Quantum state preparation of topological chiral spin liquids via Floquet engineering

We thank Prof. Marin Bukov very much for is insightful reading of the paper. We have made some revisions and added some references following his comments. We believe it has improved the quality and clarity of the paper. Below we answer point by point and describe the changes made.

Section 2

Referee Eq(3); it would be helpful if the authors define explicitly the permutation operator in terms of the spin degrees of freedom (I now see that you sort of have it in Eq(24) but an explicit reference will be useful).

Authors A sentence has been added to make the definition of $P_{ijkl}$ explicit. The expression in terms of the spin degrees of freedom has now been included in the main text.

Referee Sec 2.3, "Note that such a monochromatic drive may not be easy to implement on experimental cold atom platforms": I think there is no difficulty with the dephased monochromatic drive itself; the spin-spin interactions would be the real challenge, see e.g., https://www.nature.com/articles/s41567-020-0949-y

Authors We thank the Referee for mentioning the reference. We have rephrased the short paragraph in Sec. 2.3 on the experimental implementation, taking into account the Referee comment.

Referee Sec 2.4: before Eq (6), when the authors introduce stroboscopic times, there appear integers p, n, times t, t0, etc,; what's written is not wrong, but I think it can be simplified by removing the unnecessary info for the benefit of the readers.

Authors Following the comment we have simplified the formulation (removing $t_0$, etc...).

Referee after Eq(7), "It is interesting that the next order of the expansion is $O(1/\omega^3)$ so that we expect HF0 to capture accurately the stroboscopic motion.": this is not obvious to me; can the authors argue why there are no $1/\omega^2$ terms here?

Authors Eq. (42) of Bukov et al. suggests that, in the absence of a time independent part $H_0=0$, the expansion involves only nested commutators containing the same number of $H_1$ and $H_{-1}$ Fourier components of the Hamiltonian, i.e. odd powers of $1/\omega$. We have added a note on this.

Referee If I'm not mistaken, you might be able to compute the Floquet Hamiltonian to Eq(6) alone exactly, as follows:

1. Let $L_z$ be the generator of continuous rotations about the out-of-lattice-plane axis, i.e., the operator that rotates $H_x$ into $H_y$: $\exp(-i\pi/2L_z) H_x \exp(+i\pi/2 L_z )= H_y$; I think, $\exp(-i\pi / 2 L_z)$ is related to the permutation operator $P_{i,j,k,l}$ but we need the infinitesimal rotation generator. Not sure how crazily non-local (in terms of the spin operators) $L_z$ is, though.

2. consider now a rotation about the angle $\alpha = \omega t$: in a co-rotating frame defined by the rotation $V(t)=\exp(-i \omega t L_z)$, the circular drive will freeze and point, let's say, along $H_x$.

3. taking into account the Galilean term due to the frame transformation, in the co-rotating frame the Hamiltonian will read as $H'= \omega L_z+H_x$ (no liability regarding the correctness of signs).

This business is very similar to the two-level system in a circular drive, just the operators involved are quite a bit more complicated. To relate $H'$ to HF, you may need to do extra work, especially if the spectrum of $L_z$ is not commensurate, see Sec 2.3 of Ref [39]. In the language of $H'$, to make the chiral structure manifest, you'd have to consider the "symmetric gauge", see discussion in Sec 3.3 of Ref [39].

Authors We thank the Referee for his proposal. However, we believe that, away from the high-frequency limit, the Floquet Hamiltonian becomes more and more non local (including all sorts of chiral terms on larger and larger closed loops). It would be very surprising that tractable expressions could be obtained generically. In any case, we postpone this calculation to future studies.

Section 3

Referee Sec 3.2.1: "Since the effective Floquet dynamics is expected to be very accurate, one can use it to investigate the physics in the $t\rightarrow \infty$ limit": this is dangerous b/c the approximate effective Hamiltonian computed from the Magnus expansion does not capture heating effects. The system is expected to heat up to an infinite temperature state in the $t\rightarrow \infty$ limit for any finite $\kappa>0$, see e.g., Ref [33]. However, what's physically relevant in practice is the prethermal plateau regime which occurs at finite time (and whose lifetime can be controlled by increasing the drive frequency).

Authors We agree with the referee and have reformulated the text. We have added a comment on the role of the finite system size.

Appendix A

Referee "PEPS offer an extremely efficient variational scheme to address local Hamiltonians": it's not clear which features/properties are being addressed

Authors Beside providing a very good variational energy, PEPS can encode the topological nature of the state and the physics of the edge mode. Note and Refs. [75,76] added.

Referee it's worthwhile to mention in one sentence the drawbacks of PEPS as well, or the regime of applicability

Authors PEPS fail to describe the rapid increase of entanglement entropy (e.g. in the case of a quench, see Ref. [77]) but should still be relevant in the case of an adiabatic ramp.

Appendix C

Referee "The unitary operator $\exp(-iK(t))$ corresponds physically to a change of basis": time-dependent unitaries correspond rather to a change of reference frame; when evaluated at some fixed $t_0$, one can think of them as giving kicks to the state (i.e., changing the basis). In this sense, Floquet's theorem is a statement about the existence of a reference frame, where the dynamics is governed by a static Hamiltonian $H_F$ at all times (not only stroboscopically).

Authors We have changed "change of basis" into "change of reference frame (via a change of basis)".

Figure 2

Referee a) I find it a bit confusing that the micromotion in inset (c) seems to oscillate around the black-squares curve; however the two curves are w.r.t. different x-axes (black squares go from times $\sim$<!-- -->25 to 40, while grey micromotion curve goes from times 30.1 to 30.5). It might be better to show inset (c) as a separate panel of the figure to avoid this confusion.

Authors We have modified the figure to avoid the confusion mentioned by the Referee.

Referee b) regarding errors caused by Trotterization: I think it should be possible to evolve the state using a Runge-Kutta solver on a $4\times4$ patch.

Authors We have no expertise with such a technique but we have full control on the error introduced by the Trotterization scheme. By varying the Trotter step we can estimate the minimum number of steps per period to reach a very good accuracy, typically better than $10^{-4}$.

Figure 3

Referee is $k$ the same integer as $p$? If yes, better use $p$.

Authors We are a bit confused by the comment. Probably the Referee has mistaken $\kappa$ (defined in Eq. (11)) for $k$ ?

References

Referee non-abelian anyons have recently been observed on Honeywell's trapped ion quantum computer: https://www.nature.com/articles/s41586-023-06934-4

Authors We thank the Referee for this interesting paper which we have cited in the introduction.

Referee the authors might be interested in a recent preprint, where we discuss various aspects of state preparation under strong Floquet drives, and in particular how to speed up adiabatic ramps: https://arxiv.org/abs/2310.02728

Authors We thank the Referee for pointing out this interesting preprint. We have cited it in the final discussion.

Typos

Referee

• abstract: remarquable $\longrightarrow$ remarkable

• after Eq(1) [+ other instances]: "significantly smaller that the smallest": that $\longrightarrow$ than

• before Eq(2): Krylov's $\longrightarrow$ Krylov

• Eq(2): summation index m missing in sum subscript

• Sec 2.2, first paragraph, "It can be view as": view $\longrightarrow$ viewed

• Sec 3.1, penultimate paragraph, "p fixed": p $\longrightarrow$ p

• Sec 3.2, first paragraph, "is suddenly switch on", switch $\longrightarrow$ switched

• Sec 3.4.2, last paragraph, "As shown in Figs. 7": Figs $\longrightarrow$ Fig

• Sec 4.3: "We argue that our goal cannot be realized by branching the drive suddenly": branching $\longrightarrow$ quenching

• Sec 4.3: "the minimum ramp time to reach a good fidelity should qualitatively scales like N": scales $\longrightarrow$ scale

Authors We thank the Referee for pointing out the typos which have all been correcte]d.

Added references

Note that Refs. [19], [23] and [74] have been added.