Quantum Gross-Pitaevskii Equation

New results relevant for many particle physics of interacting 1d systems

Not enough evidence of the validity of the approach

Very difficult to read, can only be understood by the community dealing with matrix product states.

Length is not appropriate (too short)

The manuscript proposes a novel ansatz for mean-field approach to one
dimensional interacting bosons. The ansatz replaces a scalar Gross-Pitaevskii
wavefunction by a D-dimensional matrix-valued order parameter. This approach
is claimed to be reliable for treating strongly interacting regime of
Lieb-Liniger model. While I do not object the validity of this approach I feel
that its verification is somewhat insufficient. The provided calculation of
the density-density response function is not well explained and there is no
comparison with the results based on the exact solution. There is plenty of
such results in the literature either based on Luttinger liquid approach (for
all valued of interactions) or the mapping to free fermions (in the strongly
interacting regime). The authors should also explain how the dimensionality D
of the matrix-valued order parameter scales with the interactions and what is
the underlying physics.

In addition the paper is quite obscure and very difficult to read. The length
seems to be to short. I suggest incorporating the Supplementary material into
the main text and making the paper more self-contained otherwise it will only
be understood by specialists in the area of matrix product states. I recommend
major revision or resubmission to a more specialized journal.

See report above

1) Interesting result of possibly broad relevance, extending conventional (mean-field) method to deal with quantum effects.
2) Concisely and well written.

1) Some minor details need to be added (see changes).
2) Perhaps need to emphasize shorter-term applications of result (i.e. QBdG equations) given current challenges in numerical implementation of the QGPE.

The authors develop an interesting extension of the well-known Gross-Pitaevskii equation (GPE) within the context of 1D systems, the Quantum Gross-Pitaevskii equation (QGPE). This work is obviously of interest, given that whilst the GPE works excellently in describing many features of ultracold gases in 3D (where bosonic gases may form a BEC with long-range order), it is known to be invalid for 1D systems where in general there is a lack of long-range order (e.g., quasi-condensation of Bose gases in 1D) and the role of quantum correlations can be important (hence the breakdown of mean-field methods). To overcome these issues the authors develop the QGPE by a combination of the usual Dirac-Frenkel time-dependent variational principle with a cMPS (continuous matrix product state) ansatz for the wavefunction (which reduces to the usual mean-field variational ansatz under appropriate limits). Using the cMPS representation allows the QGPE to capture quantum correlations above the mean-field level, which are crucial for the description of 1D systems.

A large portion of the manuscript is devoted to the mathematical and technical basis of the QGPE, and further detail is appropriately given in the supplementary material. They also discuss that, importantly, the actual numerical implementation of the QGPE may be a non-trivial though not intractable problem. This is mainly due to the treatment of spatial derivatives (which occur within the nonlinear terms of the QGPE equation) and the appearance of low-rank matrices chracterising the boundaries of the system. Whilst not being specific, the authors do not claim that any of these problems are insurmountable in principle.

Given the current limitations of the QGPE, the authors discuss a useful implementation of the QGPE to derive a set of effective 'Quantum Bogoliubov-de Gennes equations' (QBdG). Whilst this section is concisely written, they demonstrate that the QBdG equations can be used to capture (known) physics beyond the scope of 1D mean-field methods. Specifically, they examine the response of a 1D system to a driven external potential. I would comment that, given this is the only illustrative application of the QGPE given in the manuscript (c/o the above discussed limitations of the full QGPE), that perhaps it be more emphasized within the introduction or abstract of the paper.

Lastly, the authors give a general outlook for the QGPE and possible avenues of utility. Here, I wonder if the authors could perhaps comment/speculate further on the - in principle - feasibility of the QGPE with respect to other known methods for treating the full quantum problem (in 1D). Specifically, the authors note that it would be interesting to compare the predictions of the QGPE to Ref. [59], which implements phase-space methods. Given that, in general, the various phase-space methods suffer from a range of `understood' problems such as truncation error and stability issues (beyond short-times), it would be interesting for the authors to comment whether they believe the QGPE would perhaps be a more valid/useful approach in some instances.

Overall, I believe the manuscript is well-written and presents a interesting result. I recommend publication with changes as detailed.

1) Mention that one can derive effective QBdG equations c/o the QGPE earlier in the manuscript, i.e. introduction.
2) Comment further on whether the QGPE could be a more effective approach for some systems in 1D than known methods (see above comment in report).
2) Fig. 1: \alpha is not defined explicitly in the text or caption. This should be done, particularly as currently it is effectively defined by a label in a subsequent plot.
3) Fig. 1: Similarly, \delta<\rho(x)> (= <\rho(x)> - \rho) is not explicitly defined in text or caption.
4) Fig. 1: There is a horizontal line in the lower plot of Fig. 1 which is never defined. I suspect this is the period of the density fluctuations. Either this should be commented on in the caption, labelled appropriately or removed.