SciPost Submission Page
Simulation of the 1d XY model on a quantum computer
by Marc Farreras, Alba Cervera-Lierta
Submission summary
| Authors (as registered SciPost users): | Marc Farreras |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2410.21143v2 (pdf) |
| Code repository: | https://github.com/Marc-Farreras/XYQSimulation |
| Date submitted: | 2025-03-12 12:27 |
| Submitted by: | Farreras, Marc |
| Submitted to: | SciPost Physics Lecture Notes |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
The field of quantum computing has grown fast in recent years, both in theoretical advancements and the practical construction of quantum computers. These computers were initially proposed, among other reasons, to efficiently simulate and comprehend the complexities of quantum physics. In this paper, we present a comprehensive scheme for the exact simulation of the 1-D XY model on a quantum computer. We successfully diagonalize the proposed Hamiltonian, enabling access to the complete energy spectrum. Furthermore, we propose a novel approach to design a quantum circuit to perform exact time evolution. Among all the possibilities this opens, we compute the ground and excited state energies for the symmetric XY model with spin chains of n = 4 and n =8 spins. Further, we calculate the expected value of transverse magnetization for the ground state in the transverse Ising model. Both studies allow the observation of a quantum phase transition from an antiferromagnetic to a paramagnetic state. Additionally, we have simulated the time evolution of the state all spins up in the transverse Ising model. The scalability and high performance of our algorithm make it an ideal candidate for benchmarking purposes, while also laying the foundation for simulating other integrable models on quantum computers.
List of changes
Correction paper:
“In this paper, we present the comprehensive scheme”--> “In this paper, we present a comprehensive scheme”
In page 2 of the old version ( page 2 of the new version), the first paragraph, we have changed the last 5 lines “In this context, it has become ... to avoid the large accumulation of noise.” To contextualize better our work and include some references pointed by the referees.
“This paper presents a circuit specifically designed”-->”This paper presents a circuit suitable for the NISQ era”
“These transitions occur at absolute zero”--> “These transitions occur at zero temperature”
“spin leather operator”-->”spin ladder operator”
“The quadratic Hamiltonian in fermionic annihilation and creation operators appears in more condensed matter systems notably exemplified in the Hubbard model [17]. Diagonalizing this type of Hamiltonian is a well-established procedure, leading us to the subsequent phase: the fermionic Fourier transform (fFT).”-->”Hamiltonians quadratic in fermionic annihilation and creation operators are ubiquitous in condensed matter systems, describing systems of free fermionic particles. Diagonalizing this type of Hamiltonian is a well-established process, achieved through the fermionic Fourier transform (fFT).”
In page 12 of the old version (page 14 in the new one) we have changed the first paragraph “The second issue pertains to a notational problem. In conventional … different from the description provided in this work, although the final result should remain unchanged.” This change has been done with the aim of making the text clear and less confusing.
In page 13 of the old version ( page 15 of the new one), we have changed the first paragraph “At this point, we have understood the interplay … creation operators in the Fourier Transform definition.” This change has been done with the aim of making the text clear and less confusing.
In page 13 of the old version ( page 15 of the new one), we have changed the last paragraph “Up to this point, we have found … additional fermionic SWAPS become necessary.” This change has been done with the aim of making the text clear and less confusing.
In Figure 3. We have changed the footnote following the indications of the referee to improve the grammar and structure of the text.
In page 15 of the old version(page 18 of the new version), we have changed the paragraph in the subsection n/2 Fermionic Fourier Transform. The change aims to clarify some possible confusions that could arise in the old version.
In page 18 Of the old version (page 22 of the new version), we have changed the first paragraph “The circuit scheme involves the decoupling of k and -k Fourier modes … states needed for the Bogoulibov gates.” The change aims to clarify some possible confusions that could arise in the old version.
In page 23 of the old version (page 29 of the new version), we have added in the first paragraph of the section Results and discussion the phrase: “The results show the classical simulation using the quantum computing library Qibo [Ref], for the spin chain n=4 and n = 8 using the circuits represented in Figs.13, 14, and 15.” The aim of this change is to clarify that the results of the simulation are done by a classical simulator and not by a real quantum computer.
In the conclusions of the new version, we have added one paragraph in the end, expanding more about possible uses of this circuit, different than benchmarking, specifically how it can be used, for instance, as the building model of a variational ansatz to solve the 2D-XY model.
We have eliminated some intermediate steps in some of the equations to improve the clarity and readability of the text, as pointed out by the referees. The equations we have eliminated elements or changed are:
Eq.3; Eq.31-Eq.35;Eq.37;Eq.52; Eq. 61;Eq.64
Current status:
Reports on this Submission
Report
I appreciate the efforts made by the Authors in clarifying issues and remarks arisen during the previous round of referral.
I also agree with the decision of moving the submission to Lecture Notes, given the main "revisiting" character of the work.
I am still not fully happy with certain aspects of the manuscript, but they might in most cases be matter of taste and opinion. Indeed, other Referees seem to like the over-detailed derivation of equation in the style of an undergraduate lecture note. Therefore, I would not prevent publication based on my taste. This might even turn out as a good selling point for the manuscript.
I only add a few somehow minor points below for a final round of polishing.
Requested changes
1- Eq.(13) holds also trivially for i=j, so please remove "if i≠j"
2- 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) :-)
3- 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.
4- Sentence before Eq. (52): the H_XY Hamiltonian is already non-interacting! Maybe the Authors intended to say "convert into its diagonal form"?
5- 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
6- 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?
Recommendation
Publish (meets expectations and criteria for this Journal)
Anonymous on 2025-03-12 [id 5282]
Dear Referees,
We have also prepared a copy of the resubmitted manuscript, where all revised equations and text are highlighted in red to indicate the changes made from the previous version. This is intended to facilitate the review process.
Best,
The authors
Attachment:
changes_manuscript.pdf