Simulation of the 1d XY model on a quantum computer

We are thankful for the referee’s detailed report, and we consider that it has helped to improve the readability of the new manuscript version as well as clarify some of its parts.

Comments on the text:
“ Eq.(13) holds also trivially for i=j, so please remove "if i≠j"”
We sincerely appreciate the referee's suggestion. We acknowledge that distinguishing between the cases $i \eq j$ and $i \neq j$ is redundant. We have therefore removed this distinction in the revised version.


“Below Eq. (25) it might be worth to add that quadratic fermionic Hamiltonians are also naturally appearing in the mean-field treatment of more complicated systems (i.e., their realm of relevance is even wider than genuinely non-interacting models).”
“Next sentence: fFT is only useful for translational invariant systems, whereas quadratic Hamiltonians are solvable exactly for whatever set of spatially dependent couplings (via Bogolubov Transformation and/or Fermionic Gaussian States). Excellent reviews are available, also on Scipost itself (and elsewhere, of course: Tagliacozzo and Santoro are just the first two senior authors that come to mind in this respect). Please address the reader correctly.”
We thank the referee for pointing out this confusion in the text. We have addressed both issues in only one paragraph --> “Hamiltonians that are quadratic in fermionic creation and annihilation operators are ubiquitous in condensed matter physics. They describe systems of non-interacting fermions and also arise in the mean-field treatment of more complex interacting systems. Diagonalizing such Hamiltonians is a well-established procedure, typically accomplished using spatially dependent couplings and techniques such as the Bogoliubov transformation Ref.[Phys. Rev. B 53, 8486 (1996)] Ref.[J. Stat. Mech. 2011, P07015 (2011)] and fermionic Gaussian states Ref.[SciPost Phys. Lect. Notes 54 (2022),]. In translationally invariant models, such as the XY model, the Fourier transform is particularly useful, as it partially diagonalizes the Hamiltonian by making it local in momentum space. However, this transformation often introduces anomalous terms that couple different momentum modes, thus requiring a subsequent Bogoliubov transformation to achieve full diagonalization.”


“ Sentence before Eq. (52): the H_XY Hamiltonian is already non-interacting! Maybe the Authors intended to say "convert into its diagonal form"?”
We have corrected the mistake.


“P14L276: I would have thought that \ket{\pm} was a very common notation for (\ket{0} \pm \ket{1})/\sqrt{2}, what do the Authors mean here? I do not get the reasoning about \ket{0}... I am sure to overlook something easy that could be conveyed by rephrasing the sentence.”
We thank the referee for pointing out the potential confusion. We have revised the paragraph to make it clearer and less ambiguous--> “A second issue concerns a discrepancy in notation. In quantum computing, the spin states that are eigenstates of $sigma_z$ with positive and negative eigenvalues are conventionally denoted as $\ket{\uparrow} = \ket{0}$ and $\ket{\downarrow} = \ket{1}$, respectively. In contrast, in many-body physics, the symbol $\ket{0}$ (or sometimes $\ket{\Omega}$) typically denotes the vacuum state. Since the Jordan-Wigner maps $\ket{\downarrow}$ into $\ket{\Omega}$, an $X$ gate has been introduced to keep the standard convention and avoid potential confusion. As a result, the circuit is initialized with a layer of $X$ gates applied to each qubit. “


“Figure 18: The thing that was (and is still potentially) confusing is that the plot displays M_z, which is not the order parameter for the phase transition, and not M_x (or M_x^2), which instead is... but it is all fine, as long as the point is made clear in text. It escaped my attention in the first version, therefore I raised the issue back then. But how difficult and meaningful it is to plot also the more common order parameter itself?”
We thank the referee for pointing out the potential confusion regarding the choice of order parameter. Our initial focus was on analyzing and comparing with previous works, specifically Ref. [Phys. Rev. A 79, 032316 (2009)] and Ref. [Quantum 2, 114 (2018)], which primarily considers the expectation value of M_z. Following their approach, we did not initially examine the behavior of M_x or M_x^2 , which are the order parameters in the standard XY model.
Furthermore, the Hamiltonian we simulate includes boundary conditions that differ from the standard ones, and it only reduces to the conventional XY model in the thermodynamic limit (n->inf). For small spin chains, such as those we study, it is not trivial to assume that M_x remains a valid order parameter, and a more careful analysis would be required to justify this assumption.
Given that our simulations focus on small spin chains (e.g., n=4,n=8), our goal was not to characterize the order parameter in detail. Instead, we found it more illustrative to show that the circuit captures distinct ground state behavior as a function of the circuit parameters.
Finally, we would like to emphasize that our approach allows for the straightforward computation of various expectation values. For instance, M_x can be obtained as easily as M_z, simply by applying a Hadamard gate to each qubit before measurement to rotate the basis. Expectation values of M_x^2 can also be computed efficiently, and we indeed use such techniques to evaluate the ground state energy.
To clarify this in the text, we have added the following paragraph on page 25 before chapter 4 (after the analytical value of the transverse magnetization): Notice that $M_z$ is not the conventional order parameter for the XY model [Refs. 17, 18]. Nevertheless, we choose to compute its expectation value for several reasons. First, it enables direct comparison with previous studies on the topic Ref.[10,11]. Second, although not the standard order parameter, $\langle M_z \rangle$ still captures the qualitative change in the ground state across the quantum phase transition. Finally, the Hamiltonian used in this work includes non-trivial boundary conditions which, while negligible in the thermodynamic limit, significantly affect the physics in finite-size systems. For small spin chains, such as those considered here, it is not evident that $M_x$ remains a valid order parameter, and a more detailed analysis would be required to justify its use in this context.