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

1- offers o new original approach to simulate the imaginary time evolution of quantum many-body systems on quantum hardware.
2- it can be applied to systems with sign problem and in 2D

1- The relative errors are not necessarily small and the method may not represent a killer app; i.e. it's not state of the art. (still, nice progress in the right direction).

The authors discuss a generalization of the previously developed AVQITE time evolution algorithm to the case of minimally entangled typical thermal states. They test their newly minted AVQMETTS method through numerical simulations in (small) 1D and 2D systems and they offer a number of benchmarks to characterize its performance. The idea is original and holds potential for applications in NISQ devices. I therefore recommend it for publication after some minor concerns are addressed:

1- 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?

2-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?

See point 1 in the report above.

Fidelity (nor infidelity) are never defined in the text, and are only defined in the caption of Fig. 2. It would better serve the reader to have it defined in the corresponding place in the text.

Thanks to the authors for this interesting work. Indeed, implementing imaginary-time evolution on the quantum circuit has been a challenging task for the community. In this work, the authors propose the AVQITE algorithm, which can help to measure the thermal expectation value of an observable and indicate that this algorithm can be implemented on noisy quantum simulators. However, I worry that it can be difficult to implement the authors’ method on the present quantum computer. In this case, I do not recommend the publishment of this work in SciPost Physics, unless the authors show that it is promising to carry their results on the real noisy quantum simulator. My reasons are listed as follows

(a) 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.
(b) 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.
(c) For the operation for imaginary-time evolution U_{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.

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.