SciPost Submission Page
Generalized hydrodynamics of integrable quantum circuits
by Friedrich Hübner, Eric Vernier, Lorenzo Piroli
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Lorenzo Piroli |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2408.00474v4 (pdf) |
| Date submitted: | 2025-02-04 10:13 |
| Submitted by: | Piroli, Lorenzo |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
Quantum circuits make it possible to simulate the continuous-time dynamics of a many-body Hamiltonian by implementing discrete Trotter steps of duration τ. However, when τ is sufficiently large, the discrete dynamics exhibit qualitative differences compared to the original evolution, potentially displaying novel features and many-body effects. We study an interesting example of this phenomenon, by considering the integrable Trotterization of a prototypical integrable model, the XXZ Heisenberg spin chain. We focus on the well-known bipartition protocol, where two halves of a large system are prepared in different macrostates and suddenly joined together, yielding non-trivial nonequilibrium dynamics. Building upon recent results and adapting the generalized hydrodynamics (GHD) of integrable models, we develop an exact large-scale description of an explicit one-dimensional quantum-circuit setting, where the input left and right qubits are initialized in two distinct product states. We explore the phenomenology predicted by the GHD equations, which depend on the Trotter step and the gate parameters. In some phases of the parameter space, we show that the quantum-circuit large-scale dynamics is qualitatively different compared to the continuous-time evolution. In particular, we find that a single microscopic defect at the junction, such as the addition of a single qubit, may change the nonequilibrium macrostate appearing at late time.
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
Author comments upon resubmission
We are very grateful to the Referees for their careful reading of our draft. In the following we respond to their comments.
Response to Referee 1
In the following, we present a point-by-point response to the Referee comments.
“In the past decade, many works have been dedicated to the real-time dynamics of integrable spin chains, such as the XXZ spin chain. In particular, the new theory of 'Generalized Hydrodynamics' was discovered.
Among all the works on Generalized Hydrodynamics that followed, many have focused on the so-called bipartition protocol, where two semi-infinite systems are suddenly put together, and where the predictions of Generalized Hydrodynamics are simple (because the results depend on a simple scaling variable x/t) and asymptotically exact.
Recently, 'integrable trotterizations' or 'integrable quantum circuits' have been introduced. These circuits are built from the 'diagonal-to-diagonal transfer matrices' known from the old algebraic Bethe Ansatz literature, with spectral parameters taken so that the transfer matrix is unitary and can be interpreted as an evolution operator. These integrable trotterizations have attracted a lot of interest recently in the context of digital quantum computation or simulation. In particular, the thermodynamic Bethe Ansatz for these systems was derived very recently ( by other people in Refs. [36-38] and by some of the authors in the Supplemental Material of Ref. [33]), as is nicely reviewed in Section 2 of this new paper.
No paper had yet looked at the bipartition protocol in integrable quantum circuits with Generalized Hydrodynamics. Now this is done, thanks to this new paper. The paper is correct, it can be published as it is."
We thank the Referee for their positive assessment of our work.
“Typos:
-page 4: 'Hamitlonian'
-page 5: 'masless'
-page 15: 'These points correspond to the either one of following choices'"
We thank the Referee for spotting these typos, which we have corrected in the new version of the draft.
Response to Referee 2
In the following, we present a point-by-point response to the Referee comments.
“The paper by Hübner et al. presents an interesting study of the nonequilibrium dynamics of a quantum spin chain, at the interface of three research areas that have received significant attention in recent years: (i) Trotter transitions, that is, qualitative changes in the dynamics that occur when the Trotter step size crosses a critical value; (ii) integrability and generalized hydrodynamics (GHD); (iii) quench dynamics, in particular, the bipartition protocol, which can lead to the emergence of nonequilibrium steady states. This study builds on and complements a previous work [33] that includes two of the authors of the present manuscript. The model system under investigation is an integrable quantum circuit that represents a Trotterization of the dynamics generated by the XXZ Hamiltonian. As initial states, the authors consider a single domain wall between the two halves of the spin chain in Néel, anti-Néel, or Majumdar-Ghosh states. To describe the dynamics in the space-time scaling limit, the authors build on recent developments in the theory of integrable quantum circuits, and derive the GHD equations for such systems. Apart from these technical developments, the key physical result of the paper is that for finite Trotter step sizes, the nonequilibrium macrostates appearing at late times can differ qualitatively from the continuous-time limit."
We thank the Referee for their positive assessment of our work.
"As stated above, the paper addresses a timely topic at the interface between different research areas. The results are interesting and clearly presented. I am not an expert on integrability and GHD, and, therefore, I find it hard to judge whether the paper is groundbreaking in terms of developing new techniques. However, my impression is that the foundations for the present work were laid in Ref. [33], which diminishes the claims to novelty of this work. Therefore, I believe that the paper is better suited for a less selective journal such as SciPost Physics Core."
The main criticism raised by Referee is that the novelty of our work is diminished by Ref. [33]. We do not agree with this claim, as we argue below.
First, we agree that our work does not introduce ground-breaking techniques compared to previous work on integrability. Also, it is true that Ref. [33] already highlighted the emergence of different types of behavior in integrable circuits as a function of the Trotter step. However, Ref. [33] only studied the behavior of integrable circuits in an ideal setting of translation-invariant systems, and in the late-time stationary regimes. On the contrary, our present work makes predictions on a different physical setting and studies the full time evolution (in the large space-time limit). Although one could have anticipated the emergence of peculiar behavior based on Ref. [33], it was not obvious how and to what extent it could be observed in the dynamics of simple local observables. This question is particularly relevant when considering potential benchmark implementations of integrable circuits on digital quantum devices. Our work provides a thorough study of quantum quenches in inhomogeneous settings, yielding a phenomenological overview of the possible behavior that can be observed as a function of the Trotter step. Our numerical results show very clear signatures of the Trotter transition even in the dynamics of local observables, to relatively short times.
From the technical viewpoint, we lay out the theory of GHD in integrable circuits, which was not done before. Although we do not encounter major technical complications or subtlelties in doing so, it was a priori not obvious that this would be the case. Also, our comparison of the GHD equations against independent numerics in the non-interacting regime provides convincing evidence that the GHD description applies to quantum circuits to the same degree of accuracy as previously studied continuous-time dynamics.
In summary, we believe that our work, while a natural logical continuation of the research presented in Ref. [33], is by no means a mere application of the theory developed there. Our conclusions could not have been anticipated only based on Ref. [33], and our calculations lead to results that are expected to be of interest for several communities working on non-equilibrium many-body systems, integrability, and even for future experimental work benchmarking digital quantum devices.
“1- As the authors explain, a domain wall between the Néel and anti-Néel states can be interpreted as a single localized defect, and such a defect does typically not lead to a change in the macroscopic nonequilibrium state for systems with single-site translation symmetry. In the model under consideration, this symmetry is broken due to the Trotterization. This observation, however, raises the question, whether the observed phenomenology is unique to Trotterized dynamics or could also be seen in autonomous systems with broken single-site translation symmetry."
Indeed, the Referee is right, we expect that a similar phenomenology could be observed in Hamiltonian dynamics where the Hamiltonian breaks single-site shift invariance (but preserves two-site translation symmetry). However, while this construction would appear rather ad hoc in the continuous-time limit, here it appears naturally as a consequence of increasing the Trotter step. The ability to change the global late-time behavior by a single defect in a circuit setting is also of obvious interest for benchmark experiments in current digital devices. Indeed, approximating the continuous-time evolution of a two-site shift-invariant Hamiltonian would require implementing very deep circuits, even to reach short times, which is unfeasible on current devices. Instead, existing experiments have already shown that integrable circuits can be implemented keeping sufficient coherence over a number of discrete steps that is enough to appreciate the emergence of late-time features.
We have added this discussion in the second-to-last paragraph in Sec. 4.1 (see the end of Pag. 16 and the beginning of Pag. 17 in the new version of the draft).
“2- I would welcome a discussion of micromotion. Do the calculated nonequilibrium states apply only at stroboscopic times? If that is the case, how do, e.g., the profiles of the staggered magnetization shown in Fig. 1 change during one driving period?.”
As the Referee suggests, while the quantum-circuit architecture requires that the dynamics is discrete, one could of course view the quantum-circuit dynamics as a Floquet dynamics observed at stroboscopic times. However, we are studying dynamics in the hydrodynamic limit of large time scales, namely over scales that are much larger that the single stroboscopic time step. Therefore, the calculated profiles would also exactly predict the profiles for a continuous-time evolution.
“3- The paper is clearly written and overall accessible even to non experts. However, I have found the transition from the Bethe equations (9) to the TBA in Eq. (12) rather hard to follow. As far as I understand, the Bethe equations determine the spectrum of the Floquet operator, i.e., these equations are not specific to any state of the system. However, the distribution functions appearing in Eq. (12) describe a macrostate of the system. Which state is that and how did it appear in the Bethe equations?"
This is a natural question, and we realized the text was not sufficiently clear on this point. We have added a paragraph to answer the question raised by the Referee after Eq. (18).
“4- The + sign in Eq. (54) should presumably be an =."
We thank the referee for spotting this. The + sign was correct but we forgot the =0 at the end of the equation.
Response to Referee 1
In the following, we present a point-by-point response to the Referee comments.
“In the past decade, many works have been dedicated to the real-time dynamics of integrable spin chains, such as the XXZ spin chain. In particular, the new theory of 'Generalized Hydrodynamics' was discovered.
Among all the works on Generalized Hydrodynamics that followed, many have focused on the so-called bipartition protocol, where two semi-infinite systems are suddenly put together, and where the predictions of Generalized Hydrodynamics are simple (because the results depend on a simple scaling variable x/t) and asymptotically exact.
Recently, 'integrable trotterizations' or 'integrable quantum circuits' have been introduced. These circuits are built from the 'diagonal-to-diagonal transfer matrices' known from the old algebraic Bethe Ansatz literature, with spectral parameters taken so that the transfer matrix is unitary and can be interpreted as an evolution operator. These integrable trotterizations have attracted a lot of interest recently in the context of digital quantum computation or simulation. In particular, the thermodynamic Bethe Ansatz for these systems was derived very recently ( by other people in Refs. [36-38] and by some of the authors in the Supplemental Material of Ref. [33]), as is nicely reviewed in Section 2 of this new paper.
No paper had yet looked at the bipartition protocol in integrable quantum circuits with Generalized Hydrodynamics. Now this is done, thanks to this new paper. The paper is correct, it can be published as it is."
We thank the Referee for their positive assessment of our work.
“Typos:
-page 4: 'Hamitlonian'
-page 5: 'masless'
-page 15: 'These points correspond to the either one of following choices'"
We thank the Referee for spotting these typos, which we have corrected in the new version of the draft.
Response to Referee 2
In the following, we present a point-by-point response to the Referee comments.
“The paper by Hübner et al. presents an interesting study of the nonequilibrium dynamics of a quantum spin chain, at the interface of three research areas that have received significant attention in recent years: (i) Trotter transitions, that is, qualitative changes in the dynamics that occur when the Trotter step size crosses a critical value; (ii) integrability and generalized hydrodynamics (GHD); (iii) quench dynamics, in particular, the bipartition protocol, which can lead to the emergence of nonequilibrium steady states. This study builds on and complements a previous work [33] that includes two of the authors of the present manuscript. The model system under investigation is an integrable quantum circuit that represents a Trotterization of the dynamics generated by the XXZ Hamiltonian. As initial states, the authors consider a single domain wall between the two halves of the spin chain in Néel, anti-Néel, or Majumdar-Ghosh states. To describe the dynamics in the space-time scaling limit, the authors build on recent developments in the theory of integrable quantum circuits, and derive the GHD equations for such systems. Apart from these technical developments, the key physical result of the paper is that for finite Trotter step sizes, the nonequilibrium macrostates appearing at late times can differ qualitatively from the continuous-time limit."
We thank the Referee for their positive assessment of our work.
"As stated above, the paper addresses a timely topic at the interface between different research areas. The results are interesting and clearly presented. I am not an expert on integrability and GHD, and, therefore, I find it hard to judge whether the paper is groundbreaking in terms of developing new techniques. However, my impression is that the foundations for the present work were laid in Ref. [33], which diminishes the claims to novelty of this work. Therefore, I believe that the paper is better suited for a less selective journal such as SciPost Physics Core."
The main criticism raised by Referee is that the novelty of our work is diminished by Ref. [33]. We do not agree with this claim, as we argue below.
First, we agree that our work does not introduce ground-breaking techniques compared to previous work on integrability. Also, it is true that Ref. [33] already highlighted the emergence of different types of behavior in integrable circuits as a function of the Trotter step. However, Ref. [33] only studied the behavior of integrable circuits in an ideal setting of translation-invariant systems, and in the late-time stationary regimes. On the contrary, our present work makes predictions on a different physical setting and studies the full time evolution (in the large space-time limit). Although one could have anticipated the emergence of peculiar behavior based on Ref. [33], it was not obvious how and to what extent it could be observed in the dynamics of simple local observables. This question is particularly relevant when considering potential benchmark implementations of integrable circuits on digital quantum devices. Our work provides a thorough study of quantum quenches in inhomogeneous settings, yielding a phenomenological overview of the possible behavior that can be observed as a function of the Trotter step. Our numerical results show very clear signatures of the Trotter transition even in the dynamics of local observables, to relatively short times.
From the technical viewpoint, we lay out the theory of GHD in integrable circuits, which was not done before. Although we do not encounter major technical complications or subtlelties in doing so, it was a priori not obvious that this would be the case. Also, our comparison of the GHD equations against independent numerics in the non-interacting regime provides convincing evidence that the GHD description applies to quantum circuits to the same degree of accuracy as previously studied continuous-time dynamics.
In summary, we believe that our work, while a natural logical continuation of the research presented in Ref. [33], is by no means a mere application of the theory developed there. Our conclusions could not have been anticipated only based on Ref. [33], and our calculations lead to results that are expected to be of interest for several communities working on non-equilibrium many-body systems, integrability, and even for future experimental work benchmarking digital quantum devices.
“1- As the authors explain, a domain wall between the Néel and anti-Néel states can be interpreted as a single localized defect, and such a defect does typically not lead to a change in the macroscopic nonequilibrium state for systems with single-site translation symmetry. In the model under consideration, this symmetry is broken due to the Trotterization. This observation, however, raises the question, whether the observed phenomenology is unique to Trotterized dynamics or could also be seen in autonomous systems with broken single-site translation symmetry."
Indeed, the Referee is right, we expect that a similar phenomenology could be observed in Hamiltonian dynamics where the Hamiltonian breaks single-site shift invariance (but preserves two-site translation symmetry). However, while this construction would appear rather ad hoc in the continuous-time limit, here it appears naturally as a consequence of increasing the Trotter step. The ability to change the global late-time behavior by a single defect in a circuit setting is also of obvious interest for benchmark experiments in current digital devices. Indeed, approximating the continuous-time evolution of a two-site shift-invariant Hamiltonian would require implementing very deep circuits, even to reach short times, which is unfeasible on current devices. Instead, existing experiments have already shown that integrable circuits can be implemented keeping sufficient coherence over a number of discrete steps that is enough to appreciate the emergence of late-time features.
We have added this discussion in the second-to-last paragraph in Sec. 4.1 (see the end of Pag. 16 and the beginning of Pag. 17 in the new version of the draft).
“2- I would welcome a discussion of micromotion. Do the calculated nonequilibrium states apply only at stroboscopic times? If that is the case, how do, e.g., the profiles of the staggered magnetization shown in Fig. 1 change during one driving period?.”
As the Referee suggests, while the quantum-circuit architecture requires that the dynamics is discrete, one could of course view the quantum-circuit dynamics as a Floquet dynamics observed at stroboscopic times. However, we are studying dynamics in the hydrodynamic limit of large time scales, namely over scales that are much larger that the single stroboscopic time step. Therefore, the calculated profiles would also exactly predict the profiles for a continuous-time evolution.
“3- The paper is clearly written and overall accessible even to non experts. However, I have found the transition from the Bethe equations (9) to the TBA in Eq. (12) rather hard to follow. As far as I understand, the Bethe equations determine the spectrum of the Floquet operator, i.e., these equations are not specific to any state of the system. However, the distribution functions appearing in Eq. (12) describe a macrostate of the system. Which state is that and how did it appear in the Bethe equations?"
This is a natural question, and we realized the text was not sufficiently clear on this point. We have added a paragraph to answer the question raised by the Referee after Eq. (18).
“4- The + sign in Eq. (54) should presumably be an =."
We thank the referee for spotting this. The + sign was correct but we forgot the =0 at the end of the equation.
List of changes
We have added several paragraphs to the text to address the comments of Referee 2
Current status:
Has been resubmitted