Charge-Preserving Floquet Circuits

• Charge-preserving Floquet circuits are discrete-time quantum systems that locally conserve U(1) charge through block-diagonal gate structures.
• They exhibit anomalously slow, subdiffusive thermalization with delayed entanglement growth as shown by quasienergy statistics and charge autocorrelation analysis.
• Architectural extensions such as increasing gate range, local Hilbert space, or Floquet period can restore robust chaotic thermalization.

Searching arXiv for recent and foundational papers directly related to charge-preserving Floquet circuits and nearby Floquet-code/topological contexts. Charge-preserving Floquet circuits are discrete-time, periodically driven quantum circuits in which each local gate exactly conserves a charge, most explicitly a $U(1)$ charge such as the total $z$-magnetization $S_z^{\rm tot}=\sum_j Z_j$. In the directly relevant formulation, the circuit is Floquet because the same finite sequence of layers is repeated every period, so the stroboscopic evolution is generated by a Floquet unitary $U_F(T)$, while charge preservation is enforced locally by requiring every gate to be block diagonal in the appropriate charge sectors (Jonay et al., 2022). Within this class, the best-studied minimal model is a nearest-neighbor spin-$\tfrac12$ random Floquet circuit with Haar-random $U(1)$-conserving gates, and its main significance is negative in a precise sense: despite local randomness and periodic driving, the minimal architecture thermalizes only very slowly at numerically accessible sizes and exhibits long-time subdiffusive dynamics, whereas modest enlargements of the local architecture restore robust chaotic thermalization (Jonay et al., 2022).

1. Formal definition and circuit architecture

In the minimal construction, the Floquet character is entirely kinematic: one period consists of a fixed brickwork sequence of local unitaries, and that sequence is repeated indefinitely. The one-period propagator is written as

$$U_F(T)=\prod_{t=0}^{T-1} U(t+1,t), \qquad U(t=nT,0)=U_F(T)^n.$$

Because the drive is discrete-time and periodic, energy is not conserved; the nontrivial conservation law is instead the on-site $U(1)$ charge $S_z^{\rm tot}$ (Jonay et al., 2022).

For the nearest-neighbor case with range $r=2$, one time step is a two-layer even/odd brickwork circuit,

$z$0

with nearest-neighbor gates applied first on one parity of bonds and then on the other. Charge preservation is imposed gate by gate. For spin-$z$1 qubits, each two-site unitary is block diagonal in the total $z$2-charge sector of the pair: the $z$3 and $z$4 sectors only acquire phases, while the $z$5 sector is acted on by a general $z$6 unitary. In that sense, a gate cannot change the number of up spins on the sites it touches (Jonay et al., 2022).

The directly studied random ensemble draws each symmetry block independently from the Haar measure on its unitary group. The central minimal model is specified by three constraints simultaneously: local Hilbert space $z$7, interaction range $z$8, and Floquet period $z$9. The same two-layer even/odd brickwork is therefore repeated every time step, making the model maximally local and maximally constrained within the class considered (Jonay et al., 2022).

2. Local gate structure and conserved-charge kinematics

A central technical feature of charge-preserving Floquet circuits is that the $S_z^{\rm tot}=\sum_j Z_j$0 constraint admits a useful decomposition of the local gate manifold. In the minimal $S_z^{\rm tot}=\sum_j Z_j$1, $S_z^{\rm tot}=\sum_j Z_j$2 model, the two-site gate depends on six real parameters: three phases in the $S_z^{\rm tot}=\sum_j Z_j$3 charge sectors and an $S_z^{\rm tot}=\sum_j Z_j$4-like block in the zero-charge sector. The same gate can also be parameterized as

$S_z^{\rm tot}=\sum_j Z_j$5

with a basis of $S_z^{\rm tot}=\sum_j Z_j$6-symmetric generators chosen so that the first three commute with the remaining three. The factorization implied by this commutation structure is important because it separates diagonal and transport-generating components of the dynamics (Jonay et al., 2022).

Under a Jordan–Wigner mapping, the generator basis becomes a set of fermionic operators with a transparent interpretation. In the notation of the source, $S_z^{\rm tot}=\sum_j Z_j$7, $S_z^{\rm tot}=\sum_j Z_j$8, and $S_z^{\rm tot}=\sum_j Z_j$9 correspond to local potential-like terms; $U_F(T)$0 and $U_F(T)$1 correspond to real and complex fermion hopping; and $U_F(T)$2 is a density-density interaction (Jonay et al., 2022). This fermionic re-expression is not merely cosmetic. It identifies the specific couplings that interpolate between ergodic, free-fermion, and diagonal limits, and it provides the microscopic basis for the unusually slow dynamics of the minimal circuit.

A useful consequence of this parameterization is that “charge-preserving” is stronger than a global symmetry statement about the full Floquet unitary. The conservation law is enforced locally at every gate application. In the studied ensemble, the symmetry therefore constrains both transport and operator spreading at the microscopic level, rather than emerging only after averaging or coarse graining. This suggests that slow hydrodynamic relaxation in such circuits should be understood as a structural consequence of the local gate manifold, not as an accidental feature of a particular disorder realization (Jonay et al., 2022).

3. The minimal random Floquet circuit and its anomalously slow dynamics

The most prominent result for charge-preserving Floquet circuits is that the minimal nearest-neighbor spin-$U_F(T)$3 Floquet circuit with Haar-random $U_F(T)$4-conserving gates is not robustly thermalizing at numerically accessible system sizes and displays slow subdiffusive dynamics for long times (Jonay et al., 2022). The result is notable because the model is random, local, and periodically driven, and would therefore often be expected to behave as a conventional chaotic Floquet system.

The paper diagnoses the dynamics through quasienergy level statistics, charge autocorrelation, and entanglement growth. Charge transport is characterized through

$U_F(T)$5

evaluated in fixed-magnetization states. Ordinary diffusion would give

$U_F(T)$6

but the minimal $U_F(T)$7, $U_F(T)$8, $U_F(T)$9 circuit instead exhibits subdiffusive relaxation over long time windows, meaning decay slower than $\tfrac12$0. The source does not claim a sharply extracted universal exponent, but it does state that the decay is clearly slower than diffusive on accessible scales (Jonay et al., 2022).

Entanglement growth is likewise delayed. The half-chain von Neumann entropy

$\tfrac12$1

should, in a thermalizing Floquet system, grow and saturate on a timescale of order $\tfrac12$2. In the minimal model, saturation is dramatically delayed, and the entropy approaches the $\tfrac12$3-symmetric Haar expectation only very slowly. By contrast, the $\tfrac12$4 and random-in-time circuits saturate much faster (Jonay et al., 2022).

The level-statistics analysis points in the same direction. Using the ratio of adjacent quasienergy spacings,

$\tfrac12$5

the relevant random-matrix benchmark is the GUE value $\tfrac12$6, while Poisson gives $\tfrac12$7. The $\tfrac12$8, $\tfrac12$9, $U(1)$0 model reaches GUE statistics rapidly even for small $U(1)$1, whereas the minimal $U(1)$2, $U(1)$3, $U(1)$4 circuit converges much more slowly (Jonay et al., 2022).

4. Proximate localized and integrable regimes

The slow dynamics of the minimal architecture are explained in the source by the geometry of the allowed gate manifold. The minimal circuit is not treated as a featureless chaotic ensemble; rather, it lies close to several special regimes in parameter space that are localized or nearly integrable (Jonay et al., 2022).

The first special regime is a free-fermion or Anderson localized limit. Setting

$U(1)$5

removes the interaction and yields a disordered free-fermion Floquet model that is Anderson localized in one dimension. The second special regime is a diagonal or classical localized limit. Setting

$U(1)$6

turns off hopping and makes the circuit diagonal in the $U(1)$7-basis, producing a trivially localized classical disordered spin model. The third ingredient is specifically Floquet: because the evolution is periodic, the couplings are effectively defined modulo $U(1)$8, so interactions cannot simply be increased without bound to wash out nearby special structures (Jonay et al., 2022).

To clarify the role of these nearby corners, the source studies two deformations. In the “perturbed Anderson localized model,” $U(1)$9 is held fixed while the remaining five parameters are Haar-random. Here $$U_F(T)=\prod_{t=0}^{T-1} U(t+1,t), \qquad U(t=nT,0)=U_F(T)^n.$$0 is free, $$U_F(T)=\prod_{t=0}^{T-1} U(t+1,t), \qquad U(t=nT,0)=U_F(T)^n.$$1 is also effectively free because the $$U_F(T)=\prod_{t=0}^{T-1} U(t+1,t), \qquad U(t=nT,0)=U_F(T)^n.$$2 phases cancel between even and odd layers, and $$U_F(T)=\prod_{t=0}^{T-1} U(t+1,t), \qquad U(t=nT,0)=U_F(T)^n.$$3 is the most ergodic point in this one-parameter family. In the “perturbed diagonal model,” the hopping amplitudes are tuned by

$$U_F(T)=\prod_{t=0}^{T-1} U(t+1,t), \qquad U(t=nT,0)=U_F(T)^n.$$4

with $$U_F(T)=\prod_{t=0}^{T-1} U(t+1,t), \qquad U(t=nT,0)=U_F(T)^n.$$5 for simplicity; $$U_F(T)=\prod_{t=0}^{T-1} U(t+1,t), \qquad U(t=nT,0)=U_F(T)^n.$$6 gives diagonal localized dynamics, and finite $$U_F(T)=\prod_{t=0}^{T-1} U(t+1,t), \qquad U(t=nT,0)=U_F(T)^n.$$7 restores hopping. In both deformations, the $$U_F(T)=\prod_{t=0}^{T-1} U(t+1,t), \qquad U(t=nT,0)=U_F(T)^n.$$8 circuit approaches ergodicity only slowly, whereas the $$U_F(T)=\prod_{t=0}^{T-1} U(t+1,t), \qquad U(t=nT,0)=U_F(T)^n.$$9 circuit rapidly becomes chaotic (Jonay et al., 2022).

A plausible implication is that charge-preserving Floquet circuits are highly sensitive to how symmetry, locality, and temporal periodicity are combined. In this reading, the minimal circuit is slow not because charge conservation generically suppresses thermalization, but because the smallest nontrivial $U(1)$0-symmetric gate architecture sits unusually close to free and diagonal bottlenecks in parameter space (Jonay et al., 2022).

5. Architectural extensions that restore robust thermalization

The same study maps out a broader family of charge-preserving Floquet circuits and shows that small extensions are sufficient to recover robust chaotic thermalization (Jonay et al., 2022). The three modifications are: increasing the interaction range to three-site gates, enlarging the local Hilbert space by appending an unconstrained qudit of dimension $U(1)$1, and increasing the Floquet period to $U(1)$2 by repeating two independently chosen layers rather than one.

Modification	Parameter change	Reported effect
Larger local Hilbert space	$U(1)$3	Restores rapid GUE-like level statistics
Longer-range gates	$U(1)$4	Restores robust thermalization
Larger Floquet period	$U(1)$5	Quickly becomes chaotic; diffusive scaling consistent with $U(1)$6 autocorrelation decay

The interpretation given in the source is that all three changes make the circuit less locally constrained. Enlarging the on-site Hilbert space dilutes the effect of the $U(1)$7 constraint by giving the local gates more room to mix states while still conserving charge. Three-site gates enlarge the local unitary blocks and increase connectivity within each charge sector. A two-step Floquet period increases the effective randomness in time and weakens the special periodic structure that impedes thermalization in the $U(1)$8 architecture (Jonay et al., 2022).

These observations sharply delimit what is and is not generic about the minimal model. Slow thermalization, subdiffusion, and delayed entanglement saturation are robust observations for the smallest $U(1)$9-conserving brickwork Floquet circuit. They are not, however, presented as universal properties of all charge-preserving Floquet circuits. Within the broader architectural family, modest increases in local complexity are already enough to recover the more familiar picture of rapid thermalization and random-matrix behavior (Jonay et al., 2022).

6. Relation to topological pumping and to Floquet-code circuits

The phrase “charge-preserving Floquet circuits” can be conflated with two adjacent but distinct literatures: Floquet topological transport and Floquet codes. The supplied sources make both distinctions explicit.

First, temporally disordered Floquet topological insulators provide a nearby example in which quantized charge transport persists even though Anderson localization is destroyed by temporal disorder. In that setting, the bulk becomes diffusive at long times, with $S_z^{\rm tot}$0, while the cumulative averaged pumped charge approaches quantization and the dynamic corrections vanish asymptotically, following $S_z^{\rm tot}$1 in the long-time regime (Berg et al., 2023). The broader message drawn there is that charge-preserving Floquet circuits may realize robust quantized transport without relying on Anderson localization. This suggests a conceptual overlap with the random-circuit results: diffusive bulk motion and nontrivial charge dynamics need not be mutually exclusive in periodically driven systems (Berg et al., 2023).

Second, not every “Floquet circuit” is charge-preserving in the symmetry sense used above. The dynamic circuit for the honeycomb Floquet code is a periodically repeated gauge-measurement protocol on a honeycomb lattice, with a period-3 cycle $S_z^{\rm tot}$2, and it demonstrates ancilla-free gauge measurement, improved timelike distance, automatic leakage removal, and reduced qubit overhead (Claes, 10 Jul 2025). However, the supplied text does not establish any conservation law such as $S_z^{\rm tot}$3 charge preservation, parity conservation, or a fixed excitation-number constraint. It therefore belongs to the Floquet-code literature, not to the symmetry-constrained charge-preserving Floquet-circuit literature in the narrow sense (Claes, 10 Jul 2025).

A common misconception is therefore that any periodically repeated circuit involving charge transport, Pauli measurements, or Floquet code structure is automatically a charge-preserving Floquet circuit. The sources do not support that identification. In the direct random-circuit setting, charge preservation means that every gate commutes with a global conserved charge operator, such as total $S_z^{\rm tot}$4-magnetization. In the topological-pumping setting, the emphasis is on asymptotically quantized transport under temporal disorder. In the Floquet-code setting, the emphasis is on repeated stabilizer or gauge measurements and decoding performance, without an established charge-conservation law in the supplied material (Jonay et al., 2022, Berg et al., 2023, Claes, 10 Jul 2025).