Fast Quantum State Reconstruction via Accelerated Non-Convex Programming

Round 1

Reviewer 1 Report

The author proposed the non-convex MiFGD algorithm for estimating low-rank quantum states. The proposed algorithm is a modification of the Projected factored gradient descent (ProjFGD) algorithm, which the author presented in reference [30]. The authors provide the rigorous convergence proof of MiFGD and provide evidence of faster estimation of the unknown low-rank sparse quantum state compared to the standard algorithms. The result is an interesting addition to the quantum state estimation toolbox. From the practical point of view, the method has some utility in situations where states are known to be low rank.  

However, I have some minor comments which I would like to ask the authors to bring the manuscript in the publication form.  

1. The y-axis scaling needs to be clarified in figure 2. The authors need to be consistent.  

2. Please specify the tradeoff between the two algorithms Momentum-Inspired Factored Gradient Descent (MiFGD) and Projected Factored Gradient Descent (ProjFGD), in the introduction section.  

3. CVXPY and lstsq are standard algorithms for state estimation. The author should compare their algorithms that are efficient in terms of timing. I suggest the authors compare their algorithm timing with the efficient algorithm and ADAM method given in the below references  

i)  Smolin, J. A., Gambetta, J. M. & Smith, G. Efficient method for computing the maximum-likelihood quantum state from measurements with additive Gaussian noise. Phys. Rev. Lett. 108, 070502 (2012).  

ii) Li, K., Zhang, J., Cong, S.: Fast reconstruction of high-qubit-number quantum states via low-rate measurements. Phys. Rev. A 96, 012334 (2017). The fidelity of an efficient algorithm is quite high if we assume the rank condition. It would be great for the readers if the authors provided the fidelity and timing plot of the known rank one with an efficient algorithm.  

4. The Restricted Boltzmann Machine (RBM) is designed to reconstruct the quantum state under the noise model or prepare the quantum state in the NISQ devices. To draw a fair comparison between RBM and MiFGD, the authors can consider the reconstruction of the sparse matrix under the depolarizing noise.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

This manuscript provides a new method as well as ready-to-use code for quantum state tomography. The authors have compared their method with previous approach (including traditional ones and these based on neural-network), which support that their method has a significant advantage in saving time. Time is essential for tomography of large systems, so this paper really worth publication.

Author Response

We thank the reviewer for the comments and feedback. We have checked the grammar and spells thoroughly. 

Reviewer 3 Report

The manuscript leaves rather ambiguous impression. It looks like it gives readers a good piece of  numerical modelling, some acceptable mathematics and somewhat questionable physics. To add, the authors seem conscientiously exploiting hypes and overly loud claims which serve mostly to attract attention (like using quantum computer for 8 qubit tomography).

The manuscript offers only quite moderate technical advances to the authors' 2018 paper. For the illustration of the supposedly novel technique a number of simple common rank 1 signals were chosen. The maximum of 8 qubits chosen for illustrations does not look sufficient for demonstrating essential advantages of the problem. Personally, I consider non-convex gradient descent methods as a step back to the very beginning of the quantum tomography epoch. Well, they are faster indeed. If they are proven to give results close to the true matrix and not sticking in some local minimum, that is certainly great. But one really should give an estimation of statistical and systematic noise tolerances for proving such a method to be universally applicable. Just an assurance "we observed accurate and robust reconstruction, despite the presence of experimental and statistical noise" does not look sufficiently persuasive for that.

From other side, the manuscript presents a large body of technical work which can be of interested to the specialists. So, taking into account all the pro and contra, I consider this paper still suitable for publication by a MDPI journal. However, I would still advise the authors to be more moderate in their claims. Please, explain at least these "common assumptions" under which the method is "provably" converges. Well, you know, just a few percent drift of the detector efficiency during the measurement usually is sufficient to spoil completely a nice theoretical picture.

Also, please, re-read carefully the Introductory part. It looks written rather sloppy and confusing, especially when addressing neural network QST.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf