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Energysaving fastforward scaling
by Takuya Hatomura
Submission summary
Authors (as registered SciPost users):  Takuya Hatomura 
Submission information  

Preprint Link:  https://arxiv.org/abs/2402.10683v2 (pdf) 
Date submitted:  20240907 07:07 
Submitted by:  Hatomura, Takuya 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We propose energysaving fastforward scaling. Fastforward scaling is a method which enables us to speed up (or slow down) given dynamics in a certain measurement basis. We introduce energy costs of fastforward scaling, and find possibility of energysaving speedup for timeindependent measurement bases. As concrete examples, we show such energysaving fastforward scaling in a twolevel system and quantum annealing of a general Ising spin glass. We also discuss the influence of a timedependent measurement basis, and give a remedy for unwanted energy costs. The present results pave the way for realization of energyefficient quantum technologies.
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 multipronged followup work
 Detail a groundbreaking theoretical/experimental/computational discovery
 Present a breakthrough on a previouslyidentified and longstanding research stumbling block
Author comments upon resubmission
[Reply to the Report 1]
I am grateful to the referee 1 for carefully reading the previous manuscript and for giving useful comments. I revised the manuscript according to the comments. My replies to the comments are as follows.
Regarding the Weakness 1, energysaving fastforward scaling has two implications for experiments. Suppression of the instantaneous energy cost implies the possibility of further speedup of target dynamics with limited energy scale, and suppression of the total energy cost implies a possible improvement in energy efficiency of speedup. I mention these points in the revised manuscript.
Regarding the Weakness 2, analyses on energy costs of counterdiabatic driving were already reported in Ref.1923 in the revised manuscript. Because fastforward scaling can be applied to speedup of nonadiabatic dynamics as well as speedup of adiabatic time evolution, we can regard the present results as generalization (or extension) of the previous works. Note that invariantbased inverse engineering has the same mathematical structure with counterdiabatic driving, and thus energy costs of invariantbased inverse engineering should be similar to those of counterdiabatic driving.
Regarding the Weakness 3, I showed experimentally realizable application of energysaving fastforward scaling to quantum annealing of general Ising spin glass, which is specialization of adiabatic quantum computation. Here, the measurement basis was the computational basis because we are interested in measurement outcomes with the computational basis in quantum annealing. It should be noted that the choice of the measurement basis is not arbitrary; it must be determined by target dynamics and desired measurement outcomes. It is actually difficult if the measurement basis is the energy eigenstate basis because we have to know the energy eigenstates and entanglement in the energy eigenstates introduces complicated interacting terms in the unitary $U_f(t)$, which results in nonlocal and manybody interactions of the fastforward Hamiltonian as the result of the BakerCampbellHausdorff expansion.
Regarding the Weakness 4 and the annotated PDF file, I carefully revised the manuscript.
I believe that I addressed all the requested changes in the revised manuscript and it satisfies all the criteria for publication in SciPost Physics.
[Reply to the Report 2]
I am grateful to the referee 2 for carefully reading the previous manuscript and for giving useful comments. I revised the manuscript according to the comments. My replies to the comments are as follows.
Regarding the General comment (1), I added explanation for each result. Moreover, I added derivation of the main result in Appendix.
Regarding the General comment (2), I proofread the manuscript carefully, and revised it.
Regarding Introduction (1), I removed the concerned paragraph and added a new paragraph on quantum technologies to the revised manuscript.
Regarding Introduction (2), I removed the concerned sentence and mentioned the limitation due to quantum speed limits in the next paragraph.
Regarding Introduction (3), it is rephrased.
Regarding Fastforward scaling (1), the magnification factor is given by $ds/dt$. In the previous manuscript, I wrote that the magnification factor is set as $ds/dt\ge1$, but it can be $ds/dt\ge0$. The present results hold even if slowingdown and pausing of target dynamics are included. Please see the revised manuscript.
Regarding Energy cost of fastforward scaling (1), I added the definition of the HilbertSchmidt norm to the revised manuscript.
Regarding Energysaving fastforward scaling (1), I added derivation of the squared HilbertSchmidt norm of the fastforward Hamiltonian in Appendix.
Regarding Energysaving fastforward scaling (2), I have tried to associate the present results with the stability of dynamics or control, but I could not find convincing interpretation. This is because the variance is given by using the measurement basis instead of the target dynamics. I can answer the second half of the question. The measurement basis can be the eigenbasis of a Hamiltonian as I showed in the third example. This is because $\langle\sigma\partial_t\sigma^\prime\rangle$ is not zero for the eigenbasis of a Hamiltonian, or in other words, the squared HilbertSchmidt norm of the fastforward Hamiltonian is not the variance of a Hamiltonian for the eigenbasis. If the eigenbasis is independent of time, there is no transition between the measurement basis, and thus $H_{\mathrm{FF}}(t)=0$ is enough for fastforwarding and no contradiction exists.
Regarding Examples (1), I added explanations. Moreover, I added discussion on the (sub)optimal Hamiltonians in a new section 6 Discussion.
Regarding Examples (2), I added sentences which explain difficulty in experimental realization of the fulloptimized fastforward Hamiltonian. Moreover, I discussed experimental realization of the suboptimal fastforward Hamiltonian in the new section 6 Discussion. Since we considered the computational basis as the measurement basis, we can only modify the PauliZ terms (and global phase). Therefore, I do not think that there are other useful suboptimal phases. Moreover, although the measurement basis plays an important role to decrease the energy costs, it is determined by the target state and the desired measurement outcome, and thus we do not have the choice of the measurement basis in general. For example, in quantum annealing, the measurement basis must be the computational basis or energyeigenstate basis, which are identical at the final time. We cannot generally exploit the choice of the measurement bases.
Regarding Examples (3), I addressed these points. Please see the revised manuscript.
Regarding Conclusions (1), I (mainly) addressed these points in the new section 6 Discussion. Please see the revised manuscript. As mentioned above, the present results can also be applied to the slowingdown case, $0\le ds/dt\le1$, and thus energysaving “slowdown” scaling is also possible.
Regarding Figures (1), the behavior of the spike does not seem monotonic, but it consists of monotonic envelopes and oscillations. The spike for $2\Gamma_0=0.4$ is accidentally suppressed compared with that for $2\Gamma_0=0.8$ because of the difference in the oscillation terms. This behavior of the spike is in principle irregular because it is determined by the timing of the oscillation in the monotonic envelope. In addition to this point, the total energy cost is not monotonic. I plotted it for a wide range in the revised manuscript.
Regarding Figures (2), as mentioned above, the behavior of the total energy cost is not monotonic. I changed the plot range. Please find it in the revised manuscript. I am not interested in further large $\Gamma_0$ because there will be no nonadiabatic transition, i.e., transitions in the present measurement basis, and thus the optimal fastforward Hamiltonian will be $\hat{H}_\mathrm{FF}(t)\approx0$, while the standard fastforward Hamiltonian will be $\hat{H}_\mathrm{FF}(t)=(ds/dt)\hat{H}(s)\approx(ds/dt)\Gamma_0\hat{X}$.
Regarding Figures (3), I added the Figure 3 for the study of $\delta$dependence of the instantaneous energy cost.
Regarding Figures (4), I addressed this point.
I believe that I addressed all the requested changes in the revised manuscript and it deserves to be published in SciPost Physics.
List of changes
A new section 6 Discussion is added and 7 Conclusion, which was originally 6 Summary, is also rewritten.
Derivation and explanation of equations are added.
Interpretation of figures is added.
Figure 3 is added to discuss robustness of optimization.
Typos and errors are corrected.
Vague sentences are removed.
Current status:
Reports on this Submission
Report
The revised manuscript has successfully addressed the majority of the comments and suggestions from the initial referee report, and I am satisfied with the changes made. For reference, below is a detailed evaluation of the points that were raised and whether they have been satisfactorily addressed. The overall quality of the work has improved, both in terms of clarity and depth of discussion. Therefore, I am pleased to recommend the manuscript for publication in SciPost Physics, pending any final editorial review.
   
$\textbf{Clarity in Sections IV and V (General, Comment 1):}$
The author has included additional equations and intext references to improve the clarity and reproducibility of the arguments. The derivations are now more accessible.
$\textbf{Grammatical and Orthographic Corrections (General, Comment 2):}$ The manuscript has been thoroughly proofread, and the grammatical and typographical errors highlighted in the previous report have been addressed.
$\textbf{Upper Bounds on Speedup (Introduction, Comment 2): }$ The author has addressed the request to refer to limitations imposed by quantum speed limits, and relevant literature has been cited.
$\textbf{Magnification Factor and Slowing Down (Fastforward scaling, Comment 1):}$ The author clarified the impact of the magnification factor and the possibility of applying the theory to cases where the dynamics is slowed down. This addition places the discussion on energy costs in a more general context.
$\textbf{HilbertSchmidt Norm Definition (Energy costs of fastforward scaling, Comment 1):}$ A clear and concise definition of the HilbertSchmidt norm has been provided, ensuring there is no ambiguity in the energy cost metric used throughout the manuscript.
$\textbf{Measurement Basis and Energy Costs (Energysaving fastforward scaling Comment 1 and 2):}$ Appendix added to clarify derivation. The case where the measurement basis is the eigenbasis of the Hamiltonian has been explained further.
$\textbf{Examples (Comment 1 and 2):}$ The author has expanded the discussion on the choice of measurement basis and the impact of using a different optimized phase.
$\textbf{Characterization of the Small Parameter $\delta$ (Example, Comment 3):}$ The smallness of the parameter $\delta$ has been characterized clearly, along with its relation to the other energetic scales in the problem.
$\textbf{Figures and Discussion (Figures, Comment 13): }$ The figures have been updated with more comprehensive captions, and the spike in instantaneous energy cost is now better explained. The limiting behavior of the energy cost as the transverse field strength increases is discussed in greater detail, providing a clearer understanding of the results presented in the figures and confirming that the total energy cost is not monotonic, as it initially appeared. The figure showing the robustness of the phase modulation is notably improved and insightful.
$\textbf{Conclusion and Future Perspectives (Conclusions, Comment 1):}$ The limitations and potential extensions of the study have been discussed briefly in the conclusions. The inclusion of possible directions for future work, particularly regarding experimental realization, strengthens the final section of the manuscript.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)