Measurement-induced entanglement and complexity in random constant-depth 2D quantum circuits

TL;DR

This work proves that random constant-depth 2D quantum circuits generate extensive measurement-induced entanglement between distant regions after partial measurements, overcoming previous barriers by introducing replica-free techniques that map the distillation error to a self-avoiding-walk partition function. By formulating the problem as a one-shot multi-user entanglement-of-assistance task and leveraging a multiparty splitting approach, the authors derive a rigorous lower bound on the average post-measurement entanglement that scales linearly with system size along the long axis. Applying the bound to random holographic, 4-local, and brickwork circuits, they connect high MIE to the failure of standard classical simulation methods (SEBD and boundary MPS) and prove an unconditional separation between random constant-depth quantum circuits and sublogarithmic-depth classical circuits in certain architectures. The results thus support a form of quantum advantage for shallow circuits and provide a versatile framework for analyzing entanglement and complexity in 2D quantum dynamics, with potential extensions to noisy settings, MBQC, and tensor-network contraction problems.

Abstract

We analyse the entanglement structure of states generated by random constant-depth two-dimensional quantum circuits, followed by projective measurements of a subset of sites. By deriving a rigorous lower bound on the average entanglement entropy of such post-measurement states, we prove that macroscopic long-ranged entanglement is generated above some constant critical depth in several natural classes of circuit architectures, which include brickwork circuits and random holographic tensor networks. This behaviour had been conjectured based on previous works, which utilize non-rigorous methods such as replica theory calculations, or work in regimes where the local Hilbert space dimension grows with system size. To establish our lower bound, we develop new replica-free theoretical techniques that leverage tools from multi-user quantum information theory, which are of independent interest, allowing us to map the problem onto a statistical mechanics model of self-avoiding walks without requiring large local Hilbert space dimension. Our findings have consequences for the complexity of classically simulating sampling from random shallow circuits, and of contracting tensor networks: First, we show that standard algorithms based on matrix product states which are used for both these tasks will fail above some constant depth and bond dimension, respectively. In addition, we also prove that these random constant-depth quantum circuits cannot be simulated by any classical circuit of sublogarithmic depth.

Figures (11)

• Figure 1: After preparing a state using a geometrically local depth-$d_C$ circuit, we measure degrees of freedom in $B$ and look at the (bipartite) entanglement of the post-measurement state between unmeasured regions $A$ and $C$. This figure shows one particular choice of geometry $ABC$ which we will often refer back to, however for most of this work the choice is arbitrary.
• Figure 2: Flow of logic leading to our primary technical finding, \ref{['thm:EntBound']}, and in turn our main result---that the post-measurement states $\ket{\phi^{AC}_{s_B}}$ possess extensive measurement-induced entanglement (scaling with the linear system size $L_x$). Top row: After measuring on $B$, we consider the effect of some distillation protocol, whose aim is to generate a state that is close to the maximally entangled state $\ket{\Phi_{d'}}$ of rank $d'$. Overall, this procedure constitutes an entanglement of assistance task. Right panel: The self-avoiding walk partition function is known to exhibit an ordering transition driven by the temperature $\beta^{-1}$. Here, $\beta$ is set by the amount of entanglement in the pre-measurement state, which is determined by circuit depth and/or qudit dimension. As a result, we find regimes where we can lower bound MIE in the form \ref{['eq:LinearMIE']}.
• Figure 3: Circuit architectures that we explicitly consider in this work. Here, the order of gates runs from bottom to top. (a) A 2D brickwork circuit, where 2-local gates (blue rectangles) are applied between pairs of qudits in an arrangement that depends on $t \mod 4$, where $t = 1, \ldots, d_C$ is a discrete time index labeling each layer. (b) 4-local plaquette circuit. By combining groups of 4 two-qudit gates in the 2D brickwork architecture, depth-$2d_C$ brickwork circuits can be thought of as special cases of depth-$d_C$ 4-local circuits.
• Figure 4: (a) Blocking sites into rectangular cells of dimension $4d_C \times 2d_C$. (b) Treating all the qudits within each cell as a single degree of freedom, any state prepared by $d_C$ layers of gates in the architecture shown in Fig. \ref{['fig:2dArch']}(b) can be written as a strictly local isoTNS on the triangular lattice.
• Figure 5: (a) In the holographic circuit architecture, a bipartite entangled state $\ket{\omega_e}$, denoted by dark yellow circles, is prepared on each bond, which is usually taken to be maximally entangled. Then, at each site, a random unitary is applied (light yellow squares). When projective measurements are made in the bulk, one obtains a random holographic state (b), where the physical legs of each bulk tensor have been projected out. In this geometry, the physical legs of the holographic tensor network reside on the top and bottom boundaries. The distribution of holographic tensor network states generated in this way is equivalent to Eq. \ref{['eq:HologDist']}.

Theorems & Definitions (14)

• Corollary 1: Informal
• Corollary 2: Informal
• Definition 3: IsoTNS Zaletel2020
• Definition 4: Strictly local correlations
• Proposition 5
• Lemma 6
• Lemma 7
• Theorem 8
• Corollary 9
• Proposition 10