Entanglement Domain Walls in Monitored Quantum Circuits and the Directed Polymer in a Random Environment

Monitored quantum dynamics reveal quantum state trajectories, which exhibit a rich phenomenology of entanglement structures, including a transition from a weakly monitored volume-law-entangled phase to a strongly monitored area-law phase. For one-dimensional hybrid circuits with both random unitary dynamics and interspersed measurements, we combine analytic mappings to an effective statistical mechanics model with extensive numerical simulations on hybrid Clifford circuits to demonstrate that the universal entanglement properties of the volume-law phase can be quantitatively described by a fluctuating entanglement domain wall that is equivalent to a “directed polymer in a random environment” (DPRE). This relationship improves upon a qualitative “mean-field” statistical mechanics of the volume-law-entangled phase [Ruihua Fan, Sagar Vijay, Ashvin Vishwanath, and Yi-Zhuang You, Phys. Rev. B 103, 174309 (2021), Yaodong Li and Matthew P. A. Fisher, Phys. Rev. B 103, 104306 (2021)]. For the Clifford circuit in various geometries, we obtain agreement between the subleading entanglement entropies and error-correcting properties of the volume-law phase (which quantify its stability to projective measurements) with predictions of the DPRE. We further demonstrate that depolarizing noise in the hybrid dynamics near the final circuit time can drive a continuous phase transition to a non-error-correcting volume-law phase that is not immune to the disentangling action of projective measurements. We observe this transition in hybrid Clifford dynamics, and obtain quantitative agreement with critical exponents for a “pinning” phase transition of the DPRE in the presence of an attractive interface.

Popular Summary

The measurement of a quantum-mechanical system irreversibly alters its state while yielding a stochastic outcome for the measured observable due to quantum indeterminacy. This intriguing feature of quantum mechanics has striking consequences for a quantum many-body system, composed of a large number of constituent quantum-mechanical degrees of freedom, which is repeatedly, albeit infrequently, measured.

In this work, we demonstrate that evolution of a quantum many-body system under infrequent observatio—a monitored quantum dynamics—can give rise to correlations that do not naturally arise in quantum matter in thermal equilibrium. We demonstrate these results by showing that patterns of quantum entanglement generated during a monitored evolution are related to a well-studied problem in classical statistical physics known as the directed polymer in a random environment, which seeks to understand the properties of energetically optimal paths in a random potential landscape. This remarkable relationship sheds light on the specific patterns of quantum many-body entanglement developed during monitored evolution, which can be immune to the disentangling effects of infrequent observations. We verify the quantitative predictions of this mapping through extensive numerical simulations.

Figure 1

The hybrid circuit model and minimal cut. In most of this paper we consider hybrid circuits as shown in the figure. The unitary gates (gray blocks) are arranged in a brickwork, and are sampled randomly and independently from either the Haar unitaries or the Clifford unitaries. The measurements (white circles) are projective, performed independently at each location with probability $p$. Superimposed is a “minimal cut” of the circuit geometry, that serves as a domain wall separating the subregion $A$ (orange) from $\overline{A}$ (blue) with minimum energy [given by the number of unmeasured links (red) that it crosses]. Although the minimal cut provides a heuristic picture of the “entanglement domain wall” for the vN entropy, the two should be distinguished—see the main text for more discussions.

Figure 2

“Pinning” phase transition. Depolarizing channels close to the final time of the hybrid dynamics leads to an effective attractive potential for the entanglement domain wall. An Ising domain-wall description of the entanglement [9, 10], is shown above, which yields a quantitatively inaccurate description of the transition. The numerically observed transition in Sec. 2f is consistent with a “pinning” phase transition of a DPRE to an attractive interface [22].

Figure 3

Directed polymer in a finite strip, with heights $Y\gg L_A^{\zeta }$ (left) and $Y\ll L_A^{\zeta }$ (right), respectively.

Figure 4

(a) The mean subleading entanglement entropy $⟨S_A^{\mathrm{sub}}(Y)⟩$ [see Eq. (12)] in the Clifford circuit for various values of $L_A$ and $Y$, collapsed against the scaling function $\mathrm{\Phi }(\eta )$ [defined in Eq. (19)]. (b) The scaling function $\mathrm{\Phi }(\eta )$ extracted from $F_A^{\mathrm{sub}}(Y)$ in a numerical simulation of directed polymers, rescaled and plotted on top of the same scaling function extracted from (a). In both panels, the insets are the same data, but plotted on a log-log scale. The directed polymers have length $L_A\le 32768$ and live at zero temperature [see Eq. (10)]. For the Clifford circuit data we take the best fits to the exponents, $\beta =0.34$, $\zeta =0.63$; whereas for the directed polymers we find the best fits are $\beta =0.37$, $\zeta =0.66$.

Figure 5

(a) The sample-to-sample deviation of entanglement entropy $\delta S_A(Y)$ [see Eq. (14)] in the Clifford circuit for various values of $L_A$ and $Y$, collapsed against the scaling function $\mathrm{\Psi }(\eta )$ [defined in Eq. (21)]. (b) The scaling function $\mathrm{\Psi }(\eta )$ extracted from $\delta F_A(Y)$ in a numerical simulation of directed polymers, rescaled and plotted on top of the same scaling function extracted from (a). The directed polymers have length $L_A\le 32768$ and live at zero temperature [see Eq. (10)]. For both the Clifford circuit and the directed polymers we take the best fits $\beta =0.33$, $\zeta =0.66$.

Figure 6

(a) Two configurations of DPRE that contributes to $S_{A-B}$ in Eq. (22), where $A-B$ consists of two disjoint segments $[0,L_A/2]$ and $[L_A/2,L_A]$, highlighted in blue. The calculation follows from a general prescription in Refs. [5, 6], where the directed polymers act as “domain walls” that separate boundary regions of different colors. (b) The mean values of several observables for the DPRE at zero temperature, which can all be related to $⟨I_{B,\overline{A}}⟩$ for the geometry in (a). They show the same powerlaw decay $L_A^{-\mathrm{\Delta }}$, where $\mathrm{\Delta }\approx 1.25$. (c) The mean mutual information $⟨I_{B,\overline{A}}⟩$ computed from the random Clifford circuit in the volume-law phase, for the geometry in (a). Here we take $L_B=2$, and $L_A\le 480$.

Figure 7

(a) Two configurations of DPRE that contribute to $S_{A-B}$ in Eq. (28). The left plot is a point-to-point polymer that separates $A-B$ from $B\cup \overline{A}$ (as required by the assigned boundary conditions), and the right plot has two point-to-line polymers for the same boundary condition. (b) The mean values of several observables for the DPRE at zero temperature, which can all be related to the DPRE mutual information [see Eq. (28)] $⟨I_{B,\overline{A}}⟩$ for the geometry in (a). This confirms Eq. (30). (c) The mean mutual information $⟨I_{B,\overline{A}}⟩$ computed from the random Clifford circuit for the geometry in (a), with open spatial boundary condition. Here we take $L_B=2$, and $L_A\le 480$.

Figure 8

(a) The mean DPRE entanglement entropy $⟨S_A⟩$ and the mean DPRE mutual information $⟨I_{A,R}⟩$. The nonmonotinicity in $⟨S_A⟩$ at $L_A\lesssim L$ and the vanishing of $⟨I_{A,R}⟩$ at small $L_A$ are signatures of the error-correcting properties of the weakly monitored phase [10]. (b) The contiguous code distance $d_{\mathrm{cont}}$ extracted from (a), namely the value of $L_A$ when $⟨I_{A,R}⟩=ϵ$, for a few different values of $ϵ$. We find that $d_{\mathrm{cont}}\propto L^{\beta }$, a result consistent with Clifford numerics in Ref. [10].

Figure 9

(a) The mean half-cut mutual information $⟨I_{A,\overline{A}}⟩$ ($L_A=L/2$) versus $L$ in the random Clifford circuit for various probabilities of the depolarizing channel, $p^{\mathrm{dep}}$. Here, we see clear evidences of a pinning transition. (b) Collapse of $⟨I_{A,\overline{A}}⟩$ according to Eqs. (38) and (39), where we choose $\beta \approx 0.36$, $\nu \approx 3.00$, and $p_c^{\mathrm{dep}}\approx 0.10$. In our numerics, we take all the depolarizing channels to happen $a=5$ time steps before the circuit terminates, and again the bulk measurement rate is $p=0.08\approx 0.5p_c$.

Figure 10

(a) Performing a Haar average over each two-site unitary gate in the hybrid dynamics in the calculation of the von Neumann entanglement of a subsystem $A$ yields an emergent lattice magnet with $S_n$ “spins” residing at the sites of a honeycomb lattice. Integrating out the spins on one sublattice yields the lattice model described in the text. (b) The Haar average in Eq. (41) yields the ratio of two partition functions that differ only in their boundary conditions, by the “insertion” of a domain wall at the boundaries of the $A$ subsystem for the spins in the lattice magnet located at the final time slice. The domain wall is labeled by the cyclic element of the permutation group $(12\dots n)$.

Figure 11

Lattice magnet with deterministic measurements. In the limit of large local Hilbert-space dimension with deterministically applied projective measurements, the replicated description of the entanglement is given by a uniform sum over all $S_n$ domain-wall configurations, which end at the boundaries of the $A$ subsystem, and are directed along the spatial direction. Shown is a schematic configuration of domain walls in the $n=4(S_4)$ lattice magnet. When these domain walls intersect, the total transpositions appearing in the decomposition of the “ingoing” and “outgoing” domains is conserved, yielding a description as a partition function for $(n-1)$-directed walkers with an attractive interaction.

Figure 12

$S_3$ lattice magnet with randomly located measurements. The leading contribution to the replicated description of the von Neumann entanglement with $n=3$ replicas comes from paths taken by the domain wall $(123)\in S_3$ through the bulk of the lattice magnet as shown in (a). In the large-$q$ limit considered in the text, all such paths appear with the same weight. The leading correction to this behavior comes from the “splitting” of this domain wall, shown in (b), while all other processes are further parametrically suppressed in powers of $q^{-1}$. The replicated description may be thought of as a pair of labeled paths with an attractive interaction (c).

Figure 13

A configuration of intersecting paths, corresponding to a configuration of domain walls in the $S_4$ lattice magnet is shown. Here, ${\sigma }_1{\sigma }_2{\sigma }_3=(1234)$, and each domain wall ${\sigma }_i$ is an elementary transposition. The intersection points alternate between either the top or the bottom path intersecting the middle path at the points labeled in blue.

Figure 14

Weights in the calculation of $\overline{\mathrm{Tr}{\rho }_A^3}$. The red line denotes one of three elementary transpositions in the permutation group $S_3$. Two lines denote any distinct pair of transpositions.

Figure 15

Splitting of an $S_3$ domain wall. The leading correction to the behavior is given by the process shown in (a). The remaining processes are parametrically smaller than this leading correction. As an example, the weights of processes (b)–(d), relative to the weight of a composite $(123)$ domain passing through the region are given by (b) $J_1^2/J_8^2=g^2{q}^{-2\alpha }+\cdots$, (c) $J_6{J}_5^2/(J_1{J}_8^2)=2q^{-3}+\cdots$, and (d) $J_1{J}_2/J_8^2=g^3{q}^{-3\alpha }+\cdots$.

Figure 16

Splitting of an $S_n$ domain wall. In the large-$q$ limit with randomly located measurements, and with $n-2<\alpha <n-1$, we evaluate the dominant splitting process at downward-facing triangle, with ${\tau }_n=(12\dots n)$. We find that this process is one where the number of transpositions in the decomposition of the permutations ${\sigma }_1$ and ${\sigma }_2$ is given by $|{\sigma }_1|=1$, $|{\sigma }_2|=n-1$. Other splitting processes are parametrically smaller in $q^{-1}$.

Figure 17

Universal scaling functions $\mathrm{\Phi }(\eta )$ and $\mathrm{\Psi }(\eta )$ extracted from models $A$ and $B$ (see text) and compared to those from a DPRE simulation (same data in Figs. 4 and 5). In fitting to Eqs. (19) and (21), we take $\beta =0.33$ and $\zeta =0.66$ for both models $A$ and $B$. These plots should be compared with Figs. 4 and 5.