Venue: NeurIPS 2022 Conference

Reviews:

final_score: rating: 4: Borderline reject: Technically solid paper where reasons to reject, e.g., limited evaluation, outweigh reasons to accept, e.g., good evaluation. Please use sparingly.
review_content: summary: The paper proposes a quantum version of Soft-Bayes for the maximum likelihood QST task, since Soft-Bayes enjoys fast per-iteration time complexity. The proposed algorithm is a stochastic first order algorithm, and compared to [64], it has better per-iteration complexity on the order of O(D^3), but worse iteration complexity. The paper provides numerical results comparing the proposed algorithm, RrhoR, and Monotonous FW. 
strengths_and_weaknesses: Strengths
- The paper is clearly written, and the theoretical results are well-supported. The numerical results, though not ideal for the proposed algorithm, are interesting and appreciated. 

Weaknesses
- The iteration complexity and per-iteration complexity both involve D, which is exponential in the number of q-bits. Though for the ML QST task this dependence may be inevitable, it would be helpful for the authors to explain why this problem formulation makes sense when the number of q-bits is large.
- The idea of using an online portfolio selection algorithm for a QST task is also explored in [64], and the extension of Soft-Bayes to the quantum setting doesn’t seem to involve new techniques. 
- The numerical results are unfortunately not very convincing, as the method obtains higher error than some of the baseline algorithms.  

questions: Is it possible to do ML QST when the number of q-bits is large? Why does this problem formulation make sense with exponential dependence on the number of q-bits?
limitations: Yes.
review_title: null

final_score: rating: 5: Borderline accept: Technically solid paper where reasons to accept outweigh reasons to reject, e.g., limited evaluation. Please use sparingly.
review_content: summary: This paper studies state tomography from the perspective of online learning. State tomography is an important problem in quantum computing. Its main objective is to arrive at an approximated description of an unknown quantum state by measuring several copies of that state. The main approach of this paper is based on soft Bayes ML estimation in which the objective is to minimize a log-loss defined between the true density operator and the estimated one. The paper then relates this problem to minimizing regret in online learning (portfolio selection) where soft Bayes is a promising approach. Then, the paper proposes a quantum analogous to soft Bayes. Using classically developed tools, the authors provide guaranteed results for their algorithms. Lastly, they provide numerical demonstrations.  
strengths_and_weaknesses: The main strength of the paper in my opinion is the quantum soft Bayes which leads to a quantum learning algorithm with provable guarantees. This is a nice idea with possible future applications. The analysis (especially Theorem 3.3) is interesting.

There are two major weaknesses of the paper:

1)  The paper is not clear in the problem formulation.  It is unclear to some extent as to why we need to use the online approach for state tomography where we can have all the samples at once. 

2)  The proposed approach does not seem to be scalable as the number of iterations grows exponentially with the number of qubits. 
questions: Why do we need to use online learning for state tomography?
limitations: The main limitations of the proposed approach are (a) scalability as the number of the iterations grows exponentially with the number of qubits, and (b) numerical experiments do not seem to show the advantage of the proposed algorithm.
review_title: null

final_score: rating: 4: Borderline reject: Technically solid paper where reasons to reject, e.g., limited evaluation, outweigh reasons to accept, e.g., good evaluation. Please use sparingly.
review_content: summary: The submission proposes Stochastic-Q-Soft-Bayes to accomplish the quantum state tomography task. To achieve an $\epsilon$-regret, the required runtime complexity scales with $O(D^4logD/\epsilon^2)$. Numerical simulations on a 4-qubit state demonstrate the effectiveness of the proposed method. 
strengths_and_weaknesses: Strengths:
The submission addresses an important topic in quantum computing, i.e., devising efficient algorithms to complete quantum state tomography. The paper is well organized and the central idea is well presented.

Weaknesses:
1. It has been proved that $n = \Omega(d^2/\epsilon^2)$ copies are necessary to achieve the estimation error $\epsilon$ [IEEE Trans. Inf. Theory, 63(9):5628–5641, 2017.]. Nevertheless, the submission shows that the proposed algorithm can achieve an $\epsilon$-error using $O(D^4logD/\epsilon^2)$ copies. This contradiction originates from the definition of the `error'. Conventionally, the estimation error between the estimated state $\hat{\rho}$ and the target $\rho$ is defined as $\|\rho-\hat{\rho}\|_{tr}$, whereas the submission adopts the regret to describe the `error'. Consequently, the achieved results are incomparable with prior literature. The authors should clarify this issue. 

2. To address the applicability of Stochastic-Q-Soft-Bayes. Simulation results on large-qubit states are desirable. Current results fail to demonstrate the potential advantages of Stochastic-Q-Soft-Bayes.

 


questions: See the comments above. 
limitations: See the comments above.
review_title: null

final_score: rating: 3: Reject: For instance, a paper with technical flaws, weak evaluation, inadequate reproducibility and incompletely addressed ethical considerations.
review_content: summary: In this paper the authors consider the problem of quantum state tomography (QST). Here there is an unknown n qubit quantum state rho (i.e., its a complex matrix of size 2^n x 2^n) and the goal is to learn rho given copies to the state. The sample complexity of QST has completely been pinned down to Theta(d^2) where d=2^n. Apart from the theoretical guarantees there are also some heuristic works that have looked at QST and one such method which is relevant to this work is that of maximum-likelyhood estimation. So computing the ML estimator  amounts to solving a convex optimization problem, however the main issue for using maximum likelehood estimator for QST is  negative log-likelihood function in ML QST is not Lipschitz and not smooth as well.  In contrast in this paper, the authors develop a stochastic first-order optimization technique for ML QST inspired by techniques from online learning. They moreover choose to quantumize  the soft-Bayes algorithm which is an iterative algorithm  whose round by round complexity is linear in the dimension of the state, i.e., d  and the rate of convergence of the procedures presented in this paper are similar to ones known from literature. Finally the authors provide some empirical plots implementing their learning procedures.

I think the study of QST using MLE is nice, but i dont think this paper is fit for NeurIPS for a few reasons
a) The main contributions of this paper are in a very narrow subject and i dont think this narrow subject  might be of interest to classical ML (which is the primary audience of NeurIPS) and quantum computing (since the novelty of this paper is limited to improving certain heuristic-based approach for QST).
b) In my opinion, the main contributions of this paper needs more motivation. For example, QST is a completely solved subject, in the sense the sample complexity has been pinned down. Then why should one look at solving QST using regression, MLE etc? Is there any solid motivation for considering these approaches? 

Given the niche contributions which are limited in applicability, i think this paper isnt strong enough to be accepted for NeurIPS.
strengths_and_weaknesses: Mentioned earlier.
questions: Mentioned earlier.
limitations: Mentioned earlier.
review_title: null

Meta-review: metareview: Overall: The paper propose a stochastic first-order algorithm that computes an -approximate ML estimate for the QST problem.

Reviews: The paper received four reviews. Borderline reject (less confident), Borderline accept (confident), Borderline reject (absolutely confident), Reject (confident). Overall, from the reviews there is not a reviewer that champions the paper for acceptance.

Main issues raised are:
- Clarity of presentation, notation
- Scalability/applicability
- Less relevant to the ML community.

After rebuttal: While the authors have been active responding to the reviewers' comments, the rebuttal discussion was relatively silent. While the AC has reached out to find additional reviewers, this effort was unsuccessful. One of the reviewer was responsive, but the outcome was that the paper still lacks significance and applicability within the ML community. This suggests that it might be preferable the paper be submitted to a near future conference venue (and maybe a more theoretical one), following these suggestions + corrections.

Confidence of reviews: The reviewers are fairly confident in their reviews. The thorough reviews among the four definitely get more weight than the rest of the reviews.

Overall, the paper feels to be in good state but none of the reviewers feels extremely confident championing the paper for acceptance at this venue. We highly suggest the authors to consider near future ML conferences or more quantum-related conferences for resubmission
confidence: Certain
Decision: Reject