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High-fidelity realization of the AKLT state on a NISQ-era quantum processor

by Tianqi Chen, Ruizhe Shen, Ching Hua Lee, Bo Yang

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Submission summary

Authors (as registered SciPost users): Tianqi Chen
Submission information
Preprint Link: scipost_202307_00042v1  (pdf)
Data repository:
Date accepted: 2023-09-21
Date submitted: 2023-07-31 14:46
Submitted by: Chen, Tianqi
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Quantum Physics
Approaches: Theoretical, Computational


The AKLT state is the ground state of an isotropic quantum Heisenberg spin-$1$ model. It exhibits an excitation gap and an exponentially decaying correlation function, with fractionalized excitations at its boundaries. So far, the one-dimensional AKLT model has only been experimentally realized with trapped-ions as well as photonic systems. In this work, we successfully prepared the AKLT state on a noisy intermediate-scale quantum (NISQ) era quantum device. In particular, we developed a non-deterministic algorithm on the IBM quantum processor, where the non-unitary operator necessary for the AKLT state preparation is embedded in a unitary operator with an additional ancilla qubit for each pair of auxiliary spin-$1/2$'s. Such a unitary operator is effectively represented by a parametrized circuit composed of single-qubit and nearest-neighbor $CX$ gates. Compared with the conventional operator decomposition method from Qiskit, our approach results in a much shallower circuit depth with only nearest-neighbor gates, while maintaining a fidelity in excess of $99.99\%$ with the original operator. By simultaneously post-selecting each ancilla qubit such that it belongs to the subspace of spin-up $\ket{\uparrow}$, an AKLT state can be systematically obtained by evolving from an initial trivial product state of singlets plus ancilla qubits in spin-up on a quantum computer, and it is subsequently recorded by performing measurements on all the other physical qubits. We show how the accuracy of our implementation can be further improved on the IBM quantum processor with readout error mitigation.

Author comments upon resubmission

Dear Editor-in-charge,

Thank you for sending us the referee reports. All reviewers made positive recommendations of the previous version of our manuscript, particularly on its impact and performance, and only requested for changes that we believe we have now fully addressed. Referee 1 (Prof. Emanuele Dalla Torre) stated “The authors propose and demonstrate how to perform this step using a single ancilla qubit for each pair of qubits. They implement the algorithm on an IBM quantum computer and claim to achieve very high fidelities” and provide helpful suggestions for a revision. Referee 2 suggested “I would recommend the publication of this manuscript in Scipost” after we address some questions. Referee 3 indicates “The approach presented in the paper offers several advantages. It enables efficient preparation of the AKLT state on NISQ-era quantum processors by reducing the number of CX and single-qubit gates required. The variational recompilation optimizes the quantum circuit, mitigating the impact of gate errors” and just has several minor technical comments on some presentation details that I list under "Requested changes”.

We have carefully considered the suggestions and opinions of Referees 1, 2, and 3, and have added a whole section (Appendix E) on the characterization of the entanglement entropy and the prepared AKLT state, a further clarification of why other MPS-based state preparation approach could not be directly applied in our setup of the preparation of the AKLT state in the introduction part, and also added the discussion on the state fidelity definition and the variational circuit optimization algorithm in Sec. 3.3 and Sec. 4. In particular, to address the concern from Referee 1, we perform additional simulations of measuring the entanglement spectrum. Attached please find the revised manuscript as well as our response to the referees. We highlighted the major changes and additions in blue colour and listed the changes made to our manuscript at the end of the response for your kind perusal.

We believe that we have addressed all the other, minor, comments in detail in the following of this reply and we have significantly improved our manuscript accordingly. We hope that now the manuscript can be reconsidered for publication in SciPost Physics.

With our best wishes,

Tianqi Chen,
Ruizhe Shen,
Prof Ching Hua Lee,
Prof Yang Bo

List of changes

1. Added a whole new section (Appendix E) on the characterization of the entanglement entropy and the prepared AKLT state. Also, added comments on it in the conclusion.

2. Added a further clarification of why other MPS-based state preparation approach could not be directly applied in our setup of the preparation of the AKLT state in the introduction.

3. Added further discussion on the state fidelity definition, as well as the variational circuit optimization algorithm in Sec. 3.3 and Sec. 4.

4. Uploaded the dataset for the optimized circuit open on Zenodo and share the link in Ref. [87].

5. Various other technical revisions on the presentation details advised by Referee 3.

Published as SciPost Phys. 15, 170 (2023)

Reports on this Submission

Report 2 by Emanuele Dalla Torre on 2023-9-3 (Invited Report)

  • Cite as: Emanuele Dalla Torre, Report on arXiv:scipost_202307_00042v1, delivered 2023-09-03, doi: 10.21468/SciPost.Report.7717


The authors have replied to my previous report.

Concerning weakness point 1 ("The approach is not compared with other state-of-the art state preparation algorithms"), this issue was analysed in detail in a recent preprint This work showed that the number of required CNOT is $20 L$. (In my previous report I gave a lower estimate because I forgot to take into account that generic two-qubit gates need to be compiled into CNOTs). Hence, the approached proposed in the present paper is more efficient than the trivial implementation using the MPS representation. I suggest to explicitly mention this point in the paper.

Concerning weakness point 2 ("The fidelity measured in the real hardware is inaccurate"), the authors addressed my points in a satisfactory way by adding some discussion in the paper and a new appendix.

Based on this analysis, I now support the publication of this work in its present form.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Anonymous Report 1 on 2023-8-15 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202307_00042v1, delivered 2023-08-15, doi: 10.21468/SciPost.Report.7654


The revised version is a nice piece of work. In combination with the reply to all three Referees, I believe that all relevant issues have been addressed. I therefore recommend publication in SciPost Physics.

However, there are a number of minor typographic issues that I list under "Requested changes" such that they can be addressed during the production stage.

Requested changes

Minor typographic issues:
- 2 lines above Eq. (2): "recasted" $\to$ "recast".
- No indentation after equation if the embedding sentence continues (concerns in particular Eqs. (3), (4), (5), (12), (13), and (17)).
- I believe that Eq. (6) would fit on one line.
- Just before Eq. (11): inset comma before "i.e." and maybe also after. Same 3 lines below Eq. (14)
- 3rd line of Sec. 3.2: "a Hadamard gate" (delete "s" - singular).
- Remove full stop in Eq. (13) (sentence continues after the equation).
- 3rd line of caption of Fig. 2: "remain" $\to$ "remaining".
- "NISQ-era" appears twice in the first sentence on page 10. It might be stylistically better to remove this redundancy.
- 4th line of caption of Fig. 3: "show" $\to$ "shown".
- 6th line of Sec. 3.4: I believe that the full stop just before the closing parenthesis is unnecessary.
- Last line of text on page 14: Insert space between "clearly." and "According".
- 4th line of caption of Fig. 6: "circles" is misspelled and insert article "the" before "default transpile".
- 5th line of text on page 15: insert article "the" before "noiseless Qiskit aer_simulator".
- Page 16: remove full stop after "Appendix", in particular if a specific one is referenced (at least two instances).
- Last line of first paragraph on page 16: "Sec. E" $\to$ "Appendix E".
- 6th line of Sec. 5: insert "that" before "the state can be obtained".
- 7 lines further down: insert article "the" before "AKLT state".
- Footnote 6: I believe that it would be better to write "permits" (or "is able") instead of "is possible".
- Last line of Sec. 5: remove the spurious full stop before the list of references [118-123].
- Ref. [4]: why not start "Yoon" with an upper-case letter?
- Refs. [12,13,14,30,38,40,47,58,69,71,94,97,98,101,106,107,109,115,117,122]: upper-case names ("Fock", "Laughlin", "Hall", "Haldane", "Heisenberg", "Kennedy", "Lieb", "Tasaki", "Affleck", "Hamiltonian", "Wannier", "Fermi", "Liouvillian", "Ising", "Lindbladians", and "Hubbard").
- Refs. [26,45]: upper-case the acronym "IBM".
- Ref. [40]: upper-case "S" in "S=1".
- Ref. [41] is published as Phys. Rev. A 107, 032614 (2023).
- Ref. [55]: upper-case the acronym "AKLT".
- Ref. [63]: upper-case the acronym "NISQ".
- Ref. [73]: correct "$R_n$".
- Ref. [77]: upper-case the acronym "NP".
- Ref. [80]: correct "$L^1$".
- Ref. [81]: also upper-case "Carlo" even if "Monte Carlo" is the name of a place and not of a person.
- Ref. [85] is a duplicate of Ref. [59].
- Ref. [102]: the latest version has a different title. Thus, either update or cite a specific version.
- Ref. [104]: upper-case the acronym "PT".
- Ref. [130]: add correct volume and page numbers ("19, 752–759").
- 6 lines below Eq. (20): insert space between "FIG." and "3".
- Second line of Appendix C: add article "the" before "spin-1/2 basis".
- 4th line of Appendix C: remove spurious "on" before "are to".
- Add full stop at the end of Eq. (26). Same for Eqs. (29), (31), (32), (34), (35), (37), and (38).
- Break second line of Eq. (28) such that it fits. Same for Eqs. (36) and (38).
- Line above Eq. (33): "coefficients" (plural).
- 4th line of caption of Fig. 10: "hallow" $\to$ "hollow".
- 7th line of caption of Fig. 10: remove space between "Sec" and ".".
- Second line of text on page 33: insert article "the" before "Qiskit".
- 6th line of text on page 33: insert article "the" before "full calibration matrix".
- 7th line of text on page 33: insert article "the" before "IBM Q platform".
- 9th line of text on page 33: insert article "the" before "AKLT state".
- Last line of caption of Fig. 11: "is" $\to$ "are" (plural).
- New Fig. 12: Make sure that all fonts are sufficiently large such that the labels are readable in a printout.
- First line of footnote 8: insert "too" before "heavy".

I note that I am not a native English speaker either. My related comments are thus rather points to pay attention to during production.

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

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