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NISQ algorithm for the matrix elements of a generic observable
by Rebecca Erbanni, Kishor Bharti, Leong-Chuan Kwek, Dario Poletti
This is not the latest submitted version.
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Submission summary
| Authors (as registered SciPost users): | Kishor Bharti · Rebecca Erbanni · Dario Poletti |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202211_00047v2 (pdf) |
| Date submitted: | 2023-03-22 06:31 |
| Submitted by: | Erbanni, Rebecca |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
The calculation of off-diagonal matrix elements has various applications in fields such as nuclear physics and quantum chemistry. In this paper, we present a noisy intermediate scale quantum algorithm for estimating the diagonal and off-diagonal matrix elements of a generic observable in the energy eigenbasis of a given Hamiltonian. Several numerical simulations indicate that this approach can find many of the matrix elements without, at the same time, the need to prepare the energy eigenstates. Specifically, we have considered systems with one and two qubits and we showed that, while in the first case, one can initialize the ansatz parameters over a broad interval, in the latter the optimization landscape can significantly slow down the speed of convergence and one should therefore be careful to restrict the initialization to a smaller range of parameters.
Author comments upon resubmission
List of changes
We highlighted the changes in blue in both the main pdf and the resubmission letter.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 3) on 2023-5-15 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202211_00047v2, delivered 2023-05-15, doi: 10.21468/SciPost.Report.7193
Report
As stated in the initial report, the proposed concept is original, and applications of this idea are widely spread. The exaggerated claims made in the original manuscript have been appropriately toned down, resulting in a satisfactory level that now accurately portrays the developed methodologies and numerical demonstrations in an honest manner.
Survey of the literature with similar content was improved as well and is in this form acceptable. I appreciate the effort that went into the changes on page 4 where the difference are clearly worked out. This will be beneficial to a lot of readers, especially students.
I recommend publication.
Thanks for clearly marking changes in the manuscript & apologies for the delay in my report.
Minor comment:
The newly added clarification before Eq (1) is good, but might be confusing to some on first read as: Re(M) = 0.5( M + M^T) and similar for the imaginary part only holds for M Hermitian and not for general matrices.
This is implicitly given by stating that W is an observable further above, I would however change "We first note that, for a given matrix W, we can al-
ways write" to "We first note that, for a given Hermitian matrix W, we can al-
ways write". Or replace "T" with "*" and clarify, that for Hermitian matrices W^T = W^*
Report #1 by Anonymous (Referee 4) on 2023-4-4 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202211_00047v2, delivered 2023-04-04, doi: 10.21468/SciPost.Report.7001
Strengths
1.- Novel attempt to use variational quantum algorithms beyond ground-state energies.
Weaknesses
1.- No description of the sampling complexity of the algorithm, making it difficult for a reader to assess the its practical implementation.
2.- Claim of scalable method solely based on a one and two qubit numerical experiment with the two-qubit experiment is already poorly performing.
3.- Main goal of the work is difficult to extract from the abstract and introduction.
4.- Poor motivation of the work.
Report
Summary:
The manuscript "NISQ algorithm for the matrix elements of a generic observable" describes a method to calculate the matrix elements of a generic quantum observable from the eigenstates of a Hamiltonian (e.g. in the energy bases).
The main result the use of Lagragian multiplyers to extract diagonal and off-diagonal elements of a quantum observable by optimizing the parameters of constrained quantum states.
Decision:
I am unable to accept the manuscript for publication in SciPost.
The main reason for this decision is the impossibility to assess if the algorithm will be practical for even moderate size problems.
The sampling complexity analysis is almost inexistent, and a reader has a hard time knowing how many times a quantum computer needs to be called for even N=1,2 qubits.
On the same line going from 1 to 2 qubit in a simulation is not sufficient to claim "scalability", moreover when the 2 qubit experiment already show poor performance.
Despite the decision I acknowledge the novelty of the work as an attempt to use variational quantum algorithms beyond ground-state energies. I, therefore encourage the authors to further work on the idea presented in this manuscript and mature the results.
Comments:
- The manuscript is overall poorly written.
- The main goal is hard to extract from either the abstract or the introduction.
- It is unclear why someone should care about calculating off-diagonal elements of an operator in a known basis.
- The complexity of the algorithm (in terms of the number of queries to a quantum device) is impossible to extract from the manuscript. This result is critical to assess whether the protocol is practical. I suggest to pick an example with known scaling and count the queries required to implement the algorithm.
- From one and two qubit simulations is not possible to draw any conclusion on the scalability of the algorithm.