A Quantum Annealing Protocol to Solve the Nuclear Shell Model

1. the paper has a good presentation of both no-core shell model and quantum annealing protocol suitable for a general audience
2. the authors study a reasonably large variety of nuclear systems

1. the results presented in the paper are obtained from exact classical simulation, it is therefore difficult to asses the practical viability of the proposed method on real devices affected by noise
2. the estimated gate complexity of the proposed algorithm seems to suggest that the protocol might require resources beyond those expected in NISQ devices. If the algorithm were to be run on fault-tolerant quantum computers the number of required CNOT will be of little consequence and a study of the non-Clifford resource needed should be carried out instead

The present manuscript describes an application of Quantum Annealing to the problem of preparing ground states of nuclear systems described using No-Core Shell Model. The authors present results obtained with classical simulations on a variety of nuclei and estimate the gate complexity of the algorithm on digital quantum platforms. The paper is well written and the results interesting, however it does not appear to satisfy any one of the expectations required for publication in SciPost Physics: the QA annealing approach is well known and the novel elements proposed here, such as the choice of driving Hamiltonian, do not seem to be general or ground-breaking enough. I believe that the work is solid but that it might be more suitable for a different Journal.

Below my main comments on the manuscript

- The numerical results in Sec.5 are certainly interesting but in part simply confirm previous expectations, like the 1/\Delta^2 scaling of the evolution time, while it appears difficult to extrapolate the other observations on the observed performance of different systems to other systems that have not been studied. For instance, would it be possible to leverage the current results to, at least approximately, predict the cost of annealing on larger Hilbert spaces for these nuclei? What about with other nuclear interactions?

- Right after Eq.(18) the authors suggest that choosing an initial product state that minimizes the energy of the target Hamiltonian H_T may be advantageous. Of the two provided references the first has some compilation error (shows up as a ?) while Ref.[84] is not available yet. Could the authors briefly explain the reasons for the possible advantage of using low energy initial states?

- In the conclusions at the end of Sec.6 the authors suggest that QA would scale better than ADAPT-VQE as a function of D under the assumption that the number of steps required in QA is independent on D. I think this statement needs some further clarification. As expected from adiabatic algorithms, and confirmed by the results shown in Fig.5, the number of steps of evolution will scale as the inverse gap squared. The expectation is then that: if the smallest gap \Delta depends only polynomially on D, then the total cost of QA (cost per step times number of steps) will also scale polynomially.

- There seems to be a small mistake right before Eq.(22): the authors say that D CNOT are needed to implement the exponential of a Pauli string of length D/2. I tried to look for this result in Ref.[39] but couldn't find it. In general however, it is well known that an exponential of a length N Pauli string requires a staircase of 2(N-1) CNOT instead (see e.g. Fig.8 of Ref.[39]). In the case discussed in the manuscript the CNOT cost should then be D-2 instead of D.

- The explicit definitions of \mathcal{B} and \mathcal{B}_0 in Figure 1 should probably be in the main text next to where these space are defined instead of in a Figure.

- In the conclusions the authors say that the QA approach scales polynomially in the number of many-body states. I believe they meant to say single-particle states instead.

Accept in alternative Journal (see Report)

1-This paper is one of the first study of annealing method applied to nuclear shell model
2-It gives a precise quantitative study

1-This is a purely academic study because all numerics is done exactly on a classical computer and the exact solution serves to validate.
2-Numerous Ccurrent studies of annealing are on the acceleration of adiabatic methods and/or on the proof of convergence. None of these two aspects are discussed properly in the article

In this article, the possibility of using the quantum annealing method to solve the nuclear spectroscopy problem is discussed. Specifically, after discussing how the shell model wave-function can be encoded, the standard method consisting of switching the Hamiltonian from a simple Hamiltonian to the full problem one is applied on a classical computer. The methodology is rather standard, and the convergence is ensured according to the Gell-Mann and Low adiabatic theorem. Such convergence will depend on the time scale used to switch from one Hamiltonian to another.
I have mixed feelings about the work. I feel this work is rather academic since we observe in practice what is expected, e.g.,. for instance, that the convergence property depends on the timescales tau omega. And, when two levels get closer to each other, the adiabaticity breaks down, as illustrated by the oscillation of the excited state occupation probabilities. Many groups today are trying to improve adiabatic methods to get faster convergence with the aim of being able to apply quantum platforms. If I am correct, this point is briefly mentioned only when quoting Ref. [100], while this is an active field today.
Also, the tests made in the article (relative energy errors, infidelity, …) rely on the possibility of creating the exact calculation back-to-back to the adiabatic technique. Assessing the convergence toward the proper target state is also an active field today in the context of quantum computing.
So, in short, the work seems a bit disconnected from current interests and discussions of practitioners trying to apply adiabatic methods on analog machines.
On the other hand, very few works have been made in nuclear physics using quantum annealing. As far as I know, this is one of the first attempts made to assess the quality of such a strategy for the nuclear shell model. From that point of view, the present work contains many interesting information and quantitative studies. It might even be seen as a milestone before trying to attack applications on real quantum platforms.
With this last argument, I estimate that the present article should be published provided that some more discussion is made on the recent efforts made to perform calculations using real quantum platforms, at least to better present the work as an academic intermediate step and quote the ongoing works that are made to fill the gap between this type of ideal work the progress that are made to apply quantum annealing today on real quantum processors.

1-Better discuss the current work on quantum annealing regarding the criteria of convergence and the different methods to accelerate the adiabatic approach.

Ask for minor revision