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 | |
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Preprint Link: | scipost_202412_00024v1 (pdf) |
Code repository: | https://github.com/Marc-Farreras/XYQSimulation |
Date submitted: | 2024-12-12 18:02 |
Submitted by: | Farreras, Marc |
Submitted to: | SciPost Physics |
Ontological classification | |
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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 the 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.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
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The manuscript discusses a quantum circuit that embodies the exact diagonalization of the XY Hamiltonian in transverse field.
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The topic is not novel per se, given that Ref. 7 has already laid out all the necessary ingredients some 15 years ago (and Ref. 21 has discussed fermionic Fourier transforms in more details a decade ago). Moreover, the last Author has already presented in Ref. 8 results obtained with the very same quantum circuit on IBM & Rigetti machines.
It would be desirable if the text would communicate in a much clearer way what is the element of novelty that the present study brings along -- one guess that I can make is that the quality of the experimental data seems to be considerably better and in excellent agreement with theory, but this should come out of the text without requiring the reader to skim through other papers.
I am glad to see if other points can be brought by the Authors.
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The manuscript clearly aims at being very pedagogical, with lot of details about textbook calculations. Although I very much like the spirit in general, it seems to me that it wanders off into being excessively pedantic in many parts (see pages full of small intermediate manipulations of equations), while being imprecise in others (e.g., quoting the Hubbard model as an example of quadratic Hamiltonian… or stating that Eq.(52) converts the H_{XY} Hamiltonian in a non-interacting one…). It almost reads as the first version of a master thesis, that still needs a round of polishing.
By the way, I might have got lost with the notation, but I miss why the ground state of the XY model should not be in the fermionic half-filled sector (i.e., in the zero magnetization one when dealing with spins), but rather be given by Eq. (68). I am almost sure it is simply some glitch, but it proves that overwhelming the reader might also be counterproductive.
Same applies with the fact that the important discussion about periodic/antiperiodic boundary conditions depending on the fermionic population (the spin magnetization) is put aside at some point and only the even case seems to matter from there on…
Shrinking excessive details could offer a possibility of letting messages pass better.
In particular, some effort should be spent instead around Fig. 18 in explaining how a continuous (Ising) phase transition could ever exhibit discontinuities in the (transverse) magnetization, even more in a system of finite size… This looks more like some level crossing / first order phase transition, which does not sound right, but I possibly misinterpreted the figure caption.
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As a stark contrast, the text feels really to come short when it deals with the actual results on the quantum computer and their discussion. The agreement between theory curves and experimental data looks excellent, which is a remarkable news in the realm of NISQ digital simulations at present date (compare the above mentioned Ref. 8 by the last Author). Therefore I would have naively thought that this was the central message of this work, while the section reads a bit sketchy and not delving into the reasons that allowed for such a nice performance. Along the same spirit, it would be nice to read some deeper consideration on the perspective of applying the same machinery to other models, possibly hinting at foreseen roadblocks to be circumvented and not just listing a couple of model names.
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Overall I recommend the Authors to perform a major revision of their Manuscript to bring it up to the (fairly high) standards of the SciPost family, and to consider resubmitting it to the Lecture Notes series, if the element of novelty is less prominent than the pedagogical aim.
Recommendation
Ask for major revision
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The article presents a method for performing the exact diagonalization of the XY Hamiltonian through a dedicated quantum circuit. The text has a pedagogical character, with long sections dedicated to explaining, in full detail, well-known techniques (see the entire Section 2).
I believe the article can be published for its pedagogical value, as it could serve as an extremely useful reference for students approaching the topic. I believe publication in SciPost Lecture Notes could also be a viable option to consider. In any case, I would suggest a revision of the text, as it is not very well-written in some parts.
Here, I highlight some issues I found in the text, along with improved versions. However, I think there are many other issues throughout the text...
-"In this paper, we present the comprehensive scheme for..." -> "In this paper, we present a comprehensive scheme for..."
-"These transitions occur at absolute zero..." -> "These transitions occur at zero temperature..."
-"The quadratic Hamiltonian in fermionic annihilation and creation operators appears in more condensed matter systems notably exemplified in the Hubbard model" -> "Hamiltonians quadratic in fermionic annihilation and creation operators are ubiquitous in condensed matter systems, describing systems of free fermionic particles." Concerning the Hubbard model: usually this model is considered with the interaction term, so is not quadratic...
-"...leading us to the subsequent phase: the fermionic Fourier transform (fFT)." -> "Diagonalizing this type of Hamiltonian is a well-established process, achieved through the fermionic Fourier transform (fFT)."
-"...our primary objective centers around obtaining the matrix" -> "...our primary objective is to obtain the matrix"
-"Figure 3: In the diagram is shown the decomposition of the building block of Fnk shown in Eq.(60), where ϕk =−i2πkn." -> "Figure 3: The diagram illustrates the decomposition of the building block of Fnk (Eq.(60)), where ϕk =−i2πkn."
While I did not review all the equations in detail, I believe they are essentially correct.
Other questions:
-Why you say the circuit is "specifically designed for the NISQ era"? What is NISQ specific here?
-Shouldn't the order of the three unitaries in Eq. (54) be reversed? Based on Eq. (52), it seems that U_JW should be applied to H first...
-Could your circuit serve as a starting point for a variational quantum circuit designed to address more complex, interacting Hamiltonians?
Recommendation
Ask for major revision
Report
The one-dimensional anisotropic XY model in transverse magnetic field is well known. Yet, even though some of materials rehash known facts, especially the part on Jordan Wigner transformation on spin and fermions, the article still reads well: The authors have so pedagogically presented the material that it makes reading enjoyable. I like it.
Section 3 presents detailed calculations and implementation of the simulation on a quantum simulator. The algorithm for fermionic Fourier transforms is based on work by Andrew Ferris (Ref. [21] in the article). And section 4 presents time evolution of the system. Section 5 presents results of simulations on a quantum simulator.
There may be missing references: previous work on such simulation exists in literature, e.g. Quantum Information Processing 20.8 (2021): 264; Physica Scripta 97.2 (2022): 025101; Physical Review Research 6.3 (2024): 033107, Physical Review A 95.5 (2017): 052339. The authors may wish to note some of these publications and cite them if needed.
Overall, I think the authors have made sufficient attempt to cast old wine in new wineskin. While I do not find anything novel in the article, I think this article still serves as a good pedagogical guide for beginners in quantum simulation. I would recommend acceptance after minor modifications.
Minor errors:
(i) Immediately after Eq(1), there should not be an indent after an equation. I think the authors left a space after the equation in their LaTeX version.
(ii) In the paragraph above Eq(2), "spin leather operators" should read "spin ladder operators".
(iii) Check Eq (3), there is an additional plus sign due to erroneous typing.
Recommendation
Ask for minor revision