Disordered Kicked Ising Model (KIM)

• Disordered KIM is a family of periodically driven spin-½ systems where disorder is introduced via quenched fields, bond couplings, or irregular kick parameters.
• The model employs self-dual and dual-unitary constructions that enable exact spectral analyses, revealing ergodic-to-MBL transitions and finite-size scaling behaviors.
• KIM serves as a testbed for quantum chaos, robust energy charging, and advanced numerical methods, bridging theoretical insights with quantum hardware experiments.

The disordered kicked Ising model (KIM) is a family of periodically driven spin-$\tfrac12$ systems in which an Ising interaction layer is alternated with a kick layer, and disorder is introduced through quenched fields, bond couplings, or irregular kick parameters. Across the literature, it functions simultaneously as a model of quantum chaos, a testbed for Floquet many-body localization (MBL), a setting for exact self-dual or dual-unitary constructions, and, more recently, a platform for disorder-resilient quantum charging. Representative formulations include one-dimensional chains with random longitudinal fields and transverse kicks, self-dual kicked Ising batteries with coupling disorder, and hardware-oriented Floquet circuits on heavy-hex lattices with disordered transverse fields (Sierant et al., 2022, Waltner et al., 2021, Gupta et al., 18 Jun 2025, Romero et al., 21 Nov 2025, Hayata et al., 2 Oct 2025).

1. Model class and disorder mechanisms

In the one-dimensional Floquet formulation used for MBL studies, the basic unitary over one period is

$$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$

with periodic boundary conditions $\sigma_{L+1}^z\equiv \sigma_1^z$, and $h_j$ independent quenched random variables. A frequently studied parametrization sets $g=J=1/W$, so that increasing $W$ suppresses the coherent couplings relative to the random longitudinal fields (Sierant et al., 2022).

A closely related ring geometry is written as

$$\hat U = e^{-iJ\sum_n \hat\sigma_n^z\hat\sigma_{n+1}^z}\, e^{-i\sum_n h_n\hat\sigma_n^z}\, e^{-i(\pi/4)\sum_n \hat\sigma_n^x},$$

with strong disorder in the $h_n$ and the transverse kick fixed at $b_x=\pi/4$. This version is central in dual-operator analyses of ergodic versus localized behavior and in spectral-form-factor studies (Waltner et al., 2021).

In the self-dual kicked-Ising quantum-battery construction, the time-dependent Hamiltonian is written as

$$H(t)=H_0+\lambda(t)\big[H_I+H_K\sum_{t_i\in\mathcal T}\delta(t-t_i)-H_0\big].$$

Two charger variants are emphasized: $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$0 and

$$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$1

Here disorder is primarily introduced through the bonds,

$$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$2

with optional field disorder $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$3 (Romero et al., 21 Nov 2025).

A hardware-oriented two-dimensional Floquet version on a heavy-hex graph uses

$$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$4

with

$$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$5

In that setting the disorder resides exclusively in the $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$6-field layer (Hayata et al., 2 Oct 2025).

An older, distinct notion of disorder replaces quenched spatial randomness by spin-dependent irregular kick trains. There the disorder is encoded in kick strengths $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$7 and delays $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$8, generated by classical processes on the torus and transmitted along an open Ising chain through the monodromy operator of each period (Aubourg et al., 2014). Taken together, these formulations show that “disordered KIM” is not a single Hamiltonian but a structured class of Floquet Ising systems with several inequivalent disorder protocols.

2. Self-duality, dual operators, and exact structures

A central organizing feature is the self-dual, or dual-unitary, point. In the spectral-statistics formulation with interaction

$$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$9

the one-period unitary is

$\sigma_{L+1}^z\equiv \sigma_1^z$0

and the self-dual point is

$\sigma_{L+1}^z\equiv \sigma_1^z$1

At this point, the temporal dual transfer matrix becomes unitary, enabling exact counting of unimodular eigenvalues and an exact spectral-form-factor ramp in the thermodynamic limit (Gupta et al., 18 Jun 2025).

The dual formulation rewrites $\sigma_{L+1}^z\equiv \sigma_1^z$2 as the partition function of a complex-weight two-dimensional Ising model on a $\sigma_{L+1}^z\equiv \sigma_1^z$3 lattice. Disorder-averaged spectral information is then encoded in an averaged transfer matrix

$\sigma_{L+1}^z\equiv \sigma_1^z$4

with

$\sigma_{L+1}^z\equiv \sigma_1^z$5

This representation makes disorder averaging nonperturbative and exposes the role of symmetries and boundary conditions (Gupta et al., 18 Jun 2025).

A complementary dual-operator construction for the spectral form factor uses

$\sigma_{L+1}^z\equiv \sigma_1^z$6

In the thermodynamic limit only the largest-magnitude eigenvalue $\sigma_{L+1}^z\equiv \sigma_1^z$7 survives. The criterion is operational: $\sigma_{L+1}^z\equiv \sigma_1^z$8 is associated with ergodic behavior and GOE-like spectral form factors, whereas $\sigma_{L+1}^z\equiv \sigma_1^z$9 and gapped from the rest implies exponential-in-$h_j$0 growth of $h_j$1, signaling localized behavior (Waltner et al., 2021).

In the kicked-Ising quantum-battery setting, the self-dual regime occurs at

$h_j$2

or, for non-uniform kick intervals $h_j$3, whenever

$h_j$4

At these values, the Floquet unitary factorizes into Clifford unitaries and the stroboscopic dynamics is exactly represented by a Clifford quantum cellular automaton. In momentum space, each pseudomomentum mode evolves via a $h_j$5 Floquet matrix $h_j$6 with

$h_j$7

and the Cayley–Hamilton theorem yields

$h_j$8

This exact modewise structure underlies closed-form expressions for energy injection and correlators (Romero et al., 21 Nov 2025).

3. Ergodicity, localization, and Floquet-MBL diagnostics

The disordered KIM is one of the most studied Floquet settings for the ergodic-to-MBL crossover. In the one-dimensional chain with $h_j$9, several diagnostics were computed from disorder-averaged eigenstates near eigenphase $g=J=1/W$0: the adjacent-gap ratio $g=J=1/W$1, the rescaled half-chain entanglement entropy $g=J=1/W$2, the Schmidt gap $g=J=1/W$3, a spin stiffness $g=J=1/W$4, and the rescaled quantum mutual information $g=J=1/W$5. The limiting benchmarks are $g=J=1/W$6 in the ergodic regime and $g=J=1/W$7 in the localized regime. In KIM, $g=J=1/W$8 interpolates between these values, $g=J=1/W$9 deep in MBL, $W$0 exhibits a pronounced maximum near the crossover, and $W$1 and $W$2 remain much larger and weakly $W$3-dependent beyond $W$4 (Sierant et al., 2022).

Finite-size trends are notably milder than in random-field XXZ chains. For KIM, crossing points $W$5 extracted from $W$6, $W$7, and $W$8 are consistently fitted by low-order polynomials in $W$9, yielding

$$\hat U = e^{-iJ\sum_n \hat\sigma_n^z\hat\sigma_{n+1}^z}\, e^{-i\sum_n h_n\hat\sigma_n^z}\, e^{-i(\pi/4)\sum_n \hat\sigma_n^x},$$0

The characteristic scale is $$\hat U = e^{-iJ\sum_n \hat\sigma_n^z\hat\sigma_{n+1}^z}\, e^{-i\sum_n h_n\hat\sigma_n^z}\, e^{-i(\pi/4)\sum_n \hat\sigma_n^x},$$1, compared with $$\hat U = e^{-iJ\sum_n \hat\sigma_n^z\hat\sigma_{n+1}^z}\, e^{-i\sum_n h_n\hat\sigma_n^z}\, e^{-i(\pi/4)\sum_n \hat\sigma_n^x},$$2, and a finite-size-scaling analysis based on $$\hat U = e^{-iJ\sum_n \hat\sigma_n^z\hat\sigma_{n+1}^z}\, e^{-i\sum_n h_n\hat\sigma_n^z}\, e^{-i(\pi/4)\sum_n \hat\sigma_n^x},$$3 gives

$$\hat U = e^{-iJ\sum_n \hat\sigma_n^z\hat\sigma_{n+1}^z}\, e^{-i\sum_n h_n\hat\sigma_n^z}\, e^{-i(\pi/4)\sum_n \hat\sigma_n^x},$$4

consistent with the Harris criterion in one dimension (Sierant et al., 2022).

The dual-operator approach gives a dynamical interpretation of the same crossover. Near the noninteracting limit $$\hat U = e^{-iJ\sum_n \hat\sigma_n^z\hat\sigma_{n+1}^z}\, e^{-i\sum_n h_n\hat\sigma_n^z}\, e^{-i(\pi/4)\sum_n \hat\sigma_n^x},$$5, the dominant eigenvalue has the exact strong-disorder expression

$$\hat U = e^{-iJ\sum_n \hat\sigma_n^z\hat\sigma_{n+1}^z}\, e^{-i\sum_n h_n\hat\sigma_n^z}\, e^{-i(\pi/4)\sum_n \hat\sigma_n^x},$$6

with $$\hat U = e^{-iJ\sum_n \hat\sigma_n^z\hat\sigma_{n+1}^z}\, e^{-i\sum_n h_n\hat\sigma_n^z}\, e^{-i(\pi/4)\sum_n \hat\sigma_n^x},$$7, $$\hat U = e^{-iJ\sum_n \hat\sigma_n^z\hat\sigma_{n+1}^z}\, e^{-i\sum_n h_n\hat\sigma_n^z}\, e^{-i(\pi/4)\sum_n \hat\sigma_n^x},$$8, and large-$$\hat U = e^{-iJ\sum_n \hat\sigma_n^z\hat\sigma_{n+1}^z}\, e^{-i\sum_n h_n\hat\sigma_n^z}\, e^{-i(\pi/4)\sum_n \hat\sigma_n^x},$$9 asymptotics

$h_n$0

After smoothing over the period-4 oscillations set by $h_n$1, the extracted $h_n$2 decays toward $h_n$3 in the ergodic regime, and the associated Thouless time obeys

$h_n$4

For $h_n$5, the inferred threshold is $h_n$6: below it the model is localized at short times and ergodic at longer times, whereas above it the localized behavior persists for all accessible times (Waltner et al., 2021).

A distinct hardware study on a 60-qubit heavy-hex lattice identified the crossover using late-time out-of-time-ordered correlators (OTOCs). For

$h_n$7

the operational crossover scale was defined by the maximal slope of the normalized OTOC at $h_n$8 and graph distance $h_n$9,

$b_x=\pi/4$0

This number is formulation-specific rather than universal: it belongs to a different geometry and disorder protocol than the one-dimensional $b_x=\pi/4$1 estimate (Hayata et al., 2 Oct 2025).

The stability of Floquet MBL remains an open issue in the thermodynamic limit and in higher dimensions. The literature summarized here is careful on this point: finite-size extrapolations are consistent and cross-diagnostic, but they do not constitute a mathematical proof that the extrapolated $b_x=\pi/4$2 equals the true asymptotic critical disorder, and higher-dimensional stability is explicitly left open (Sierant et al., 2022, Hayata et al., 2 Oct 2025).

4. Spectral statistics and the effect of boundary conditions

The spectral form factor

$b_x=\pi/4$3

is a primary diagnostic of spectral correlations in KIM. At the self-dual point and for odd $b_x=\pi/4$4, the thermodynamic-limit result is

$b_x=\pi/4$5

This reproduces the leading COE ramp $b_x=\pi/4$6 for $b_x=\pi/4$7, but higher moments reveal a much finer universality structure (Gupta et al., 18 Jun 2025).

The decisive distinction is between periodic and open boundaries. Under periodic boundary conditions at the self-dual point, an additional conjugation symmetry,

$b_x=\pi/4$8

forces $b_x=\pi/4$9 to behave as a real Gaussian random variable. Consequently,

$$H(t)=H_0+\lambda(t)\big[H_I+H_K\sum_{t_i\in\mathcal T}\delta(t-t_i)-H_0\big].$$0

Under open boundary conditions, by contrast, $$H(t)=H_0+\lambda(t)\big[H_I+H_K\sum_{t_i\in\mathcal T}\delta(t-t_i)-H_0\big].$$1 is complex Gaussian and

$$H(t)=H_0+\lambda(t)\big[H_I+H_K\sum_{t_i\in\mathcal T}\delta(t-t_i)-H_0\big].$$2

The ratio between the periodic and open moments is therefore

$$H(t)=H_0+\lambda(t)\big[H_I+H_K\sum_{t_i\in\mathcal T}\delta(t-t_i)-H_0\big].$$3

For $$H(t)=H_0+\lambda(t)\big[H_I+H_K\sum_{t_i\in\mathcal T}\delta(t-t_i)-H_0\big].$$4, the reported ratios are $$H(t)=H_0+\lambda(t)\big[H_I+H_K\sum_{t_i\in\mathcal T}\delta(t-t_i)-H_0\big].$$5, $$H(t)=H_0+\lambda(t)\big[H_I+H_K\sum_{t_i\in\mathcal T}\delta(t-t_i)-H_0\big].$$6, and $$H(t)=H_0+\lambda(t)\big[H_I+H_K\sum_{t_i\in\mathcal T}\delta(t-t_i)-H_0\big].$$7 (Gupta et al., 18 Jun 2025).

This boundary-condition sensitivity is stronger than the level of the two-point spectral form factor itself. Under periodic boundaries, the appropriate random-matrix comparators at the self-dual point are compact symmetric-space ensembles $$H(t)=H_0+\lambda(t)\big[H_I+H_K\sum_{t_i\in\mathcal T}\delta(t-t_i)-H_0\big].$$8 for even $$H(t)=H_0+\lambda(t)\big[H_I+H_K\sum_{t_i\in\mathcal T}\delta(t-t_i)-H_0\big].$$9 and $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$00 for odd $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$01, rather than the standard COE; under open boundaries, the numerics match COE behavior. Away from self-duality, both periodic and open systems revert to COE-like higher moments. The result is not merely quantitative: boundary conditions alter the Gaussian character of $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$02 itself (Gupta et al., 18 Jun 2025).

The same transfer-matrix technology extends to correlated-disorder echoes through the Loschmidt spectral form factor. For disorder draws with correlation $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$03, writing $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$04, the large-$$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$05 self-dual result is

$$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$06

while the fourth echo moment under periodic boundaries obeys

$$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$07

Open boundaries yield the same exponential scale but different prefactors, again reflecting the boundary-condition dependence of pairing structures in the transfer matrix (Gupta et al., 18 Jun 2025).

5. Disorder-resilient charging in the self-dual kicked Ising battery

In the quantum-battery realization, the self-dual KIM supports exact stroboscopic charging laws for arbitrary system size and number of kicks. With the ground-state energy of $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$08 shifted to zero, the injected energy for the $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$09-Ising charger at the self-dual point is

$$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$10

where the pseudomomenta $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$11 depend on boundary conditions and parity sector. This single formula generates a family of parity- and boundary-dependent plateaus and resonant maxima (Romero et al., 21 Nov 2025).

The stroboscopic patterns are highly structured.

Geometry and charger	Stroboscopic condition	Energy density
$$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$12, PBC, $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$13 even	$$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$14	$$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$15
$$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$16, PBC, $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$17 even	$$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$18	$$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$19
$$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$20, PBC, $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$21 even	otherwise	$$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$22
$$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$23, PBC, $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$24 odd	$$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$25	$$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$26
$$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$27, PBC, $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$28 odd	otherwise	$$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$29
$$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$30, PBC, $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$31 odd	$$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$32	$$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$33
$$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$34, PBC, $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$35 odd	$$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$36	$$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$37
OBC, both chargers	same as PBC odd-$$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$38 case	parity-independent

For $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$39 with periodic boundaries and even $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$40, the dynamics coincides with the $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$41 periodic odd-$$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$42 case. The periodicities are $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$43 or $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$44, depending on the sector (Romero et al., 21 Nov 2025).

Disorder does not affect this regime uniformly. For $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$45, periodic boundaries, and $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$46, averaging over $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$47 disorder realizations gave two reported regimes. For $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$48, the disorder-averaged energy density follows the disorder-free curves closely, including the maxima, minima, and $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$49 plateaus. For stronger disorder, such as $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$50, the average approaches

$$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$51

with weak $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$52-dependence, which the paper interprets as a balance between kick-induced delocalization and random pinning (Romero et al., 21 Nov 2025).

The non-uniform protocol replaces periodic kicks by a schedule in a fixed time window,

$$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$53

and in the dense-kick limit,

$$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$54

For $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$55, a uniform kicked Ising circuit yields $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$56 after one kick, whereas random schedules in $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$57 rapidly approach the saturation of the continuous Ising battery; approximately $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$58 kicks suffice to match continuous evolution at $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$59 (Romero et al., 21 Nov 2025).

The same work links charging to scrambling. Using

$$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$60

it reports linearly expanding light cones in the self-dual regime and shows that, for non-uniform kicking, a short window $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$61 suppresses spreading and energy injection, while a longer window $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$62 yields clear light cones and high injected energy. The stated conclusion is that low-frequency kicking, equivalently larger $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$63, boosts energy injection by allowing entanglement growth between kicks (Romero et al., 21 Nov 2025).

6. Numerical methods and quantum-hardware realizations

The disordered KIM has also served as a methodological benchmark for large-scale Floquet numerics. The polynomially filtered exact diagonalization (POLFED) algorithm targets eigenstates of a unitary near a desired eigenphase $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$64 using the filter

$$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$65

With $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$66, $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$67, and $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$68, the method converges to roughly $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$69 eigenvectors after $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$70 Lanczos steps with $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$71, and residual norms satisfy $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$72. For KIM this extended exact treatment to $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$73, with disorder averages exceeding $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$74 realizations for $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$75 (Sierant et al., 2022).

Tensor-network simulations provide an independent route in the charging problem. TEBD/MPS calculations with ITensor and MPO cutoff $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$76 reproduce the exact parity-dependent charging profiles for both $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$77 and $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$78, under OBC and PBC, and were used for systems as large as $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$79 (Romero et al., 21 Nov 2025).

Two IBM superconducting platforms have been used for direct experimental tests. In the quantum-battery study, the device was ibm_torino with $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$80 superconducting qubits. The $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$81 charger admits an exact gate decomposition

$$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$82

and for $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$83, periodic boundaries, and up to $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$84 kicks, measurements with $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$85 shots per circuit agreed with exact predictions, including the $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$86 plateaus and the non-uniform approach to continuous-chain saturation (Romero et al., 21 Nov 2025).

In the Floquet-MBL crossover experiment, the device was ibm_fez, IBM Heron r2, with $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$87 qubits, of which $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$88 formed a heavy-hex patch. The reported median errors at job submission were CZ error $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$89, SX error $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$90, and readout error $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$91. OTOCs were measured up to $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$92 Floquet cycles with $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$93 shots per circuit, using Pauli twirling, TREX readout mitigation, manual causal-cone pruning, and two independent mitigation strategies: operator renormalization and zero-noise extrapolation. Both methods located the crossover at $$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$94 within uncertainties (Hayata et al., 2 Oct 2025).

An earlier line of work studied disorder generated not by quenched random fields but by disturbed kick trains. There the stroboscopic evolution is assembled period by period from free Ising evolution and instantaneous rank-one kicks, with disorder propagated by classical torus maps such as stationary, drifting, microcanonical, or Markovian baths. In open Ising chains, the reported effect depends strongly on coupling orientation: Ising-$$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$95 exhibits a short-time coherence plateau and pronounced edge effects, whereas Ising-$$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$96 yields the fastest relaxation to microcanonical populations and the strongest entanglement growth (Aubourg et al., 2014).

These numerical and experimental developments have made the disordered KIM unusually versatile. It is analytically tractable at self-duality, amenable to large-$$U=\exp\!\left(-i g \sum_{j=1}^{L}\sigma_j^x\right)\, \exp\!\left[-i \sum_{j=1}^{L}\left(J\sigma_j^z\sigma_{j+1}^z+h_j\sigma_j^z\right)\right],$$97 spectral computations, compatible with tensor-network simulation in structured regimes, and implementable on current superconducting hardware at scales beyond classical exact diagonalization (Sierant et al., 2022, Romero et al., 21 Nov 2025, Hayata et al., 2 Oct 2025).