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Measurement-induced entanglement phase transitions in variational quantum circuits

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

Submission summary

As Contributors: Roeland Wiersema
Preprint link: scipost_202206_00017v1
Date submitted: 2022-06-18 17:33
Submitted by: Wiersema, Roeland
Submitted to: SciPost Physics
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.

Current status:
Editor-in-charge assigned

Submission & Refereeing History

Reports on this Submission

Anonymous Report 2 on 2022-7-28 (Invited Report)


The aim of this paper is to make a connection between measurement-induced entanglement phase transitions in random circuits and variational quantum circuits proposed for quantum machine learning. The authors investigate the possibility of using measurements, which reduce entanglement in the circuits, to avoid the barren plateau problem. They propose numerical evidence for such a connection. This is a very interesting direction and the paper is generally well-written and interesting.

However, I would also like to see a clarification of the issue pointed out by the other referee, namely Equation 4 and the physical meaning of the derivatives being calculated.

The authors discuss the dependence of the output mixed state upon parameters theta. However, this mixed state, which is averaged over measurement outcomes, is well known to be insensitive to the measurement-induced entanglement transition.

After Equation 4 the authors write rho(theta) as a function of p^i and rho^i(theta). But p^i is also a function of theta. Is p^i also differentiated in Equation 4? If not, what is the physical meaning of this derivative?

Other specific comments:

Although the circuits discussed here are termed “variational”, the parameters are not optimized. Instead they are sampled uniformly, giving a random circuit ensemble, which the authors envisage as a starting point for a variational calculation. This is a potential source of confusion for the reader, so it would be useful to have a clarifying comment early on.

Page 2-3 It is stated that the HVA circuit is integrable: clarify exactly what is integrable (e.g. is it the unitary circuit for arbitrary parameter values but without measurements? or the unitary circuit with particular parameter values?)

Page 2 mapping to 2 dimensional percolation was described in ref 5 (this ref also relevant to “steady state” entanglement just below).

Page 3 it is stated that measurements are sampled uniformly. Does this mean that the standard quantum mechanical measurement probability is not used?

One of the main claims of the paper is that Fig 3 shows a transition that coincides with the measurement induced entanglement transition. However, an independent determination of a transition point (or bounds on such a point) from the data in Fig 3, is lacking.

More general comments:

Do the authors see applications of the phase transition in ensembles where the parameters are in fact variationally optimized?

The authors propose using these measurement circuits as a practical tool for variational optimization of a wavefunction. However the measurement makes the output wavefunction stochastic, as they discuss. This stochasticity may be harmful for targeting a particular state. Therefore it may be useful to also include feedback based on the measurement outcomes. E.g. it is known that some topologically ordered states can be prepared deterministically by shallow depth circuits if measurement and feedback is allowed.

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

Anonymous Report 1 on 2022-6-29 (Invited Report)


- original idea
- well motivated
- can stimulate novel work


- cryptic presentation
- a bit of overselling
- no effort in bridging between readerships


The work by Wiersema et al. attempts a connection between MIPT and variational algorithms. Their research appears well justified and poses a number of stimulating questions for the future of the field of digital quantum simulators. However, I cannot recommend the paper for publication in the current form for a number of presentation issues, that are borderline with hindering understanding, as I detail in the following. In order to provide constructive criticism, I should say that most of the issues would be probably solved by formatting the paper in a style more SciPost-friendly. This appears to me a Letter, as the authors themselves write at page 1 of the manuscript.

To start with, the introduction could be improved to a large extent. There are a number of sentences which appear too colloquial. Examples range from poor effort in connecting blocks: ‘Another important element ..’, ‘however, this is not the case’,
to statements that do not appear solid: what does it mean ‘entanglement is destroyed globally’? to the motivation given at page 1: ‘also take place in circuits of practical interest…machine learning’. The work done by the authors is to look at quantities interesting for variational algorithms, not really changing approach by looking at new circuits, as I discuss in the following. All of this may appear cosmetics but it accumulates through the manuscript and it leaves a bad feeling to the reader.

The motivation given at page 2 (first column) is instead way better summarized, yet the authors should have now the space to elaborate a bit more on barren plateaux, connecting the reader not familiar with the topic. It is hard for me to follow that part, besides getting a sense of its relevance.

From Fig 2 it appears that the authors are not solving for Floquet dynamics of an interacting integrable model interspersed with measurements (what they claim is an open question). Instead, they alternate XX, YY, XX gates. This seems very similar in spirit to what several other authors have done with Clifford circuits et similia, and indeed the authors do find analogue results.
Again, this would have been a less critical report if the related statement at the beginning of page 3 had been milder.

Finally, I appreciate the results in Fig 3, but they are presented too fast and way too cryptical. Which observable is O? why parameters thata_l are shifted? Also, am I lost or Eq 4 is linear in the density matrix? if so, how it can lnow about the MIPT? this is the whole point about such field. Perhaps, I am just confused by the too compact presentation, but these aspects should be definitely amended in the next version of the paper. In the same block, there is a typo at page 4: Fig 2 does not discuss circuit settings, but data collapse for entanglement entropies. Another hint that perhaps the authors should invest more time in the presentation of their results.

Requested changes

- enlarge the manuscript
- downplay some statements
- expand significantly the original part of the manuscript (currently page 4)
- address my perplexities on Eq 4

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

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