Simulation of the 1d XY model on a quantum computer

In Eq.(13) we have removed the distinction between $i=j$ and $i \neq j$.

Under Eq.(25) we have added a more detailed explanation about the importance of quadratic Hamiltonians in physics and how to diagonalize them--> “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), we have exchanged the term “non-interacting form” by diagonal form.

We changed the first paragraph after Eq.(56) to make it more understandable -->”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. “

In page 25, after Eq.(70) we have added a paragraph to explain the reasons why we simulate the transverse magnetization instead of the typical order parameter in the XY model-->”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.”