Quantum state preparation of topological chiral spin liquids via Floquet engineering

- timely study

- touches upon various aspects of nonequilibrium dynamics, with particular focus on Hamiltonian engineering and state preparation of chiral spin liquid states

- comparison between theory and experiment shows results are solid and justified.

- a few statements require clarification

The paper "Quantum state preparation of topological chiral spin liquids via Floquet engineering" by Mambrini and Poilblanc is an interesting and timely study, touching upon various aspects of nonequilibrium dynamics, with particular focus on Hamiltonian engineering and state preparation of chiral spin liquid states.

The paper is interesting and can be published once the authors have considered the following points:

Sec 2:

- 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)

- 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

- Sec 2.4: before Eq(6), when the authors introduce stroboscopic times, there appear integers $p$, $n$, times $t$, $t_0$, 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.

- after Eq(7), "It is interesting that the next order of the expansion is O(1/ω3) 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?

- 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 $H_y$: $\exp(-i \pi/2 L_z) H_x \exp(+i \pi/2 L_z) = H_y$; 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 $H_F$, 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].

Sec 3:

- 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 → ∞ 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 → ∞ 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).

- do PEPS/chiral PEPS offer a controlled approximation to CSL states in large systems? What about iPEPS? Are the deviations from unity in the fidelities reported in Table 1 due to the failure of the adiabatic ramp or due to the PEPS ansatz not reaching the ground state?

App A:

- "PEPS offer an extremely efficient variational scheme to address local Hamiltonians": it's not clear which features/properties are being addressed
- it's worthwhile to mention in one sentence the drawbacks of PEPS as well, or the regime of applicability

App C:

- "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).

Figs:

- Fig 2:

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 ~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.

b) regarding errors caused by Trotterization: I think it should be possible to evolve the state using a Runge-Kutta solver on a 4x4 patch.

- Fig 3: is $k$ the same integer as $p$? If yes, better use $p$.

References:

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

- 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

Typos:

abstract: remarquable --> remarkable
after Eq(1) [+ other instances]: "significantly smaller that the smallest": that --> than
before Eq(2): Krylov’s --> Krylov
Eq(2): summation index $m$ missing in sum subscript
Sec 2.2, first paragraph, "It can be view as": view --> viewed
Sec 3.1, penultimate paragraph, "p fixed": p --> $p$
Sec 3.2, first paragraph, "is suddenly switch on", switch --> switched
Sec 3.4.2, last paragraph, "As shown in Figs. 7": Figs --> Fig
Sec 4.3: "We argue that our goal cannot be realized by branching the drive suddenly": branching --> quenching
Sec 4.3: "the minimum ramp time to reach a good fidelity should qualitatively scales like N": scales --> scale

see report above