Roughening Transition in Quantum Circuits

TL;DR

This work examines a randomness driven roughening transition of the entanglement membrane in a (3+1)-dimensional Clifford circuit, revealing how disorder competes with lattice pinning to shape entanglement growth. By introducing a hyperdiamond circuit architecture and leveraging ZX calculus, the authors diagnose the transition through tilt-induced changes in the entanglement entropy across bipartitions, and develop a scaling theory that captures smooth, rough, and tilted regimes. Finite-size numerics yield a correlation-length exponent ν ≈ 1.5 and a critical tilt exponent θc ≈ 1.3, consistent with FRG predictions and previous numerical work, while providing a first direct numerical estimate of θc in this setting. The results elucidate high-dimensional entanglement dynamics with translationally invariant geometries and have implications for robust quantum memories and scalable quantum simulations in higher dimensions.

Abstract

We explore a roughening phase transition that occurs in the entanglement dynamics of certain quantum circuits. Viewing entanglement as the free energy of a membrane in a circuit-defined random environment, there is a competition between membrane smoothing due to lattice pinning and roughening due to disorder in the circuit. In particular, we investigate the randomness-induced roughening transition of the entanglement membrane in a (3+1)-dimensional Clifford circuit model, by calculating the entanglement entropy for various bipartitions. We further construct a scaling theory for membranes tilted away from lattice planes, uncovering new scaling forms and a crossover to a previously unexplored critical "tilted regime".

Figures (3)

• Figure 1: (a) Illustration of the setup and protocol. The initial state consists of two independent subsystems ($x<0$ and $x>0$), each with $2L^3$ qubits prepared in a near-maximally entangled pure state via a random unitary Clifford circuit. The entanglement membrane is initially anchored at the central cut ($x=0$) across which there is no entanglement at $t=0$. Time evolution is then performed using a 3+1-dimensional STTI Clifford circuit (illustrated as a lattice) perturbed by disorder (indicated by stars) with rate $p$. After time $t$, we compute the bipartite entanglement entropy $S(x,t)$ across cuts perpendicular to $\hat{x}$ and average this over disorder realizations. These cuts also serve as anchoring surfaces for the entanglement membrane at $(x,t)$. (b) In the smooth phase ($p<p_c$), the membrane remains pinned to the lattice. Moving the cut to $x \neq 0$ forces the membrane to take $|x|$ discrete steps, each contributing to the energy, resulting in a cusp in $S(x,t)$ at $x=0$, as in (d). (c) In the rough phase ($p>p_c$), the membrane becomes pinned by disorder, so it is rough on large scales. The sample-averaged entanglement $S(x,t)$ smooths out near $x=0$, reflecting the minimal volume connecting the anchoring surfaces. (d) Results for $p=0.05<p_c$ show the characteristic cusp at $x=0$. (e) Numerical results in the rough phase ($p=0.15>p_c$) show a smooth quadratic profile near $x=0$ without a cusp. All numerical results in this figure are for system size $L = 20$, with time evolution shown at even times up to $t = 2L$.
• Figure 2: Scaling regimes of a 3d interface with tilt $|x|/t$ and disorder strength $p$. $\xi \sim |p-p_c|^{-\nu}$ is the correlation length in an infinite system. $\xi_t = t/|x|$ is the length scale beyond which we "notice" the tilt. The crossovers between the tilted critical regime and the rough or smooth phases occur at $\xi_t \sim \xi$. The roughening phase transition at $p=p_c$ only occurs at zero tilt.
• Figure 3: Numerical results from the hyperdiamond circuit at runtime $t = 2L$ and associated finite-size scaling analysis. (a) The ratio $R_{1/2} \equiv \mathbb{E}[\Delta S(\pm 1)] / \mathbb{E}[\Delta S(\pm 2)]$ shows a crossing near $p_c \approx 0.095$. The inset shows scaling collapse with $\nu = 1.5$. (b) Finite-size scaling of $\mathbb{E}[\Delta S(\pm 1)]$, using the previously estimated $\nu \approx 1.5$. The best collapse yields $\theta_c \approx 1.3$. The inset shows the objective function evaluating collapse quality across $\theta_c$ and $\nu$. The white dotted line marks the expected regime $\theta_c \in (1, 2)$, and the brown dotted lines indicate the bounds on $\nu$ inferred from panel (a). (c) The ratio $R_{1/\delta 1} \equiv \mathbb{E}[\Delta S(\pm1)] / \sigma[\Delta S(\pm1)]$ also exhibits a crossing near $p_c \approx 0.095$, with a scaling collapse shown as the inset.