Disordered Kicked Ising Model

• Disordered Kicked Ising Model is a Floquet quantum spin system that alternates between a transverse-field kick and a disordered Ising interaction to probe many-body localization and ergodicity breaking.
• It employs both numerical and experimental techniques, including OTOCs, gap ratio statistics, and advanced POLFED algorithms, to benchmark quantum simulation protocols.
• The model offers actionable insights into finite-size scaling, error mitigation strategies, and the realization of Floquet phases, driving further exploration of quantum chaos and localization.

The disordered kicked Ising model (KIM) is a paradigmatic driven (Floquet) quantum spin system that serves as a minimal platform for controlled studies of many-body localization (MBL) and ergodicity breaking in the presence of temporal periodicity and disorder. In its standard form, the KIM comprises spin-½ degrees of freedom evolving under a stroboscopically applied sequence of non-commuting unitary operations: a transverse-field “kick” and an Ising interaction with static on-site disorder. This construction, realized both in 1D chains and higher-dimensional lattices, enables detailed numerical, analytical, and experimental investigation of the mechanisms underpinning MBL and quantum chaos in non-autonomous settings, as well as systematic benchmarking of quantum simulation protocols and error-mitigation strategies in state-of-the-art quantum hardware (Hayata et al., 2 Oct 2025, Sierant et al., 2022).

1. Model Definition and Floquet Structure

The KIM is defined via a time-dependent Hamiltonian that alternates between a transverse-field drive and a longitudinal Ising layer, with periodicity $T$. In the heavy-hex lattice realization, the Hamiltonian over one period is: $$H(t) = \begin{cases} H_x & 0 \leq t < T/2 \ H_z & T/2 \leq t < T \end{cases}$$ where

$$H_x = \sum_{i=1}^N B_x(i) X_i, \quad H_z = J \sum_{\langle i,j\rangle} Z_i Z_j + B_z \sum_{i=1}^N Z_i.$$

Here, $X_i$ and $Z_i$ are Pauli operators on site $i$, $B_x(i)$ (drawn i.i.d. from $[B_{x0}-W, B_{x0}+W]$) parameterizes on-site disorder, $J$ is the Ising coupling, and $B_z$ a uniform longitudinal field. The disorder strength $$H(t) = \begin{cases} H_x & 0 \leq t < T/2 \ H_z & T/2 \leq t < T \end{cases}$$0 tunes the system between ergodic (chaotic) and localized (MBL-like) phases.

The associated Floquet operator for one period is

$$H(t) = \begin{cases} H_x & 0 \leq t < T/2 \ H_z & T/2 \leq t < T \end{cases}$$1

so stroboscopic evolution proceeds as $$H(t) = \begin{cases} H_x & 0 \leq t < T/2 \ H_z & T/2 \leq t < T \end{cases}$$2. The standard 1D KIM varies this structure slightly: $$H(t) = \begin{cases} H_x & 0 \leq t < T/2 \ H_z & T/2 \leq t < T \end{cases}$$3 with $$H(t) = \begin{cases} H_x & 0 \leq t < T/2 \ H_z & T/2 \leq t < T \end{cases}$$4 and $$H(t) = \begin{cases} H_x & 0 \leq t < T/2 \ H_z & T/2 \leq t < T \end{cases}$$5 parameterizing kick and Ising strengths and $$H(t) = \begin{cases} H_x & 0 \leq t < T/2 \ H_z & T/2 \leq t < T \end{cases}$$6 uniformly random on-site fields, $$H(t) = \begin{cases} H_x & 0 \leq t < T/2 \ H_z & T/2 \leq t < T \end{cases}$$7. This construction ensures energy is not conserved, in contrast to autonomous spin chains.

2. Digital Quantum Simulation and Experimental Realization

Floquet KIMs are amenable to digital simulation on quantum processors, as the entire Floquet period coincides with a minimal, non-Trotterized quantum circuit layer. Each period is implemented as two sequential layers: all single-qubit $$H(t) = \begin{cases} H_x & 0 \leq t < T/2 \ H_z & T/2 \leq t < T \end{cases}$$8 (SX) rotations for the kick, followed by parallelized two-qubit $$H(t) = \begin{cases} H_x & 0 \leq t < T/2 \ H_z & T/2 \leq t < T \end{cases}$$9 interactions and $$H_x = \sum_{i=1}^N B_x(i) X_i, \quad H_z = J \sum_{\langle i,j\rangle} Z_i Z_j + B_z \sum_{i=1}^N Z_i.$$0 rotations for the Ising and field terms. Heavy-hex connectivity supports efficient edge coloring, allowing maximal parallelization of two-qubit gates for Ising interactions.

A recent large-scale implementation utilized a 60-qubit patch on the IBM Heron r2 (“ibm_fez”) device, with circuit construction mapped directly onto hardware-native gates. Error mitigation was achieved using operator renormalization (self-calibration, via normalization of out-of-time-ordered correlators) and zero-noise extrapolation (ZNE) via gate folding, with raw observables extrapolated to the zero-noise limit to benchmark against self-calibrated quantities (Hayata et al., 2 Oct 2025).

3. Ergodic–MBL Crossover and Diagnostic Observables

The ergodic–MBL transition in the disordered KIM is characterized using both dynamical and eigenstate-based diagnostics:

• Out-of-time-ordered correlators (OTOCs): For a local X perturbation at site 1 and late-time $$H_x = \sum_{i=1}^N B_x(i) X_i, \quad H_z = J \sum_{\langle i,j\rangle} Z_i Z_j + B_z \sum_{i=1}^N Z_i.$$1 measurements at distance $$H_x = \sum_{i=1}^N B_x(i) X_i, \quad H_z = J \sum_{\langle i,j\rangle} Z_i Z_j + B_z \sum_{i=1}^N Z_i.$$2, the normalized OTOC,

$$H_x = \sum_{i=1}^N B_x(i) X_i, \quad H_z = J \sum_{\langle i,j\rangle} Z_i Z_j + B_z \sum_{i=1}^N Z_i.$$3

with $$H_x = \sum_{i=1}^N B_x(i) X_i, \quad H_z = J \sum_{\langle i,j\rangle} Z_i Z_j + B_z \sum_{i=1}^N Z_i.$$4 and $$H_x = \sum_{i=1}^N B_x(i) X_i, \quad H_z = J \sum_{\langle i,j\rangle} Z_i Z_j + B_z \sum_{i=1}^N Z_i.$$5 defined via stroboscopic sequences of conjugated operators and identity, robustly discriminates between fast operator spreading (chaos) and restricted growth (MBL).

• Spectral statistics (gap ratio): For Floquet eigenphases $$H_x = \sum_{i=1}^N B_x(i) X_i, \quad H_z = J \sum_{\langle i,j\rangle} Z_i Z_j + B_z \sum_{i=1}^N Z_i.$$6, the gap ratio $$H_x = \sum_{i=1}^N B_x(i) X_i, \quad H_z = J \sum_{\langle i,j\rangle} Z_i Z_j + B_z \sum_{i=1}^N Z_i.$$7, with $$H_x = \sum_{i=1}^N B_x(i) X_i, \quad H_z = J \sum_{\langle i,j\rangle} Z_i Z_j + B_z \sum_{i=1}^N Z_i.$$8 distinguishes chaotic (COE: $$H_x = \sum_{i=1}^N B_x(i) X_i, \quad H_z = J \sum_{\langle i,j\rangle} Z_i Z_j + B_z \sum_{i=1}^N Z_i.$$9) and localized (Poisson: $X_i$0) regimes (Sierant et al., 2022).
• Eigenstate entanglement: Bipartite entanglement entropy $X_i$1 and normalized entropy $X_i$2 (by random-matrix averages) diagnose transition from volume to area law as $X_i$3 increases.
• Schmidt gap $X_i$4 and mutual information $X_i$5: Both show sharp features at the transition: $X_i$6 finite in MBL, $X_i$7 peaking near criticality.
• Spin-stiffness autocorrelation $X_i$8: Captures persistent memory in the MBL regime.

For 2D heavy-hex KIM, late-time, mid-cone OTOC (at $X_i$9, $Z_i$0) acts as a practical “order parameter.” As disorder strength $Z_i$1 increases, this OTOC interpolates steeply from nearly $Z_i$2 (chaotic) to nearly $Z_i$3 (MBL), with the maximally fast change marking a crossover at $Z_i$4 (Hayata et al., 2 Oct 2025). In the 1D KIM, finite-size scaling of various indicators produce a crossover near $Z_i$5 (Sierant et al., 2022).

4. Advanced Numerical Techniques — POLFED

Numerical investigation of moderate to large 1D KIM systems is enabled by the POLFED (Polynomially Filtered Exact Diagonalization) algorithm. This approach applies spectral filtering via a geometric-sum polynomial in $Z_i$6, selecting eigenstates within a target quasienergy window, followed by block-Lanczos iteration to converge on the desired eigenpairs. This method permits exploration of system sizes up to $Z_i$7, substantially beyond conventional shift-invert diagonalization. Observables such as gap statistics, entanglement, mutual information, and autocorrelations can then be accurately extracted in the critical and MBL regimes (Sierant et al., 2022).

5. Finite-Size Scaling and Comparison to Autonomous Chains

The finite-size crossover and scaling behavior are central to distinguishing genuine MBL from finite-system crossovers. In KIM, the ergodic-regime boundary $Z_i$8 exhibits an initial linear drift with $Z_i$9 that sublinearizes and bends towards stabilization beyond $i$0. The effective critical point $i$1, identified via the crossing of diagnostic curves, converges polynomially in $i$2 to a thermodynamic value $i$3. Finite-size scaling with $i$4 collapses data for various indicators with $i$5, satisfying the Harris/Chayes bound for 1D disordered systems.

Comparison with autonomous models (e.g., random-field XXZ and transverse-field Ising chains) highlights reduced finite-size effects in Floquet KIM: characteristic crossover lengths $i$6 are considerably smaller (KIM: 28 versus XXZ: 50), suggesting that the absence of energy conservation in KIM leads to earlier stabilization of MBL signatures (Sierant et al., 2022). Abelian symmetries, such as $i$7 or $i$8, have minor influence by comparison.

6. Implications, Robustness, and Extensions

These studies collectively support the stability of Floquet MBL and underscore the utility of KIM for controlled exploration of nonequilibrium quantum dynamics. Robust error mitigation in experimental quantum simulation is demonstrated via agreement of independent OTOC normalization and ZNE extrapolation, even in disorder-driven crossovers. The suppression of finite-size drifts suggests that Floquet KIM may provide a firmer basis for realizing distinct Floquet phases, including Floquet time crystals and Floquet-protected topological order, stabilized by localization.

Ongoing directions include applying improved spectral filtering or tensor-network–augmented approaches to access larger system sizes; extending exploration to higher-dimensional driven MBL; and analyzing the role of non-Abelian symmetries or long-range interactions within the Floquet-MBL paradigm (Hayata et al., 2 Oct 2025, Sierant et al., 2022).