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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, YongXin Yao
This Submission thread is now published as
Submission summary
Authors (as registered SciPost users):  Yongxin Yao 
Submission information  

Preprint Link:  https://arxiv.org/abs/2301.02592v2 (pdf) 
Date accepted:  20230720 
Date submitted:  20230524 14:09 
Submitted by:  Yao, Yongxin 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
Scalable quantum algorithms for the simulation of quantum manybody 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 problemspecific quantum circuits, which are suitable for noisy intermediatescale 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 systemsize scaling of the circuit complexity. We further map out the finitetemperature phase transition line of the twodimensional transverse field Ising model. Finally, we study the impact of noise on AVQMETTS calculations using a phenomenological noise model.
Author comments upon resubmission
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 statebased 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. DEAC02CH11358 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 (http://energy.gov/downloads/doepublicaccessplan).
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 problemspecific 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 $1060\%$ 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 referredto as canonical thermal pure quantum states''(CTPQS) (doi=``10.1007/9789811015069\_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 arealaw entanglement (up to logarithmic violations for gapless systems), in contrast to the volumelaw 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 systemsize 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 problemspecific ground state ans\"atze comparable to other stateoftheart approach such as qubitADAPTVQE. 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 zeronoise 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 nextnearest device.
Reply: For digital quantum computers with restricted qubit connectivity, such as the superconducting transmon qubitbased processors, any nonlocal gate can still be implemented with the aid of a sequence of swap gates, because singlequbit 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 heavyhexagon connectivity) have a sublattice matching the connectivity requirements of PBC. More generally, some quantum computing platforms (e.g. trapped ions, see below) have alltoall connectivity.
Referee's comment: For the operation for imaginarytime evolution $U_\text{AVQITE}$, the authors consider the decomposition based on alltoall 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 pseudoTrotter step as Eq. 4 in terms of the special geometry of today’s quantum device.
Reply: Trappedion quantum computers, such as Quantinuum’s System Model H1 and H2, allow to reorder and reconfigure the chains of ions within the architecture, enabling alltoall connectivity. For the sake of simplicity, our CNOTgate analysis is based on hardware with alltoall connectivity. Some difference is expected when the analysis is switched to specific superconducting qubit devices; however, similar leading behaviour of systemsize scaling should still hold. We updated the following line in the caption of Fig.2: ``\textcolor{blue}{$N_\text{cx}$ is calculated assuming alltoall qubit connectivity as in trappedion 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 currentday 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)