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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 Submission thread is now published as

Submission summary

Authors (as registered SciPost users): Kishor Bharti · Rebecca Erbanni · Dario Poletti
Submission information
Preprint Link: scipost_202211_00047v3  (pdf)
Date accepted: 2023-08-28
Date submitted: 2023-07-06 05:39
Submitted by: Erbanni, Rebecca
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Quantum Physics
Approach: Theoretical


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 without explicitly preparing its eigenstates. By means of numerical simulations we show that this approach finds many of the matrix elements for the one and two qubits cases. Specifically, 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

Our reply contains again two new plots, so we uploaded it along with the main file in the zip folder attached to this resubmission.

List of changes

We highlighted the changes in blue in the main pdf and the resubmission letter.

Published as SciPost Phys. 15, 180 (2023)

Reports on this Submission

Anonymous Report 2 on 2023-8-9 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202211_00047v3, delivered 2023-08-09, doi: 10.21468/SciPost.Report.7633


1. High/Improved readability
2. Main Ideas explained with sufficient detail
3. Sufficient comparisons to related methods
4. Data is presented nicely
5. Calculations on the simple demonstrations are quite extensive
6. No overstated claims anymore
7. Almost all points of previous reports addressed


1. Only 1 and 2 qubit demonstrations (remains a weakness)
2. Scalability unclear (analysis improved though)


The current version of the article presents a conceptually intriguing idea in a well-organized manner. The primary focus revolves around the estimation of matrix elements for general observables within the eigenbasis of a specified Hamiltonian. Unlike the conventional approach, which entails solving for eigenstates followed by matrix element computation, the proposed methodology directly constructs a corresponding objective function.

Although approaches like this exist in the literature (as research on variational optimization has always been quite rich) these concepts might remain unfamiliar to a substantial portion of the quantum computing community. The present work is primarily dedicated to elucidating the key components for constructing suitable objective functions.

The concept is illustrated on a single and two-qubit example. Admittedly, these instances lean towards extreme simplicity, functioning more as preliminary demonstrations rather than serving as definitive numerical proofs of the approach's applicability. The authors do also not claim such things (in this revised version).

Acknowledging the reservations expressed by colleagues in other reports, I share their skepticism towards deriving meaningful numerical insights from experiments involving just 1-2 qubits. I think however, that in the present case, it is more forgivable than in most other works with similar shortcomings. What speaks for the presented data beyond a mere didactical demonstration is that potential obstacles can already be detected at the two-qubit level, indicating that more work needs to be done to make the approach practicable. One can for example already see effects of (simulated) device-noise and shot-noise.

Fair comparison to alternative approaches (e.g. sequential VQEs + measurement matrix element) are at this stage out of scope for this work, as there is a high dependence on a number of different parameters intrinsic to the involved methologies (choice of test systems, choice of VQE ansatz, choice of the ansätze for the methodology introduced here, .... and many more). The authors choice to not do further numerics in this direction but rather discuss the differences to other prominent methods at high-level in the text is therefore understandable.

Requested changes

From context I assume that qiskit was used to simulate the results ("IBM QPU simulator"). It should be cited accordingly.

  • validity: good
  • significance: high
  • originality: high
  • clarity: high
  • formatting: excellent
  • grammar: excellent

Anonymous Report 1 on 2023-8-3 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202211_00047v3, delivered 2023-08-03, doi: 10.21468/SciPost.Report.7603


1.- Improved writing and readability.


1.- The scaling calculation only concerns the number of measurements for the overlaps, but still lacks on the sampling complexity, and how it affects the algorithm.
2.- Only 2 qubit experiments with poor performance to claim scalability, unacceptable.


I am unable to accept this paper for publication. I acknowledge the improvements on the manuscript but the work is still not mature enough to be published.
My main reasons are based ont he following points:
- Claiming scalability based on numerical evidence with only one and two qubit experiments is not enough.
- The scaling analysis only concerns one part of the algorithm, but ignores the rest.

  • validity: low
  • significance: ok
  • originality: good
  • clarity: ok
  • formatting: acceptable
  • grammar: acceptable

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