Operator space fragmentation in perturbed Floquet-Clifford circuits

TL;DR

This work analyzes how operator localization persists or breaks down in one-dimensional Floquet-Clifford circuits when perturbed by random non-Clifford gates. By constructing and classifying wall configurations (1-, 2-, and k-walls) and proving the existence of local conserved charges within walls, the authors show that operator space fragments into stable subspaces (fragmentation) for p<1, with a calculable localization length mu(p) and an entanglement bottleneck across fragment boundaries. They derive analytical bounds on wall probabilities, fragmentation in the thermodynamic limit, and entanglement behavior, complemented by exact numerics on spectral form factors and entanglement for finite systems. The results reveal a robust, tunable localization-to-ergodicity transition controlled by p, providing a concrete, analytically tractable toy model for operator dynamics with potential realizations on current NISQ devices.

Abstract

Floquet quantum circuits are able to realise a wide range of non-equilibrium quantum states, exhibiting quantum chaos, topological order and localisation. In this work, we investigate the stability of operator localisation and emergence of chaos in random Floquet-Clifford circuits subjected to unitary perturbations which drive them away from the Clifford limit. We construct a nearest-neighbour Clifford circuit with a brickwork pattern and study the effect of including disordered non-Clifford gates. The perturbations are uniformly sampled from single-qubit unitaries with probability $p$ on each qubit. We show that the interacting model exhibits strong localisation of operators for $0 \le p < 1$ that is characterised by the fragmentation of operator space into disjoint sectors due to the appearance of wall configurations. Such walls give rise to emergent local integrals of motion for the circuit that we construct exactly. We analytically establish the stability of localisation against generic perturbations and calculate the average length of operator spreading tunable by $p$. Although our circuit is not separable across any bi-partition, we further show that the operator localisation leads to an entanglement bottleneck, where initially unentangled states remain weakly entangled across typical fragment boundaries. Finally, we study the spectral form factor (SFF) to characterise the chaotic properties of the operator fragments and spectral fluctuations as a probe of non-ergodicity. In the $p = 1$ model, the emergence of a fragmentation time scale is found before random matrix theory sets in after which the SFF can be approximated by that of the circular unitary ensemble. Our work provides an explicit description of quantum phases in operator dynamics and circuit ergodicity which can be realised on current NISQ devices.

Figures (8)

• Figure 1: A segment of the brickwork Floquet circuit considered in this work. We define a unitary evolution operator according to Equation (\ref{['eq:floquet_unitary']}) on a one-dimensional qubit chain. The gates consist of randomly sampled entangling Clifford gates (rectangles) and random $\mathrm{SU}(2)$ rotation gates (coloured circles) applied stochastically with probability $p$. Same colours represent the same gate.
• Figure 2: Equivalence classes of $\mathcal{C}_2$ with respect to product Cliffords as tensor diagrams. Identity-like, $\mathrm{CZ}$-like, $\mathrm{SWAP}$-like and $\mathrm{FSWAP}$-like gates shown respectively. Each leg represents a single qubit degree of freedom with multiple lines showing tensor product spaces. Boxes show elements of $\mathcal{C}_1$. Sampling the rectangular boxes under uniform measure on $\mathcal{C}_1$ respects the multiplicity of classes in Haar-sampling of ${\mathcal{C}}_2$Grier2022Mele2023.
• Figure 3: Representation of $k$-walls. We model the circuit environment as the injection of operators into the left ($L$) and right ($R$) qubit subspaces of the wall. Localisation in the circuit can be understood by constraining the inner gates to stop the spreading of arbitrary injected operators at the arrow locations. This ensures that localising gates can be stably embedded into larger circuits.
• Figure 4: In the $p=0$ (Clifford) limit, the two Floquet operators shown in the figure give rise to the same evolution operator, except at finite time boundaries. The Floquet operator in "staircase" form (right-hand side) reflects the causal structure of operator spreading.
• Figure 5: General form of $1$-walls by attaching two members of the $\mathrm{CZ}$-class. The loop on the central qubit captures the periodic nature of the circuit. Two compatible $\mathrm{CZ}$-like gates form a one-wall if their single-qubit Clifford degrees of freedom preserve the conserved Pauli on the central qubit as per \ref{['lemma:conserved_pauli']}. Sampling under uniform $\mathcal{C}_1$ measure gives the same probability of sampling the configuration on a) and b), by Haar-invariance.

Theorems & Definitions (19)

• Definition 4.1: $k$-walls
• Lemma 4.1: $0$-walls
• proof
• Lemma 4.2
• proof
• Lemma 4.3
• proof
• Lemma 4.4: Local conserved charges
• proof
• Lemma 4.5: Internal orthogonality