Paper: Near-Optimal Quantum Algorithm for Minimizing the Maximal Loss (ICLR-2024)

Reviewer comment (Theory):
There is still a gap between the query-complexity of the algorithm and the shown lower bound in terms of the dependence on $\epsilon$, but this is not a significant weakness of the paper.

Author response:
Please refer to our general response for your comment on the gap between our upper bound and lower bound.

Reviewer comment (Theory):
I wonder if one can state a general meta-theorem of the form: "For a wide class of computational/optimization problems, if we have a classical algorithm that solves it and which satisfy some properties, then we can find a corresponding quantum algorithm with a quadratic speedup".

Author response:
Regarding your question on a potential meta-theorem, we agree that it is indeed a good research question to ask.  
There have been many quantum algorithms on important subroutines adopted in many classical algorithms.  
For example, our paper exploits a quantum speedup for multiple sampling from the same distribution found by Hamoudi [1].  
However, the possibility of deriving a meta-theorem appears uncertain to us. Distinct quantum approaches and techniques, albeit that the source of the speedup might be the same, are needed for different scenarios with different settings.  
In particular, conceptually Gao et al. [2] and Bouland et al. [3] both achieve speedup on sampling, but the sampling tasks are slightly different, leading to entirely different algorithms compared to the one used in our algorithm.  
Generally, we think it is nontrivial to summarize types of sampling tasks with similar quantum speedup due to the diversity and specificity of scenarios.

Reviewer comment (Novelty):
A disadvantage is that the algorithm in a sense takes an existing classical algorithm and replaces a step in it with a suitable, known, quantum algorithm. So novelty is a bit low.

Author response:
We appreciate your positive feedback and detailed comments!

Please refer to our general response for your comment on the novelty of our algorithm.

Reviewer comment (Theory):
The major unsatisfactory point is the unmatched upper and lower bound.

Author response:
Please refer to our general response for your comment on the unmatched upper and lower bound.

Reviewer comment (Writing):
The presentation is fine in general, but it could be improved (e.g., the English is a bit awkward sometimes).

Author response:
We also thank you for detailed suggestions regarding presentation - they have been fixed in the revised version.

Reviewer comment (Reproducibility):
I understand that you first find the \(K\) largest elements of the distribution, and then you set the probability of the rest of the elements equal to the probability of the \(K\)-th element — this changes the distribution. Do you need some rejection sampling procedure afterwards? (Is the answer in Figure 1? Then it would be better to move the figure to the main paper.)

Author response:
Regarding your question about Algorithm 2,
we did not use any rejection sampling for the preparation of the states.
The unitary $\mathcal{D}$ that is built in the step 3 of our Algorithm 2 is used only as a subroutine in the circuit that is shown in Figure 1.
With several additional gates, in Figure 1 we build a circuit that prepares a state that contains the desired and an ancillary garbage state.
We then apply the amplitude amplification algorithm to prepare the desired state, as shown in step 5 of Algorithm 2.
The reason we first only consider the first $K$ largest element is to guarantee that the amplitude amplification can extract the state we want efficiently: in Eq. (26) we lower bound the coefficient before the target state, guaranteeing a total cost of $\tilde{O}(\sqrt{NK})$ according to Proposition 3.
We have made updates correspondingly in the paper to improve the presentation of Algorithm 2.