Automated construction of $U(1)$-invariant matrix-product operators from graph representations

We thank the referee for his or her careful reading of the manuscript and for the positive assessment "the work is sound and that the results are interesting and will be of use to the MPS community." In the following, we address the issues raised. We have uploaded an updated version including these changes to the arXiv, where the manuscript is accessible under the identifier 1706.05338v2.

Referee:
"I believe the authors should be more careful with their notation. For example, works on tensor methods are often unavoidably awash in a sea of indices, however I found tracking the meaning of indices in this manuscript more difficult than usual. Some of this may be due to typos. In the following I give a list of places where I had issues with the notation."

We thank the referee for this remark. We have carefully taken care of the various points concerning notation and removed some typos. We hope the manuscript is clearer now.

"This work hinges on understanding the operation of finite state machines...."

We have now included a description of finite state machines and sketch the construction for a specific example (Hamiltonian of the XX model).

"The authors should do more to place their work in context with other recent work on efficient construction of generic MPOs, especially Hubig \emph{et al}. PRB 95, 035129 (2017). "

We now explain in more detail how our work relates to prior work in the end of section "Numerical behavior of the variance for antiferromagnetic S=1 Heisenberg chains". We state that the approach of Hubig et al is complementary to ours. The additional advantages of our FSM approach is that the MPO is generated from an abstract graph operation, so that the representation is exact, leading to the higher accuracy.
Note that to our understanding, Hubig et al. make use of non-abelean symmetries in their paper, so that it is not possible to directly compare to the bond dimensions used by us.

"Also in Sec. 5, it would be useful to know if the chain had open or periodic boundary conditions, and if the ground state was obtained using an iterative (i.e. sweeping) MPS method such as DMRG rather than "a standard Lanczos algorithm". At the end of the second paragraph of that section the authors refer to "perfect linear scaling", but the choice of axes in Fig. 8b makes this far from clear."

We apply open boundary conditions, and the ground state is obtained by sweeping through the lattice and locally optimizing the site tensors using a Lanczos algorithm. Both is now stated in the text.
We now also show linear fits in Fig. 8b, which clearly demonstrate the linear scaling.

"In the conclusion the authors refer to their algorithm as "optimized". It might be worth saying what quantity is extremal here?"
In the main text we now address this point - the 'optimization' refers to minimizing the number of floating point operations when calculating expectation values of operator products, as needed in the evaluation of the variance.

"Finally there are some other typos:"
We have corrected typos throughout the manuscript.

Requested changes:
All requested changes are included.

We have included several changes throughout the paper in reply to the referee's request, as detailed in our reply above. In more detail, the major changes are:

- We have added numbers to the equations, unified the notation of the tensors, and corrected typos throughout the paper.

- Figs. 1 and 2 have been updated for clarity

- Sec. 3.1 / Fig. 3 provide the example of constructing a FSM for the XX model

- We modified the discussion in Sec. 5 to better relate to the work of Hubig et al.