Understanding and utilizing the inner bonds of process tensors

We thank the Referee for this remark. Indeed, the two references are concerned with how to define or measure thermodynamic quantities in the quantum setting without explicit reference to strong coupling. We have added a statement as well as a few references about the importance of methods that can treat strong system-environment couplings.

We thank the Referee for this interesting question. Indeed, we are not aware of any formal and useful (tight) bound for the error on any physical observable that is directly related to PT-MPO compression strategies and parameters.
Physical observables are obtained by eventually contracting the PT-MPO with a set of system propagators. PT-MPO are however designed to be independent of the particular choice of system propagators. One could in principle consider an adapted metric based on, e.g., a certain statistical distribution of system propagators. In this sense, the choice of considering the overall norm distance is unbiased and treats every possible system intervention with the same right.

Because of their novelty, error propagation in MPOs in the time domain are much less well understood than MPO representations in many-body physics. While analytical insights are lacking, one can turn to numerical experiments. For instance, in Appendix B of [https://arxiv.org/pdf/2405.16548] it was found that convergence of PT-MPOs has very similar trends whether one measures it via the PT-MPOs norm distance or one considers the changes in a concrete system observable. This suggests that minimising the norm distance is indeed a sensible criterion for PT-MPO compression.

We now provide more details on the rationale behind our choice of PT-MPO compression in Sec. III.B.

We thank the Referee for their positive evaluation.

We are grateful to the Referee for their report and for raising this question. We agree that, as suggested in https://arxiv.org/pdf/1008.3477, for MPO compression with SVDs to be locally optimal, it would be necessary to first bring the MPO into the proper canonical form, which can be done, e.g., by sweeping with SVDs while not truncating in the forward sweep.

However, here we instead utilise a much faster, and whilst slightly suboptimal, yet entirely adequate strategy. This builds on the zip-up algorithm for MPS-MPO or MPO-MPO contractions by Stoudenmire and White [New J. Phys. 12, 055026 (2010)], which is also discussed in the review article mentioned by the Referee. Specifically, performing truncation already during forward sweeps leads to a significantly faster algorithm in practice. While it is not guaranteed to lead to optimal compression---in the sense that it produces minimal bond dimensions for a given accuracy---the results are nonetheless often found to be close to optimal. Additional errors incurred by this sub-optimality can be compensated by using a smaller compression threshold, yielding slightly larger bond dimensions. Overall, considering the trade-off between accuracy and computation time, the zip-up approach typically outperforms the nominally optimal compression, and has thus become the most common technique employed for PT-MPOs, such as in the first constructive PT-MPO algorithm for spin-boson models by Jorgensen and Pollock [Phys. Rev. Lett. 123, 240602 (2019)]. Another example is the precursor to PT-MPO methods, the TEMPO algorithm by Strathearn et al. [Nature Communications 9, 3322 (2018)].

The reason for the speed-up is that typical SVD routines require O(M^2N) operations, where M and N are the smaller and larger dimension of the matrix to be decomposed, respectively. When combining two MPOs with inner bond dimensions chi_1 and chi_2, for SVDs in the non-truncating sweep, one would have N=M=chi_1*chi_2. By contrast, the zip-up scheme involves SVDs on matrices where the small dimension M is already of the order of the inner bond dimension after compression chi. Thus, a relative speed-up by a factor of (chi_1*chi_2/chi)^2 is achieved.

It should be noted that the speed-up is so dramatic that, in most of our examples, convergence is out of reach if we do not use compression already in the forward sweep. However, it is possible to interpolate between nominally optimal compression and the zip-up approach, by using a smaller threshold for the forward sweep compared to the backward sweep. This is discussed in detail in Appendix A of Cygorek et al. [Phys. Rev. X 14, 011010 (2024)]. This also provides a practical solution strategy if suboptimal compression should ever result in issues for extracting environment observables.

Regarding the need for sweeps in the backward direction after the compression in the forward sweep, we point out that this is typically not only a gauge transformation. As also discussed in https://arxiv.org/pdf/1008.3477, even starting from the canonical form, truncating sweeps with SVDs are only locally optimal but truncation later in the chain is not independent of truncation earlier in the chain. Therefore, in many-body physics, one finds it necessary to sweep multiple times or to employ additional, e.g., variational techniques. Moreover, even the gauge transformation is useful, as it preconditions the MPO to a form much closer to the canonical form required for the next forward sweep. One way to see the importance of the backward sweep is that only when it is included, can one interpolate between nominally optimal compression and the zip-up scheme as described above.

We now discuss these subtle points in detail in Sec. III.B.