Quantum Entanglement Growth under Random Unitary Dynamics

Characterizing how entanglement grows with time in a many-body system, for example, after a quantum quench, is a key problem in nonequilibrium quantum physics. We study this problem for the case of random unitary dynamics, representing either Hamiltonian evolution with time-dependent noise or evolution by a random quantum circuit. Our results reveal a universal structure behind noisy entanglement growth, and also provide simple new heuristics for the “entanglement tsunami” in Hamiltonian systems without noise. In 1D, we show that noise causes the entanglement entropy across a cut to grow according to the celebrated Kardar-Parisi-Zhang (KPZ) equation. The mean entanglement grows linearly in time, while fluctuations grow like $(\text{time}{)}^{1/3}$ and are spatially correlated over a distance $\propto (\text{time}{)}^{2/3}$. We derive KPZ universal behavior in three complementary ways, by mapping random entanglement growth to (i) a stochastic model of a growing surface, (ii) a “minimal cut” picture, reminiscent of the Ryu-Takayanagi formula in holography, and (iii) a hydrodynamic problem involving the dynamical spreading of operators. We demonstrate KPZ universality in 1D numerically using simulations of random unitary circuits. Importantly, the leading-order time dependence of the entropy is deterministic even in the presence of noise, allowing us to propose a simple coarse grained minimal cut picture for the entanglement growth of generic Hamiltonians, even without noise, in arbitrary dimensionality. We clarify the meaning of the “velocity” of entanglement growth in the 1D entanglement tsunami. We show that in higher dimensions, noisy entanglement evolution maps to the well-studied problem of pinning of a membrane or domain wall by disorder.

Popular Summary

Quantum mechanics involves distinctive nonlocal correlations known as “entanglement,” which imply that the state of a system cannot be reduced to separate states of its constituent parts. Quantitative measures of entanglement have recently illuminated many areas of theoretical physics, ranging from the classification of phases of matter to the question of how our classical world emerges from the microscopic quantum-mechanical “soup.” But despite remarkable theoretical, computational, and experimental progress in studying entanglement, a theory explaining entanglement generation in generic many-body systems has been lacking. Here, we propose that general results for entanglement growth can be obtained by studying certain “minimal models” composed of quantum circuits made of random gates.

We present a detailed picture of entanglement growth in these minimal models, whose dynamics are random both temporally and spatially. We show that our findings can be mapped to paradigmatic problems in classical statistical mechanics, including the growth of a classical surface and an elastic energy minimization problem for a membrane. We then extract “universal” lessons that apply to generic many-body systems. These lessons include leading-order scaling forms for entanglement growth, information about the relation of the “speed” of thermalization to other characteristic speeds, and easily visualizable heuristics for entanglement growth.

We expect that our findings will pave the way for studies focused on entanglement in higher dimensions.

Figure 1

The KPZ “triumvirate” is made up of three very different problems in classical statistical mechanics which all map to the KPZ universality class. As we discuss, each of them can be usefully related to entanglement in $1+1\mathrm{D}$.

Figure 2

Spin chain with open boundary conditions. $S(x)$ denotes the entanglement entropy (von Neumann or Rényi depending on context) between the part of the chain to the left of bond $x$, indicated by the box, and the part to the right of bond $x$.

Figure 3

Dynamical update in the solvable model. Application of a random unitary $U$ to a randomly chosen pair of adjacent spins.

Figure 4

Surface growth model for entanglement $S(x,t)$ across a cut at $x$, in the large-$q$ limit. Applying a unitary to bond $x$ can increase the height of the surface locally [Eq. (19)], corresponding to dropping a “block” of height $\mathrm{\Delta }S=1$ or $\mathrm{\Delta }S=2$.

Figure 5

Entanglement growth in the large-$q$ model. Effect of applying a random unitary to the central bond, for four choices of the initial local entropy configuration of three adjacent bonds.

Figure 6

Any cut through the unitary circuit that separates the legs to the left and right of $x$ (on the top boundary) gives an upper bound on $S(x,t)$. The best such bound is given by the minimal cut (note that the cut shown here is not the minimal one). Finding the minimal cut in a random network is akin to finding the lowest-energy state of a polymer in a random potential landscape.

Figure 7

Spreading of stabilizer operators defining the quantum state (Sec. 4). Each blue particle marks the right end point of some stabilizer (the rightmost spin on which it acts). Blue particles hop predominantly to the right. Whenever a particle enters the right-hand region ($A$) the entanglement $S_A$ increases by one bit. The particle density is described by the noisy Burgers equation, which maps to KPZ. A “hole” (empty circle) marks the left-hand end point of some stabilizer.

Figure 8

Left: The initial product state represented in terms of the fictitious particles. Right: A state with maximal $S(x)$.

Figure 9

Fluctuations at late times, after saturation of $⟨S(x)⟩$, in the Clifford case. When $x\ll L/2$ it requires a rare fluctuation (fighting against the net drift) to remove a particle from region $A$, leading to an exponentially small $S_{\max }(x)-⟨S(x)⟩$.

Figure 10

Infinite chain with regions $A$, $B$, $C$ marked. $B$ is of length $l$ while $A$ and $C$ are semi-infinite. The mutual information between $A$ and $C$ is nonzero so long as $l<2v_{E}t$: correlations exist over distances up to $2v_{E}t$, not $v_{E}t$.

Figure 11

Bottom: Infinite chain with finite regions $A$ and $C$ each of length $d$, separated by distance $l$. Top: The mutual information between $A$ and $C$ in the case $d>l$. In the opposite regime the mutual information vanishes.

Figure 12

Sequence of minimal cut configurations (red lines) determining the entropy of region $B$ in Fig. 11. (a) gives way to (b) when $2v_{E}t=l$ and (b) gives way to (c) when $2v_{E}t+l=2d$.

Figure 13

Bottom: Semi-infinite chain with regions $A$, $B$ (length $l_A$ and $l_B$, respectively), and $C$ adjacent to the boundary. Top: The mutual information between $B$ and $C$ for this geometry, for the two regimes indicated.

Figure 14

Schematic structure of a layer in the quantum circuits used for simulations.

Figure 15

The von Neumann entropy $S(x,t)$ for a system of length $L=459$, as a function of $x$, for several successive times ($t=340$, 690, 1024, 1365, 1707, 2048, and 4096), in the Clifford evolution. This shows that the state evolves from a product state to a near-maximally-entangled one. Prior to saturation the entanglement displays KPZ-like stochastic growth. $S(x,t)$ is in units of log 2.

Figure 16

The von Neumann entropy $S(x,t)$ in units of log 2, far from the boundaries, in a system of length $L=1025$ at various times (from bottom to top $t=170$, 340, 512, and 682) evolved with the Clifford evolution scheme. $\xi$ schematically shows the typical correlation length Eq. (8), which grows in time like $t^{1/z}$.

Figure 17

Top: Growth of the mean entanglement with time for the Clifford evolution with only cnot gates (in units of log 2). The solid red curve is a fit using Eq. (49). The exponent $\beta$ is found to be $\beta =0.33\pm 0.01$, in agreement with the KPZ prediction $\beta =1/3$. Dashed line shows asymptotic linear behavior. Bottom: Growth in the fluctuations in the entanglement with time. The dashed line shows the expected asymptotic behavior, $w(t)\sim t^{\beta }$, with $\beta =1/3$. The fit includes a subleading correction: Eq. (49), with $\beta =0.32\pm 0.02$. Error bars denote the $1\sigma$ uncertainty.

Figure 18

The logarithmic derivative of the width $dw/d\log t$ versus time for the Clifford evolution. The universal behavior with exponent $t^{1/3}$ is observed at shorter time scales compared with Fig. 17.

Figure 19

Correlation function $G(r)=⟨[S(r)-S(0){]}^2{⟩}^{1/2}$ at time $t=512$, 1024, and 2048 for the Clifford evolution, showing excellent agreement with the KPZ prediction $G(r)\sim r^{\chi }$ with $\chi =1/2$ in the regime $r\ll \xi (t)$.

Figure 20

The entropy across the center of the chain (in units of log 2) divided by $v_{E}t$ versus $L/2v_{E}t$ for various fixed values of $t$. This plot converges nicely to the scaling form in Eq. (51).

Figure 21

The average size $W$ of a growing Pauli string as a function of time for two protocols, cnot evolution (subscript “cnot”) and the full Clifford evolution (subscript “Cliff”). The correspondence with the dashed lines, showing the average entanglement entropy multiplied by four, is consistent with $v_E=v_B/2$. (Taken from a system of size $L=1024$.)

Figure 22

Top: Growth of the mean entanglement as a function of time for the universal and phase gate set fitted to Eq. (49), with $\beta$ set to $1/3$. The dashed line shows the expected asymptotic behavior for comparison. Error bars indicate 1 standard deviation ($1\sigma$) uncertainty.

Figure 23

The logarithmic derivative of the width $dw/d\log t$ versus time for the phase and universal evolution protocols. For comparison, we plot the universal behavior with exponent $t^{1/3}$ in gray (dashed line). (The derivative is calculated using three data points. Errors are estimated from maximal and minimal slopes obtained within 1 standard deviation from the averaged data points.)

Figure 24

Correlation function $G(r)=⟨[S(r)-S(0){]}^2{⟩}^{1/2}$ at three values of the time for the phase (top) and universal (bottom) gate sets, showing good agreement with the KPZ exponent value $\alpha =1/2$.

Figure 25

Minimal membrane picture for the entanglement of two regions in $d=2$.

Figure 26

Minimal membrane for a disk-shaped region in $d=2$.

Figure 27

Top: Growth of the mean entanglement in units of log 2 as a function of time for the random Clifford evolution (only cnot gates). The red solid curve is a fit using the form Eq. (49). Dashed line shows asymptotic linear behavior. Bottom: Growth in the fluctuations in the entanglement with time. The exponent $\beta$ is found to be ${\beta }_w=0.3\pm 0.04$, in agreement with the KPZ prediction $\beta =1/3$. The dashed line shows the expected asymptotic behavior, $w(t)\sim t^{\beta }$ with $\beta =1/3$.

Figure 28

Observed probability distribution for the entanglement entropy across the central bond of chain of length $L=2048$ at time $t=2048$ under the full Clifford dynamics, fitted to two probability distributions. Top: Best fit to the Tracy-Widom (TW) distribution with $\beta =1$. A fit to TW with $\beta =2$ is not shown, but is indistinguishable at the scale of the figure. Bottom: Best fit to the Gaussian. Clearly, the Tracy-Widom distribution fits the data better than Gaussian, as the latter shows systematic deviation. The $1-R^2$ values for the fits are $2.1\times {10}^{-4}$ for ${\mathrm{TW}}_{\beta =1}$, $2.0\times {10}^{-4}$ for ${\mathrm{TW}}_{\beta =2}$, and $1.6\times {10}^{-3}$ for Gaussian.