Liouvillian Gap in Dissipative Haar-Doped Clifford Circuits

TL;DR

We address the question of intrinsic relaxation in open quantum circuits by studying a dissipative Haar-doped Floquet Clifford model with an iSWAP-class backbone. We use a weight-truncation Pauli-dynamics framework to obtain analytic lower and upper bounds on the Liouvillian gap and complement them with numerics in the weak and strong dissipation regimes. We show that the undoped circuit has a gap that grows as $Δ ∼ γ N$, while fully doped circuits yield a finite gap $Δ = γ + 2$ in the thermodynamic limit; with nonzero Haar-doping density, several spatial patterns realize system-size independent gaps, including dense and block-staggered constructions, supported by explicit finite-dissipation bounds. The results connect the dissipative gap diagnostic to Haar-like scrambling thresholds, and the bounds depend only on spatial doping structure, making the conclusions robust to Haar rotations being quenched or re-sampled per period.

Abstract

Quantum chaos is commonly assessed through probe-dependent signatures such as spectral statistics, OTOCs, and entanglement growth, which need not coincide. Recently, a dissipative diagnostic of chaos has been proposed, in which an infinitesimal coupling to a bath yields a finite Liouvillian gap in chaotic systems, marking the onset of intrinsic relaxation. This raises a conceptual question: what is the minimal departure from Clifford dynamics needed for this intrinsically relaxing behavior to emerge? In this work, we investigate the dynamics under the Floquet two-qubit Clifford circuit interleaved with a finite density of Haar-random single-site gates, followed by a depolarizing channel with strength $γ$. For Floquet Clifford circuits built from an \textit{i}SWAP-class two-qubit gate, our analysis identifies two distinct regimes for the Liouvillian gap in the thermodynamic limit, exemplified by the undoped and fully doped extreme cases. In both regimes, the dissipative diagnostic signals chaotic behavior, differing only in how the gap scales with system size. In the undoped circuit, the gap scales as $Δ\sim γN$, whereas in the fully doped circuit it remains finite as $N\to\infty$. We find that the doping density $p_h$ governs the crossover: as $p_h\to 0$, any spatial structure remains undoped-like, whereas for finite $p_h$ certain structures can enter a finite-gap regime. These results are analytically established in the strongly dissipative regime $γ\gg 1$ by deriving lower bounds on the gap as a function of $p_h$ and explicit finite-gap constructions, and their extension toward $γ\to 0$ is supported by numerics. Importantly, our analytic treatment depends only on the spatial doping structure, so the same gap scaling persists even when the Haar rotations are independently resampled each Floquet period.

Figures (8)

• Figure 1: Summary of analytic results. Dissipative Haar-doped Floquet circuit and the weight-truncation framework for bounding the Liouvillian gap. (a) One Floquet period consists of a two-layer Clifford brickwork built from a fixed iSWAP-class gate on odd and even bonds (blue), with single-qubit Haar rotations applied immediately after each layer on a chosen set of $n_h$ qubit lines (orange), followed by an onsite depolarizing layer on all qubits (yellow). In (b)-(d), each long rectangle represents the qubit lines, with blue and orange segments indicating undoped and doped sites, respectively. (b) Diagram of Liouvillian gap $\Delta$ versus dissipation strength $\gamma$ for representative doping patterns at the thermodynamic limit. Solid lines show exact results for fully doped (orange); dashed lines show analytic upper bounds for dense doping (light purple) and block-staggered doping with spacing $k$ between doped (green). Unlike the doped patterns, whose singularities remain independent of system size, the undoped circuit (blue) exhibits a singularity that diverges as the system size $N$ increases. (c,d) Lower- and upper-bound constructions. Gray capacitors denote Pauli-weight truncation projectors inserted between Floquet steps. (c) Representative truncation of an initially local $Y$ operator (green) once its weight exceeds $w_{\mathrm{t}}=6$ and vanishes after five steps. (d) Under truncation, return cycles can only be supported by Pauli strings confined to a small consecutive block, illustrated by a three-site string $YXZ$ that translates by two sites per step and returns after $N/2$ steps.
• Figure 2: Summary of numerical results. Ensemble-averaged Liouvillian gap $\langle\Delta\rangle$ of the DHFC as a function of the dissipation strength $\gamma$ at (a) full and (b) staggered doping for $N\in\{4,6,8\}$. Dashed lines (with accompanying formulae) give the analytic results for the gap in the thermodynamic and strong dissipation limits. As $N$ increases, the discrepancy between $\langle \Delta \rangle$ and the analytic result decreases; the finite-$N$ scaling of our numerically obtained $\langle \Delta \rangle$ is shown at (c) weak, (d) intermediate, and (e) strong dissipation strengths $\gamma$.
• Figure 3: Two-qubit Clifford mapping patterns. Local-Clifford classification of two-qubit Clifford gates from their action on single-site Paulis. The $16$ two-qubit Paulis are placed on a $4\times4$ grid, and the images of $\{X_1,Y_1,Z_1,X_2,Y_2,Z_2\}$ under conjugation are marked, with red (blue) indicating support on qubit 1 (2). Two patterns occur. (a) Trivial pattern, where single-site Paulis remain single-site supported. (b) Nontrivial pattern, where two single-site Paulis per qubit map to two-site Paulis within a common row or column and the third maps to the intersection single-site Pauli.
• Figure 4: Orbit-averaged Pauli weights for dissipative brickwork Floquet Clifford circuits. For each nontrivial Pauli string $S$, we compute the orbit-averaged weight $\bar{w}(S)$ and plot the value normalized by the system size, $\bar{w}(S)/N$, sorted in increasing order and labeled by the relative index. Black-circled markers indicate the smallest values for each $N$. (a) Generic two-qubit Clifford gate set. (b) CZ class gate set. (c) iSWAP-class gate set. (d) iSWAP-class circuit with all two-qubit gates fixed to $i\mathrm{SWAP}(H\otimes H)$. A small low-$\bar{w}$ sector is visible in (a) and (b), while it is absent in the iSWAP class cases (c) and (d).
• Figure 5: Weight distribution of the eigenoperator. Pauli-weight content of the Liouvillian-gap eigenoperator for $N=6$. Distributions are computed from the Pauli-basis expansion with squared-amplitude weights. Each panel shows $\gamma=0.1$ (blue), $\gamma=1.0$ (orange), and $\gamma=3.0$ (green); $\gamma=0.1$ corresponds to weak dissipation and $\gamma=3.0$ to strong dissipation. (a) Non-doped circuit: support is concentrated at weight $w=3$ independent of $\gamma$. (b) 4-doped contiguous pattern and (c) fully doped limit; at $\gamma=3.0$ the fully doped case concentrates at $w=1$ whereas the 4-doped case retains a dominant $w=2$ contribution.

Theorems & Definitions (23)

• Definition 1: weight-$w$ eigenmode-free
• Corollary 1
• Lemma 1
• proof
• Lemma 2
• proof
• Corollary 2
• Lemma 3
• proof
• Lemma 4