Practitioners’ Rule of Thumb for Quantum Volume

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The authors present a formula that aims to reproduce the quantum value based on the number of two-qubit gates and their average fidelity. Although the formula agrees with the largest values reported by various vendors so far, it slightly overestimates the values of the previous years. The general validity of the formula remains unclear, though it represents an interesting empirical observation. Additionally, the assumptions underlying the formula are quite extensive and hard to verify. to make the work publishable, the authors should conduct tests on at least two different real quantum hardware for specific problem classes.  

Author Response

Comment 1: The authors should conduct tests on at least two different real quantum hardware for specific problem classes.  

Response 1: We sincerely appreciate the referee’s recognition of the adequacy of our methods and presentation for this journal. Regarding the suggestion to conduct tests on real quantum hardware, we would like to clarify that the quantum volumes reported in Table 1 are derived from extensive experimental runs performed by the quantum computing manufacturers themselves. To further validate our approximations, we have now compared our results with the numerical solution of an expression reported by the IBM group in Ref. [2]. This comparison demonstrates that our approach aligns more closely with experimental data and has the key advantage of providing an analytical expression, which enhances interpretability and generalizability beyond purely numerical solutions.

Reviewer 2 Report

Comments and Suggestions for Authors

A useful paper for increasingly larger QPUs. The results seem to tie with reported values but the derivation of the main equation (2) needs further elaboration along with the rationale for why it is stated that there are N² native gates when there are only twice the number of native gates. A few more clarifications would also be useful for the use of noisy simulators in checking QV and the significance of QV for narrow but deep circuits.

Author Response

Comment 1: Why it is stated that there are N² native gates when there are only twice the number of native gates. 

Response 1: We appreciate the referee’s comment, which highlights an issue in our presentation. To enhance clarity, we now introduce the variable $n_{2q} = N \times N / 2 = N^2 / 2$, which explicitly counts the total number of 2-qubit gates. Consequently, the total number of native gates is given by $n = 2n_{2q} = N^2$. This revision ensures consistency and improves the readability of our notation.

Comment 2: A few more clarifications would also be useful for the use of noisy simulators in checking QV and the significance of QV for narrow but deep circuits.

Response 2: Following the first part of the referee’s suggestion, we now explicitly reference an expression for QV obtained in Ref. [2] using noisy simulations. We demonstrate that our analytical approach not only yields a more compact and accessible result but also generalizes to a wider range of connectivity models, including both all-to-all and square-lattice architectures.

Regarding the significance of QV for narrow but deep circuits, we have chosen not to expand on this point, as the standard definition of QV in the literature primarily pertains to square circuits. The focus of our work is to provide a simple yet broadly applicable analytical expression that practitioners can readily use.