of heavy outputs found in a certain number of trial initial state $|00\dots 0\rangle$ ($m$-spins), right?

  4. Then if 3 is correct, $h_d$ can be computed from the (normalized) histogram just constructed. However the authors give the formula

$$ \frac{n_h-2\sqrt{n_h(n_s-n_h/n_c)}}{n_c n_s}>2/3, $$

to validate if $h_d$ has been achieved. How is this formula found? Is it some numerical (e.g. Simpson) rule?

  1. On the other, one can measure how well the $U'$ resembles $U$ by computing the fidelity $F_{\mathrm{avg}}(U,U')$. Where does this quantity enters for computing $V_Q$? $\log_2 V_Q =\mathrm{argmax}_m \; \mathrm{min}(m,d(m))$ doesn't tell much...
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    $\begingroup$ these are good questions but, as it stands, the post is way too broad. The stackexchange format is best-suited for laser-focused questions. Feel free to edit your post to focus it on a specific question. You can open different posts to ask different questions $\endgroup$ – glS Apr 21 at 10:41
  • $\begingroup$ Please ask just one question. $\endgroup$ – user1271772 Apr 22 at 11:37

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