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Adaptive variational quantum minimally entangled typical thermal states for finite temperature simulations

by João C. Getelina, Niladri Gomes, Thomas Iadecola, Peter P. Orth, Yong-Xin Yao

This Submission thread is now published as

Submission summary

Authors (as registered SciPost users): Yongxin Yao
Submission information
Preprint Link:  (pdf)
Date accepted: 2023-07-20
Date submitted: 2023-05-24 14:09
Submitted by: Yao, Yongxin
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Computational
  • Quantum Physics
  • Statistical and Soft Matter Physics
Approaches: Theoretical, Computational


Scalable quantum algorithms for the simulation of quantum many-body systems in thermal equilibrium are important for predicting properties of quantum matter at finite temperatures. Here we describe and benchmark a quantum computing version of the minimally entangled typical thermal states (METTS) algorithm for which we adopt an adaptive variational approach to perform the required quantum imaginary time evolution. The algorithm, which we name AVQMETTS, dynamically generates compact and problem-specific quantum circuits, which are suitable for noisy intermediate-scale quantum (NISQ) hardware. We benchmark AVQMETTS on statevector simulators and perform thermal energy calculations of integrable and nonintegrable quantum spin models in one and two dimensions and demonstrate an approximately linear system-size scaling of the circuit complexity. We further map out the finite-temperature phase transition line of the two-dimensional transverse field Ising model. Finally, we study the impact of noise on AVQMETTS calculations using a phenomenological noise model.

Author comments upon resubmission

Dear Editor,

Many thanks for returning the reviews of our paper entitled "Adaptive variational quantum minimally entangled typical thermal states for finite temperature simulations.'' The reviewer in the invited report recommends our manuscript for publication with minor concerns to be addressed. The reviewer in the contributed report finds our work interesting, and suggests noisy simulations of a minimal model to be added in the manuscript.

To address the questions we have revised our manuscript to include
(a) A new section (VIII) for AVQMETTS simulations with noise model.
(b) Discussions to clarify some details of AVQITE, typical pure state-based approaches, among others.

We are confident that these substantial improvements address all the comments and questions. We therefore would like to seek further consideration of our work in SciPost Physics.

We look forward to hearing from you soon.

Yours sincerely,
Yongxin Yao on behalf of all the authors

Notice: This work was produced by Iowa State University under Contract No. DE-AC02CH11358 with the U.S. Department of Energy. Publisher acknowledges the U.S. Government license and the provision to provide public access under the DOE Public Access Plan (

List of changes

Response to the Invited Report

Referee's comment: The authors refer to previous work for details on the algorithm and only briefly describe it without much depth. This would be fair if it wasn't because the method is quite new, leaving room for confusion. For instance, a naive question would be: Does the quantum circuit have to be optimized for each initial random state? I assume that's the case but, otherwise, if the circuit is obtained before hand, how is it done? Can the authors please clarify the necessary points to make this more clear?

Reply: It is correct that each different initial random state, specifically the classical product state in this study, requires a unique AVQITE calculation. However, reusing AVQITE circuits is also feasible and helpful to reduce the quantum resource demand for AVQMETTS because of the duplicated CPSs during calculations.

To spell out the point clearly, we inserted the following sentence during the first exposition of AVQITE in Sec. II: ``\textcolor{blue}{We note that the resulting AVQITE circuit is problem-specific and tied to the initial state $\ket{i}$.}'' We also added the following lines in the description of AVQMETTS flowchart:``\textcolor{blue}{As the AVQITE circuit is associated with the initial state, each distinct CPS requires a unique AVQITE calculation. However, reusing AVQITE circuits is also feasible and can help minimize the quantum resource demand for AVQMETTS calculations. This is due to the fact that a CPS obtained from state collapse after a thermal step may be identical to one that was sampled in a previous step due to the inherent structure of the distribution of METTS. In the numerical simulations reported below, we observe $10-60\%$ of CPSs are sampled for more then once, depending on the system size and temperature.}''

Referee's comment: It is not obvious why the algorithm should use initial product states (hence, METTS). Wouldn't it equally work with any initial random state? See, for instance, the approach referred-to as canonical thermal pure quantum states''(CTPQS) (doi=``10.1007/978-981-10-1506-9\_3'') and references therein. Or is it that one uses initial product states to facilitate the implementation in a quantum device? On the other hand: if one starts from an (entangled) initial random state, how will it affect the depth of the circuit and the performance of the method?

Reply: Indeed, we use initial product states to facilitate the implementation in a quantum device. Furthermore, the circuit complexity for METTS preparation is bounded by that of the ground state, which has area-law entanglement (up to logarithmic violations for gapless systems), in contrast to the volume-law entanglement of a generic random state. The preparation of a truly random state is exponentially hard on quantum computers, hence quantum algorithms utilizing such states are not scalable.

We added the following lines in the Introduction section to clarify the point: ``\textcolor{blue}{We note that, while it might seem straightforward and appealing to leverage the statistical approach based on TPQ states in a quantum algorithm, the necessary initial step of random state preparation is known to be exponentially hard on quantum computers~[40].}''

Response to the Contributed Report

Referee's comment: For any model on a noisy quantum computer, a critical factor of robust simulations is the depth of a quantum circuit. Generally, we focus on the number of CX gates. Such things are shown in FIG3 and FIG4 of this work. Here, an important point is that the number of CX gates required by the authors’ method is about hundreds, which is not tolerable for the present noisy quantum simulator, e.g., IBM Quantum device. In this case, the authors need to show how to use optimization methods to compress their circuit to a safe depth.

Reply: The number of CNOT gates for circuit simulations of quantum systems will generally grow with system size. Our work demonstrates a favorable approximately linear system-size scaling for the AVQMETTS calculations of the studied spin models. The AVQITE algorithm we use to prepare METTS has been demonstrated to generate highly compact problem-specific ground state ans\"atze comparable to other state-of-the-art approach such as qubit-ADAPT-VQE. We agree that executing circuits with hundreds of CNOTs is challenging on present hardware, but not impossible given the rapid development of quantum computing technology. For instance, a recent publication (Y. Kim, et al, Nature Physics 19, 752–759 (2023) from the IBM group has demonstrated hardware calculations with circuits containing up to 1080 CNOTs.

We added the following line in the concluding remarks: ``\textcolor{blue}{Recently, practical error mitigation techniques such as zero-noise extrapolation have been demonstrated on quantum hardware to scale to circuits containing up to $26$ qubits and $1080$ CNOTs~[85,86].}''

Referee's comment: For the models shown in FIG1, the authors consider periodic boundary conditions (PBCs). Here, I suppose that the authors need to take the geometry of real noisy quantum simulators into account and show how to realize a model under PBCs based on a next-nearest device.

Reply: For digital quantum computers with restricted qubit connectivity, such as the superconducting transmon qubit-based processors, any nonlocal gate can still be implemented with the aid of a sequence of swap gates, because single-qubit gates plus the CNOT gates between nearest neighbors constitute a universal complete gate set. Therefore, except for some overhead of swap gates, the models with PBC can be simulated on digital quantum hardware. Moreover, in some circumstances the swap overhead is unnecessary. For example, for 1D models many devices (such as IBM QPUs with heavy-hexagon connectivity) have a sublattice matching the connectivity requirements of PBC. More generally, some quantum computing platforms (e.g. trapped ions, see below) have all-to-all connectivity.

Referee's comment: For the operation for imaginary-time evolution $U_\text{AVQITE}$, the authors consider the decomposition based on all-to-all qubit connectivity. I do not think any present noisy quantum simulator supports this kind of geometry. If the authors would like to convince that their method is suitable for the present quantum device, they need to do the decomposition of their pseudo-Trotter step as Eq. 4 in terms of the special geometry of today’s quantum device.

Reply: Trapped-ion quantum computers, such as Quantinuum’s System Model H1 and H2, allow to reorder and reconfigure the chains of ions within the architecture, enabling all-to-all connectivity. For the sake of simplicity, our CNOT-gate analysis is based on hardware with all-to-all connectivity. Some difference is expected when the analysis is switched to specific superconducting qubit devices; however, similar leading behaviour of system-size scaling should still hold. We updated the following line in the caption of Fig.2: ``\textcolor{blue}{$N_\text{cx}$ is calculated assuming all-to-all qubit connectivity as in trapped-ion quantum processors...}''

Referee's comment: In sum, There is a gap between the noiseless and noisy simulation. To faithfully convince the community, noisy simulations of a minimal case might be needed. According to the above reasons, I cannot recommend this manuscript for publication and think the authors need to address a critical question why their algorithm is suitable for
the current-day quantum device.

Reply: To address the referee's comment, we added a new section VIII on noisy AVQMETTS simulations of a $N=2\times 3$ TFIM. In the noisy simulation, we consider a realistic noise model and investigate the impact of noise on the thermal energies. We refer the referee to the main text for the details.

Published as SciPost Phys. 15, 102 (2023)

Reports on this Submission

Anonymous Report 2 on 2023-6-21 (Invited Report)


The authors have significantly improved the manuscript and addressed all the issues raised in my original report. I therefore recommend it for publication as is.

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

Anonymous Report 1 on 2023-5-30 (Invited Report)


In this revision, the authors have worked on my comments carefully. In particular, they consider a realistic noise model and investigate the noise on simulations. The paper has significantly improved. Thus, I now recommend it for publication.

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

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