Deterministic Quantum East Circuit

• The paper introduces a circuit defined by fixed local gates and an asymmetric causal structure that enables unidirectional facilitation, leading to disorder-free localization and exact dynamical results.
• At the deterministic point, the model employs CNOT permutation gates yielding an exact quench solution with finite-time thermalization and submaximal, controlled entanglement growth.
• Extensions include measurement-conditioned fractal correlations and cyclic entanglement spreading, with potential implementations on superconducting qubits, trapped ions, and Rydberg platforms.

to specific statements. Since article can draw primarily on (Bertini et al., 2023) and (Bertini et al., 2023), with (Li et al., 8 Sep 2025, Chen et al., 2023), maybe (Ruiz et al., 2023) as methodological context. Need ensure cite no more than once per section. One cite per section maybe.

Need perhaps perform arXiv search. Let's use tool. A deterministic quantum East circuit is a disorder-free, directionally constrained quantum circuit that realizes the East kinetic rule in discrete time: local evolution is facilitated by a neighboring excitation, so the circuit explores Hilbert space through a strongly asymmetric causal structure rather than through random fields or random gates. In the literature this notion appears in several closely related forms: as a depth-2 Floquet brickwork Trotterization of the quantum East Hamiltonian, as a special “deterministic point” where the local gate becomes a CNOT permutation gate, and as a sequential boundary-driven Clifford circuit with repeated ancilla measurements. Across these constructions, the defining features are fixed local gates, unidirectional facilitation, and nontrivial many-body dynamics ranging from disorder-free localization and finite-time subsystem thermalization to measurement-conditioned fractal correlations (Bertini et al., 2023).

1. Canonical circuit definition

The standard starting point is the continuous-time quantum East model on a chain of $2L$ spin-$1/2$ degrees of freedom,

$$H(a)=\sum_{j=1}^{2L-1} P_j\left(aX_{j+1}-I\right)+aX_1-I, \qquad P_j=\frac{I+Z_j}{2},$$

where $P_j$ projects onto $\ket{\uparrow}_j$. The East kinetic constraint is encoded by the factor $P_j$: the spin at site $j+1$ is acted on by $X_{j+1}$ only if site $j$ is in the $\ket{\uparrow}$ state. This is the direct quantum analogue of the classical East rule that a spin can flip only if its left neighbor is excited (Bertini et al., 2023).

The discrete-time Floquet version is a fixed-depth brickwork circuit with single-period operator

$1/2$0

with

$1/2$1

$1/2$2

and local two-site gates

$1/2$3

Equivalently,

$1/2$4

If the left spin is $1/2$5, the gate is the identity on the right spin; if the left spin is $1/2$6, it applies a nontrivial single-qubit unitary to the right spin. The left boundary is unconstrained through $1/2$7, which acts as a source of excitations (Bertini et al., 2023).

In this Floquet setting, “deterministic” has a precise meaning. There is no disorder, no temporal randomness, and no measurements or probabilistic post-selections; the same unitary $1/2$8 is applied at every period. The circuit is translation-invariant up to the boundary, and the Trotter limit reproduces the continuous-time propagator,

$1/2$9

2. Deterministic point and exact solvability

A particularly important specialization is the “deterministic point” of the Floquet quantum East model, where the local facilitated gate becomes a CNOT permutation gate and the circuit reduces to a deterministic Floquet quantum East model (DFQE) (Bertini et al., 2023). In that representation the local unitary is

$$H(a)=\sum_{j=1}^{2L-1} P_j\left(aX_{j+1}-I\right)+aX_1-I, \qquad P_j=\frac{I+Z_j}{2},$$0

with $$H(a)=\sum_{j=1}^{2L-1} P_j\left(aX_{j+1}-I\right)+aX_1-I, \qquad P_j=\frac{I+Z_j}{2},$$1. This is a CNOT with the right qubit as control and the left qubit as target: if the right qubit is $$H(a)=\sum_{j=1}^{2L-1} P_j\left(aX_{j+1}-I\right)+aX_1-I, \qquad P_j=\frac{I+Z_j}{2},$$2, nothing happens; if it is $$H(a)=\sum_{j=1}^{2L-1} P_j\left(aX_{j+1}-I\right)+aX_1-I, \qquad P_j=\frac{I+Z_j}{2},$$3, the left qubit is flipped. The gate is therefore a permutation of computational-basis states, and basis-state dynamics is deterministic.

The brickwork Floquet operator is built from two staggered layers of these CNOT gates. The circuit is a Clifford quantum cellular automaton with a facilitated rule encoded directly in gate orientation. Although the precise left/right convention differs from the brickwork formulation above, both descriptions implement a unidirectional facilitation condition at the gate level. This suggests that the defining invariant is not a single orientation convention but the asymmetric causal structure of the constraint.

At the deterministic point, the quench dynamics can be solved exactly by “space evolution,” obtained by folding the circuit in space-time and replacing time evolution of observables by powers of a space transfer matrix. For broad classes of product initial states, the associated influence states have a dimer-product structure in time, which yields exact formulas for local observables, reduced density matrices, and Rényi entropies (Bertini et al., 2023).

Two results are structurally central. First, a finite block $$H(a)=\sum_{j=1}^{2L-1} P_j\left(aX_{j+1}-I\right)+aX_1-I, \qquad P_j=\frac{I+Z_j}{2},$$4 embedded in a solvable environment thermalizes exactly to the infinite-temperature state in a time scaling linearly with block size: $$H(a)=\sum_{j=1}^{2L-1} P_j\left(aX_{j+1}-I\right)+aX_1-I, \qquad P_j=\frac{I+Z_j}{2},$$5 for a block of $$H(a)=\sum_{j=1}^{2L-1} P_j\left(aX_{j+1}-I\right)+aX_1-I, \qquad P_j=\frac{I+Z_j}{2},$$6 qubits, with a faster $$H(a)=\sum_{j=1}^{2L-1} P_j\left(aX_{j+1}-I\right)+aX_1-I, \qquad P_j=\frac{I+Z_j}{2},$$7 thermalization time for a more restrictive subclass of solvable initial states. Second, entanglement grows ballistically but below the dual-unitary bound. The paper proves that the entanglement of a block grows at most at one-half the maximal speed allowed by locality, and that if the block is initially prepared in a classical configuration, the speed of entanglement is one quarter of the maximum (Bertini et al., 2023).

3. Disorder-free localization in the Floquet circuit

Away from the deterministic Clifford point, the Floquet quantum East model exhibits a disorder-free localization transition in the $$H(a)=\sum_{j=1}^{2L-1} P_j\left(aX_{j+1}-I\right)+aX_1-I, \qquad P_j=\frac{I+Z_j}{2},$$8 plane. For each fixed $$H(a)=\sum_{j=1}^{2L-1} P_j\left(aX_{j+1}-I\right)+aX_1-I, \qquad P_j=\frac{I+Z_j}{2},$$9, there is a critical $P_j$0 separating a localized or dynamically inactive regime from a thermal or ergodic regime, with numerical behavior

$P_j$1

where $P_j$2 and $P_j$3 is numerically inferred (Bertini et al., 2023).

The analytic mechanism is expressed most transparently in a ladder representation unitarily equivalent to the brickwork circuit. For a quench from the fully polarized all-down state, the leading perturbative amplitude for a contiguous block of $P_j$4 up spins satisfies

$P_j$5

so the competition is between the local suppression factor $P_j$6 and the combinatorial growth of the $P_j$7-deformed binomial coefficient. If $P_j$8 is irrational, the paper shows

$P_j$9

so the combinatorial factor remains bounded in time and the perturbation stays exponentially localized in space,

$\ket{\uparrow}_j$0

If $\ket{\uparrow}_j$1 is rational, the $\ket{\uparrow}_j$2-Lucas theorem implies exponential combinatorial growth and a linearly propagating front, hence delocalization (Bertini et al., 2023).

This yields the central finite-$\ket{\uparrow}_j$3 statement: the localization transition persists for discrete times and is present for any finite value of the time step apart from a zero-measure set. The exceptional set consists of rational multiples of $\ket{\uparrow}_j$4,

$\ket{\uparrow}_j$5

Small denominators generate particularly strong deviations from localization and “Arnold-tongue-like” irregular structures in the phase diagram. In the Trotter limit $\ket{\uparrow}_j$6, consistent with the continuous-time Hamiltonian model, although the paper emphasizes that the limits $\ket{\uparrow}_j$7 and $\ket{\uparrow}_j$8 do not commute (Bertini et al., 2023).

4. Dynamical diagnostics and eigenstate structure

The primary quench protocol begins from the fully polarized state

$\ket{\uparrow}_j$9

All bulk conditional gates act trivially on this state, so only the left boundary term initiates dynamics; this makes the protocol a boundary quench with a direct operator-spreading interpretation. The main diagnostics are the rightmost-up-spin distribution $P_j$0, the von Neumann entanglement entropy $P_j$1, and the inverse participation ratio (IPR) extracted from the time-averaged Loschmidt echo (Bertini et al., 2023).

In the localized regime, $P_j$2 stays concentrated near the boundary and does not develop a front at $P_j$3; $P_j$4 saturates rapidly and remains bounded, with a localized entanglement peak near the left edge; and the IPR remains $P_j$5-independent, consistent with dynamics confined to a finite-dimensional edge subspace. In the thermal regime, the support of $P_j$6 extends ballistically with maximum front velocity $P_j$7 sites per period due to the brickwork geometry, $P_j$8 develops a propagating entanglement front and volume-law saturation, and the IPR decays as $P_j$9, as expected for thermal eigenstates (Bertini et al., 2023).

The Floquet paper does not give a closed-form eigenstate classification, but it links the localized dynamics to an exponentially large family of non-thermal eigenstates already known in the continuous-time quantum East model. Those states are described as weakly entangled and localized near the boundary or within finite regions, and earlier Hamiltonian analysis established that their number is exponentially large in system volume. A plausible implication is that the Floquet localized phase inherits a similarly large non-thermal sector, even though explicit Floquet LIOM formulas are not given (Bertini et al., 2023).

At the deterministic point, the exact quench solution sharpens these observations from a complementary angle. For classical initial blocks and for flat $j+1$0-product states, the Rényi entropies obey

$j+1$1

while for a special subclass of solvable initial states

$j+1$2

These piecewise-linear laws make the relation between facilitation and entanglement-speed reduction explicit (Bertini et al., 2023).

5. Sequential boundary-measured circuits

A distinct but closely related construction is the sequential deterministic quantum East circuit with boundary measurements. Here the system is a chain of $j+1$3 qubits coupled, at each time step, to a fresh ancilla qubit. The ancilla interacts through a fixed sequence of CNOT and SWAP gates,

$j+1$4

followed by a SWAP sequence, and is then projectively measured in the computational basis. The circuit is fully deterministic in the sense that gate placement and gate choice are fixed; the only randomness arises from measurement outcomes (Li et al., 8 Sep 2025).

The averaged one-step channel $j+1$5 is CPTP, unital, and has the maximally mixed state as its unique steady state, so typical long-time trajectories are trivial: the bulk state is a product of maximally mixed qubits and ancilla outcomes are IID Bernoulli. The nontrivial structure emerges under large-deviation conditioning of the measurement record. Tilting trajectories by a counting field $j+1$6 produces a tilted channel $j+1$7 with exact scaled cumulant generating function

$j+1$8

Although this SCGF is simple, the dominant left eigenmatrix is highly structured,

$j+1$9

where $X_{j+1}$0 is a triangular superoperator built from folded CNOTs (Li et al., 8 Sep 2025).

The conditioned state is diagonal in the computational basis and therefore separable, but its correlations are long-ranged and fractal. The key operator identity is

$X_{j+1}$1

so the support pattern is exactly the Sierpiński triangle. One-point and two-point functions can then be written in closed form, and the paper exhibits connected correlators such as

$X_{j+1}$2

which remain $X_{j+1}$3 even as $X_{j+1}$4. The corresponding Doob-transformed channel $X_{j+1}$5 is obtained exactly and is shown to equal the Petz recovery map of a complementary time-reversed channel,

$X_{j+1}$6

This provides an exact many-body realization of rare-trajectory control by boundary measurements (Li et al., 8 Sep 2025).

6. Extensions, implementations, and conceptual scope

The East construction has also been extended to inhomogeneous East/West heterojunctions. In that setting the Hamiltonian

$X_{j+1}$7

combines a left East segment, a central hopping contact term, and a right West segment with reversed facilitation. A Trotterized circuit realization consists of on-site gates implementing $X_{j+1}$8, directional controlled-$X_{j+1}$9 rotations implementing East or West kinetic constraints, and a two-site hopping gate at the junction. The control parameters are the initial occupation pattern $j$0, the site-dependent chemical potential $j$1, and the drive-region size $j$2 (Chen et al., 2023).

Within this heterojunction geometry, the paper reports cyclic entanglement entropy spreading, directional entanglement flow toward the junction, and an approximate superposition principle for entanglement generated by East and West sources. The number of initial $j$3 states is treated as an effective temperature, while $j$4 governs both energy cost and dynamical speed. This does not define a unique canonical deterministic quantum East circuit, but it shows that the East mechanism can be modularized into directional subcircuits whose entanglement contributions add in a simple way (Chen et al., 2023).

From an implementation standpoint, the Floquet East circuit is composed entirely of one-qubit rotations and nearest-neighbor two-qubit conditional gates, with circuit depth per Floquet period scaling as $j$5. The papers explicitly note superconducting qubits, trapped ions, and neutral-atom or Rydberg platforms as relevant architectures, and state that the Floquet localization transition is currently observable in state-of-the-art platforms for digital quantum simulation (Bertini et al., 2023).

A common misconception is to reduce the subject to a single circuit definition. The literature instead supports a broader usage. The phrase covers at least four technically distinct but structurally aligned constructions: the Trotterized Floquet circuit, the CNOT deterministic point, the sequential boundary-measured Clifford circuit, and the East/West heterojunction generalization.

Variant	Defining ingredients	Characteristic result
Floquet quantum East model	Depth-2 brickwork, conditional two-site gates, boundary rotation	Disorder-free localization transition
Deterministic point	CNOT permutation gate in brickwork Floquet circuit	Exact quench solution and finite-time subsystem thermalization
Sequential boundary-measured circuit	CNOT/SWAP chain with repeated ancilla measurements	Sierpiński-fractal conditioned correlations
East/West heterojunction	Inhomogeneous East and West segments with hopping contact	Directed cyclic entanglement spreading

Taken together, these works define the deterministic quantum East circuit not as a single gate list but as a research program on directionally facilitated, disorder-free quantum circuits. Its significance lies in showing that simple asymmetric constraints can generate sharply distinct dynamical phenomena: localization without disorder, exact but submaximal entanglement spreading, finite-time local thermalization, and long-range measurement-conditioned correlations prepared from boundary control alone (Bertini et al., 2023).