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Measurement-induced entanglement phase transitions in variational quantum circuits

by Roeland Wiersema, Cunlu Zhou, Juan Felipe Carrasquilla, Yong Baek Kim

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Submission summary

Authors (as registered SciPost users): Roeland Wiersema
Submission information
Preprint Link: scipost_202206_00017v2  (pdf)
Date submitted: 2022-12-07 22:52
Submitted by: Wiersema, Roeland
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Quantum Physics
Approach: Computational


Variational quantum algorithms (VQAs), which classically optimize a parametrized quantum circuit to solve a computational task, promise to advance our understanding of quantum many-body systems and improve machine learning algorithms using near-term quantum computers. Prominent challenges associated with this family of quantum-classical hybrid algorithms are the control of quantum entanglement and quantum gradients linked to their classical optimization. Known as the barren plateau phenomenon, these quantum gradients may rapidly vanish in the presence of volume-law entanglement growth, which poses a serious obstacle to the practical utility of VQAs. Inspired by recent studies of measurement-induced entanglement transition in random circuits, we investigate the entanglement transition in variational quantum circuits endowed with intermediate projective measurements. Considering the Hamiltonian Variational Ansatz (HVA) for the XXZ model and the Hardware Efficient Ansatz (HEA), we observe a measurement-induced entanglement transition from volume-law to area-law with increasing measurement rate. Moreover, we provide evidence that the transition belongs to the same universality class of random unitary circuits. Importantly, the transition coincides with a “landscape transition” from severe to mild/no barren plateaus in the classical optimization. Our work paves an avenue for greatly improving the trainability of quantum circuits by incorporating intermediate measurement protocols in currently available quantum hardware.

Author comments upon resubmission

We want to thank the referees for their careful review and the suggestions for improving the manuscript. In
addition to addressing their comments we have rewritten a large portion of the original paper.

List of changes

1. Restructured the text, expanded some of the sections and adapted the formatting to be suitable for a
SciPost article.
2. Added Appendix D where we propose a possible practical algorithm that uses intermediate measurement
in an optimization setting.
3. Added Appendix E where we perform a finite-scaling analysis of the critical exponents through the
gradient variance.
4. Added Figure 1 to illustrate the general measurement setup we consider to support the mathematical
exposition in the main text and appendix.

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 2 on 2022-12-15 (Invited Report)


The authors appear to have made a serious attempt to address the comments of the referees and to take better account of the challenges in relating the transition they discuss to practical algorithms.

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Anonymous Report 1 on 2022-12-12 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202206_00017v2, delivered 2022-12-12, doi: 10.21468/SciPost.Report.6291


1- clear goal
2- well presented
3- intriguing bridge between two communities


1- unclear whether the findings of the paper can really have useful impact on q. algorithms


The authors have seriously considered my remarks, rewriting large part of the paper. The manuscript does indeed address a topic suited to SciPost standards and I am happy to recommend the paper for acceptance.

I have one last important physical question: the volume law barrier the authors talk about may have in my opinion little to do with MIPTs. The latter relies on post-selecting trajectories, and without this procedure, the asymptotic state is an infinite temperature one, with boring entanglement.
Otherwise, if MIPTs could be relevant for their variational algorithms, then we would have found a remarkable application of measurement induced transitions, which are very elusive as the authors know.

Could they comment on this issue? I think it is important since it seems to undermine a bit their original motivation

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Author:  Roeland Wiersema  on 2022-12-21  [id 3172]

(in reply to Report 1 on 2022-12-12)

We agree that the MIPT relies on the post-selected trajectories which are hard to access in a variational algorithm. We concede this point just above Section 5 after exploring what a variational algorithm that includes remixed trajectories could look like in appendix D. As mentioned by the referee, such an algorithm would effectively be starting with with an infinite temperature state, with rather boring entanglement that would make optimization difficult. Hence although the original motivation for this work was to indeed find a way to utilize the MIPT in a variational algorithm, we found that this may be difficult in practice.

However, we believe that our numerical results for the barren plateau effect with respect to the individual trajectories of the measured circuits support the idea that the trainability of variational algorithms is intricately linked to the amount of entanglement produced in the circuit. We therefore hope that our work serves as further motivation to investigate this connection in more detail.

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