Floquet Clifford Circuits: Dynamics & Structures

• Floquet Clifford circuits are exactly solvable systems defined by periodic Clifford unitaries that map Pauli strings to Pauli strings, ensuring classical simulability.
• They showcase distinct dynamical regimes, including strong localization in one dimension and ballistic operator spreading in two dimensions, along with operator-space fragmentation.
• Different architectures—from local disordered models to nonlocal fast-scrambling designs—provide insights into thermalization, spectral diagnostics, and dissipative behavior.

Floquet Clifford circuits are periodic time evolutions in which one Floquet period is implemented by a Clifford unitary, either as a finite-depth circuit on a qubit lattice or as repeated application of a translationally invariant Clifford quantum cellular automaton. Their defining structural property is that Clifford evolution maps Pauli strings to Pauli strings under conjugation, so operator dynamics stays within the Pauli group and is classically efficiently simulable via the stabilizer or symplectic formalism. Within that exactly solvable setting, the literature identifies sharply different dynamical regimes, including strong localization, ballistic operator growth, operator-space fragmentation, mixing and thermalization to the infinite-temperature state, finite- and divergent-gap dissipative relaxation, and fast-scrambling behavior in nonlocal architectures (Farshi et al., 2022, Kapustin et al., 1 Jan 2026, Ma et al., 27 Mar 2026).

1. Definition and formal frameworks

A Clifford unitary is a unitary operator that normalizes the Pauli group under conjugation. In the finite-$n$ setting, the $n$-qubit Pauli group is

$$P(n)=\{\lambda P_1\otimes\cdots\otimes P_n \mid \lambda\in\{\pm 1,\pm i\},\; P_j\in\{I,X,Y,Z\}\},$$

and an $n$-qubit Clifford operator $C$ satisfies $CPC^{-1}\in P(n)$ for every $P\in P(n)$. Selinger’s algebraic treatment provides a unique normal form for every Clifford operator, together with a terminating rewrite system and a generators-and-relations presentation based on $\omega$, $H$, $S$, and controlled-$n$0 (Selinger, 2013).

In disordered lattice realizations, Floquet Clifford dynamics is defined by periodic time evolution $n$1, where the one-period unitary is a local Clifford circuit. In the one- and two-dimensional random models, the gates are sampled once in space and then reused every period, so the disorder is quenched rather than annealed. This distinction is important in localization problems, because randomness is fixed within the Floquet period rather than resampled in time (Farshi et al., 2022).

A second framework replaces finite-depth circuit layers by a locality-preserving automorphism on an infinite lattice. In the Clifford quantum cellular automaton formulation, the Floquet dynamics is the repeated application of a fixed translationally invariant automorphism $n$2, so the evolution is $n$3 on $n$4. Translation invariance allows the action on Pauli monomials to be encoded by a Laurent-polynomial matrix $n$5 over $n$6, with

$n$7

This converts Floquet operator dynamics into linear algebra over a finite field and makes ergodic and thermalization questions amenable to exact analysis (Kapustin et al., 1 Jan 2026).

2. Circuit architectures and algebraic organization

Several distinct circuit architectures fall under the label “Floquet Clifford circuits.” In the two-dimensional random model, one Floquet period is a depth-2 circuit on a spin-$n$8 lattice, with independently sampled uniformly random $n$9-qubit Clifford gates acting on $$P(n)=\{\lambda P_1\otimes\cdots\otimes P_n \mid \lambda\in\{\pm 1,\pm i\},\; P_j\in\{I,X,Y,Z\}\},$$0 plaquettes. Its one-dimensional analogue is a brickwork circuit of random $$P(n)=\{\lambda P_1\otimes\cdots\otimes P_n \mid \lambda\in\{\pm 1,\pm i\},\; P_j\in\{I,X,Y,Z\}\},$$1-qubit Clifford gates sampled from $$P(n)=\{\lambda P_1\otimes\cdots\otimes P_n \mid \lambda\in\{\pm 1,\pm i\},\; P_j\in\{I,X,Y,Z\}\},$$2. Both architectures are local and disordered, but their long-time dynamics differs strongly with spatial dimension (Farshi et al., 2022).

A related one-dimensional architecture adds non-Clifford perturbations to a nearest-neighbour Clifford brickwork Floquet circuit. One period is

$$P(n)=\{\lambda P_1\otimes\cdots\otimes P_n \mid \lambda\in\{\pm 1,\pm i\},\; P_j\in\{I,X,Y,Z\}\},$$3

where the $$P(n)=\{\lambda P_1\otimes\cdots\otimes P_n \mid \lambda\in\{\pm 1,\pm i\},\; P_j\in\{I,X,Y,Z\}\},$$4 are sampled from the non-product two-qubit Clifford classes and the $$P(n)=\{\lambda P_1\otimes\cdots\otimes P_n \mid \lambda\in\{\pm 1,\pm i\},\; P_j\in\{I,X,Y,Z\}\},$$5 are stochastic single-qubit perturbation gates. The paper distinguishes $$P(n)=\{\lambda P_1\otimes\cdots\otimes P_n \mid \lambda\in\{\pm 1,\pm i\},\; P_j\in\{I,X,Y,Z\}\},$$6-like, $$P(n)=\{\lambda P_1\otimes\cdots\otimes P_n \mid \lambda\in\{\pm 1,\pm i\},\; P_j\in\{I,X,Y,Z\}\},$$7-like, $$P(n)=\{\lambda P_1\otimes\cdots\otimes P_n \mid \lambda\in\{\pm 1,\pm i\},\; P_j\in\{I,X,Y,Z\}\},$$8-like, and $$P(n)=\{\lambda P_1\otimes\cdots\otimes P_n \mid \lambda\in\{\pm 1,\pm i\},\; P_j\in\{I,X,Y,Z\}\},$$9-like two-qubit Clifford classes, with $n$0-like gates supplying the microscopic structures that generate localization walls, whereas $n$1 and $n$2 favor more ballistic spreading (Kovács et al., 2024).

Another architecture fixes the two-qubit backbone and modifies only selected qubit lines. In the dissipative Haar-doped construction, each Floquet period consists of a two-layer brickwork of the fixed Clifford gate

$n$3

followed by Haar-random single-qubit gates on a chosen subset of sites and then an onsite depolarizing layer of strength $n$4. If $n$5 qubit lines are doped in a system of size $n$6, the doping density is $n$7. This architecture is designed to isolate how much deviation from strictly Clifford dynamics is needed to change the Liouvillian gap in the thermodynamic limit (Kim et al., 3 Feb 2026).

A more nonlocal realization appears in circuits inspired by matrix-model dynamics. There, each discrete time step is a global permutation of matrix indices followed by a layer of disjoint four-qubit interaction gates,

$n$8

The four-qubit gate $n$9 is itself Clifford, built from $C$0, $C$1, and CNOT gates, and the permutation is chosen to mimic matrix-model index contractions while remaining compatible with an experimentally motivated interaction pattern. This produces a classically tractable Floquet Clifford “cartoon” of a nonlocal scrambling system (Ma et al., 27 Mar 2026).

3. Operator spreading, localization, and fragmentation

One of the sharpest results for Floquet Clifford circuits is the dimensional contrast between one and two spatial dimensions. In the two-dimensional random model, localization is ruled out rigorously by proving that some local Pauli operators reach the boundary of the light cone at ballistic speed. The localization criterion is framed as

$C$2

and the theorem states that, for a single-site Pauli operator, with probability at least $C$3 its evolution is non-identity on some sites of the light-cone boundary for all half-integer times $C$4. The proof reduces boundary propagation to a random directed graph, compares it to an independent-arrow graph, and uses a dual-wall counting argument to show that the blocking-wall probability is at most about $C$5, leaving probability at least $C$6 for indefinite propagation. This establishes the absence of localization in two dimensions (Farshi et al., 2022).

The one-dimensional analogue behaves oppositely. There, random left-blocking and right-blocking walls arrest operator growth, and any operator trapped between them remains confined forever. The appendix gives

$C$7

so $C$8 acts as an upper bound on the localization length. Numerical operator spreading is consistent with this picture: $C$9 in one dimension, with $CPC^{-1}\in P(n)$0, whereas in two dimensions $CPC^{-1}\in P(n)$1 inside the light cone and drops sharply outside it. The result is a strong Anderson-type localization in one dimension and delocalized ballistic spreading in two dimensions (Farshi et al., 2022).

Perturbing the one-dimensional Floquet-Clifford brickwork circuit away from the Clifford limit does not immediately destroy localization. Instead, the dynamics exhibits operator space fragmentation: the Pauli operator algebra decomposes into disjoint invariant sectors generated by wall configurations. A $CPC^{-1}\in P(n)$2-wall is a local circuit substructure of width $CPC^{-1}\in P(n)$3 that prevents operator growth across a central region, and for $CPC^{-1}\in P(n)$4-walls the left and right internal subspaces coincide and are one-dimensional, so every irreducible $CPC^{-1}\in P(n)$5-wall conserves some Pauli operator. The paper further proves that walls in local Clifford circuits are two-sided and that their internal subspaces are $CPC^{-1}\in P(n)$6-orthogonal. For random single-qubit perturbations applied with probability $CPC^{-1}\in P(n)$7, localization remains stable for all $CPC^{-1}\in P(n)$8, with stopping probability

$CPC^{-1}\in P(n)$9

and a tunable localization length

$P\in P(n)$0

Larger walls are exponentially rare, so $P\in P(n)$1- and $P\in P(n)$2-walls dominate the localization physics (Kovács et al., 2024).

4. Entanglement and spectral diagnostics

Entanglement growth in Floquet Clifford circuits tracks the same distinction between localized and delocalized operator dynamics. In the one-dimensional disordered model, the half-system entanglement entropy saturates at a small value,

$P\in P(n)$3

essentially independent of system size. In the two-dimensional model, $P\in P(n)$4 grows linearly at short times and saturates at the maximal half-system value,

$P\in P(n)$5

Within the stabilizer tableau formalism this is accompanied by the operator-density contrast already seen in $P\in P(n)$6: exponential spatial decay in one dimension and a nearly flat, fully scrambled interior in two dimensions (Farshi et al., 2022).

Spectral diagnostics show that Clifford dynamics can display nontrivial chaos-like structure without matching Haar-random behavior. The spectral form factor is

$P\in P(n)$7

For the two-dimensional random Clifford circuit, the spectral form factor behaves like that of quasi-free fermions with chaotic single-particle dynamics, exhibiting an exponential ramp

$P\in P(n)$8

up to a time

$P\in P(n)$9

after which it fluctuates around a plateau. In one dimension, by contrast, the early growth crosses over to a plateau on a timescale essentially independent of system size, consistent with localized operator orbits and enhanced quasi-energy degeneracies (Farshi et al., 2022).

Fragmented Floquet-Clifford circuits give a different spectral signature. In localized regimes, the spectral form factor deviates strongly from the circular unitary ensemble benchmark, and because fragments evolve almost independently the paper argues for an effective quadratic early-time contribution summarized by

$\omega$0

Sample-to-sample fluctuations remain anomalous, signaling non-ergodicity and persistent recurrences. The same wall structures that localize operators also constrain entanglement: for stabilizer states across a $\omega$1-wall,

$\omega$2

and after Haar-averaged single-qubit perturbations the ensemble-averaged steady-state entropy across a wall is estimated to be about $\omega$3 bits. At $\omega$4, where all qubits are perturbed every step, these bottlenecks disappear and the smoothed spectral form factor crosses over to CUE-like behavior after a finite fragmentation or Thouless-like timescale (Kovács et al., 2024).

5. Ergodicity and thermalization in Clifford-Floquet QCA

In translationally invariant Clifford quantum cellular automata, the central distinction is between periodic and nonperiodic dynamics. For the associated linear cellular automaton, mixing holds iff

$\omega$5

Equivalently, no nontrivial Pauli monomial is periodic under iteration. A stronger support-growth condition is strong diffusivity, defined by $\omega$6 for every nonzero $\omega$7, while weak diffusivity requires unbounded growth only along a density-$\omega$8 subsequence. Weak diffusivity is equivalent to the absence of solitons, expressed by

$\omega$9

Both strong and weak diffusivity imply mixing (Kapustin et al., 1 Jan 2026).

A structural theorem collapses the usual ergodic hierarchy for Clifford QCAs: $H$0 The equilibrium state is the infinite-temperature tracial state, so mixing means that correlators of nontrivial Pauli observables decay to zero. The mechanism is direct in the Pauli basis: if a local observable is expanded as $H$1, then

$H$2

and in a product state

$H$3

Support growth therefore suppresses all non-identity Pauli coefficients and leaves only the infinite-temperature value (Kapustin et al., 1 Jan 2026).

This yields rigorous thermalization statements. In one dimension with one qubit per site, a soliton-free Clifford QCA weakly thermalizes every translation-invariant product state. In arbitrary dimension and for any number of qubits per unit cell, a soliton-free QCA weakly thermalizes every $H$4-generic product state, defined by

$H$5

and strong diffusivity upgrades weak thermalization to strong thermalization. The results extend to short-range entangled states of the form $H$6, where $H$7 is a finite-range QCA: if $H$8, then soliton-free dynamics weakly thermalizes $H$9, and strongly diffusive dynamics thermalizes it strongly. The paper also emphasizes that weak and strong thermalization are genuinely distinct, and that support growth in Clifford-Floquet systems can be highly nonuniform, including logarithmic growth along certain subsequences (Kapustin et al., 1 Jan 2026).

6. Perturbations, dissipation, and fast-scrambling extensions

Dissipation supplies a distinct chaos diagnostic for Floquet Clifford circuits. In the Haar-doped iSWAP-class brickwork model, the observable is the Liouvillian gap $S$0, the slowest nontrivial decay rate of the one-period channel obtained by appending onsite depolarization to the unitary Floquet step. In the undoped case $S$1,

$S$2

whereas in the fully doped case $S$3,

$S$4

Thus the undoped iSWAP-class circuit has a gap that grows linearly with system size, while the fully doped circuit has a finite thermodynamic-limit gap. In the strong-dissipation regime the analysis uses the Pauli-weight action

$S$5

and a low-weight truncation criterion based on nilpotency of the projected map. For arbitrary doping patterns with $S$6, the paper derives

$S$7

for some constant $S$8. A finite thermodynamic-limit gap therefore requires $S$9, that is, a nonzero doping density $n$00. The analytic bounds depend only on the spatial doping pattern, so the same scaling persists whether Haar rotations are fixed each period or independently resampled (Kim et al., 3 Feb 2026).

Nonlocal Floquet Clifford circuits motivated by matrix models furnish a different extension of the subject, aimed at fast scrambling rather than localization or dissipation. In these models, a global permutation of matrix indices is followed by a layer of identical four-qubit Clifford gates, producing a structured but nonlocal interaction graph. Operator size grows exponentially at early times and saturates near the random-Clifford value, with scrambling time of order $n$01. The numerics report

$n$02

for Rule 1 and

$n$03

for Rule 2, compared with the random-unitary benchmark $n$04. Stabilizer entanglement entropy grows and saturates at the maximal value $n$05 for $n$06, with lower bound

$n$07

The same circuits also realize a simplified Hayden–Preskill recovery protocol in which stabilizer quantum error correction replaces postselection; in the case $n$08, the recoverable regime obeys

$n$09

These constructions show that Floquet Clifford dynamics can be classically simulable while still exhibiting multiple signatures of fast scrambling in experimentally motivated architectures (Ma et al., 27 Mar 2026).

Across these models, Floquet Clifford circuits occupy a technically distinctive position. They are exactly solvable because Pauli evolution remains linear, yet they realize sharply different nonequilibrium phases: one-dimensional blocking-wall localization, two-dimensional ballistic delocalization, operator-space fragmentation, weak and strong thermalization in deterministic Clifford QCAs, divergent and finite Liouvillian-gap regimes under local dissipation, and fast-scrambling behavior in nonlocal circuits. A recurring misconception is that classical simulability forces trivial dynamics. The results surveyed here show instead that, within Floquet Clifford systems, solvability and rich many-body dynamics coexist rather than exclude one another (Farshi et al., 2022, Kapustin et al., 1 Jan 2026).