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arXiv:2510.12743 · PRX Quantum

Measurement-Induced Entanglement in Noisy 2D Random Circuits

Zhi-Yuan Wei, Jon Nelson, Joel Rajakumar, Esther Cruz, Alexey Vyacheslavovich Gorshkov, Michael J. Gullans, Daniel Malz

Published Jun 17, 2026• 0 citations
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Abstract

We study measurement-induced entanglement (MIE) generated by column-by-column sampling of noisy 2D random circuits of size N and depth T . Focusing primarily on Clifford circuits and using the operator entanglement S op of the sampling-induced boundary state as a proxy for computational complexity, first, we reproduce in the noiseless limit a finite-depth transition from area- to volume-law scaling at a threshold depth T c = 6 . In contrast, in the presence of single-qubit depolarizing noise at any constant rate p > 0 , we find that the operator entanglement S op obeys an area law, with its maximum value scaling approximately linearly with T / p in the regime T > T c . By analyzing the spatial distribution of stabilizer generators, we observe exponential localization of stabilizer generators; these both account for the scaling of the maximal S op and imply an exponential decay of conditional mutual information across buffered tripartitions, which we also confirm numerically. Together, these results indicate that constant local noise destroys long-range MIE in 2D random Clifford circuits, and that a tensor-network-based algorithm can efficiently sample from noisy 2D random Clifford circuits (i) at sub-logarithmic depths T = o ( log N ) for any constant noise rate p = Ω ( 1 ) , and (ii) at constant depths T = O ( 1 ) for noise rates p = Ω ( log − 1 N ) . Finally, we turn to depth T = 4 Haar-random and measurement-based quantum computing-type circuits, providing evidence that MIE in noisy 2D Haar-random circuits exhibits the same qualitative behavior as in random Clifford circuits, and that noise destroys the volume-law scaling of MIE in non-Clifford circuits.

DOI
10.1103/yfzm-5rr6

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We study measurement-induced entanglement (MIE) generated by column-by-column sampling of noisy 2D random circuits of size N and depth T . Focusing primarily on Clifford circuits and using the operator entanglement S op of the sampling-induced boundary state as a proxy for computational complexity, first, we reproduce in the noiseless limit a finite-depth transition from area- to volume-law scaling at a threshold depth T c = 6 . In contrast, in the presence of single-qubit depolarizing noise at any constant rate p > 0 , we find that the operator entanglement S op obeys an area law, with its maximum value scaling approximately linearly with T / p in the regime T > T c . By analyzing the spatial distribution of stabilizer generators, we observe exponential localization of stabilizer generators; these both account for the scaling of the maximal S op and imply an exponential decay of conditional mutual information across buffered tripartitions, which we also confirm numerically. Together, these results indicate that constant local noise destroys long-range MIE in 2D random Clifford circuits, and that a tensor-network-based algorithm can efficiently sample from noisy 2D random Clifford circuits (i) at sub-logarithmic depths T = o ( log N ) for any constant noise rate p = Ω ( 1 ) , and (ii) at constant depths T = O ( 1 ) for noise rates p = Ω ( log − 1 N ) . Finally, we turn to depth T = 4 Haar-random and measurement-based quantum computing-type circuits, providing evidence that MIE in noisy 2D Haar-random circuits exhibits the same qualitative behavior as in random Clifford circuits, and that noise destroys the volume-law scaling of MIE in non-Clifford circuits.
We study random constant-depth quantum circuits in a two-dimensional (2D) architecture. While these circuits only produce entanglement between nearby qubits on the lattice, long-range entanglement can be generated by measuring a subset of the qubits of the output state. It is conjectured that this (MIE) proliferates when the circuit depth is at least a constant critical value d ∗ . For circuits composed of Haar-random two-qubit gates, it is also believed that this coincides with a in the classical hardness of sampling from the output distribution. Here, we provide evidence for a quantum advantage phase transition in the setting of random circuits. Our work extends the scope of recent separations between the computational power of constant-depth quantum and classical circuits, demonstrating that this kind of advantage is present in canonical random circuit sampling tasks. In particular, we show that in any architecture of random shallow Clifford circuits, the presence of long-range MIE gives rise to an unconditional quantum advantage. In contrast, any depth- d 2D quantum circuit that satisfies a short-range MIE property can be classically simulated efficiently and with depth O ( d ) . Finally, we introduce a 2D depth- 2 “coarse-grained” circuit architecture, composed of random Clifford gates acting on O ( log  ( n ) ) qubits, for which we prove the existence of long-range MIE and establish an unconditional quantum advantage.
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The process by which open quantum systems thermalize with an environment is both of fundamental interest and relevant to noisy quantum devices. As a minimal model of this process, we consider a qudit chain evolving under local random unitaries and local depolarization channels. After mapping to a statistical mechanics model, the depolarization (noise) acts like a symmetry-breaking field, and we argue that it causes the system to thermalize within a timescale independent of system size. We show that various bipartite entanglement measures---mutual information, operator entanglement, and entanglement negativity---grow at a speed proportional to the size of the bipartition boundary. As a result, these entanglement measures obey an area law: Their maximal value during the dynamics is bounded by the boundary instead of the volume. In contrast, if the depolarization only acts at the system boundary, then the maximum value of the entanglement measures obeys a volume law. We complement our analysis with scalable simulations involving Clifford gates, for both one- and two-dimensional systems.
Measurement-only circuits provide a minimal setting in which repeated local projections can either generate or suppress many-body entanglement, giving rise to measurement-induced phase transitions and dynamical regimes, that might have no unitary counterpart. Here we investigate entanglement and information transitions in one-dimensional measurement-only Clifford circuits with long-range two-qubit parity checks. By tuning both the measurement range and density per layer, we uncover a broad set of phases whose classification requires probes beyond entanglement entropy, such as mutual information, tripartite mutual information, purification from an ancilla, and Bell-cluster statistics. We map phase diagrams using large-scale Clifford simulations for two protocols: a random-basis design in which each measurement is randomly chosen from {XX,YY,ZZ}\lbrace XX,YY,ZZ \rbrace{XX,YY,ZZ}, and a single-basis design in which the basis is fixed within each layer but varies between layers, hence introducing more structure to the circuit. We map the trajectory-averaged entanglement entropy to a two-dimensional statistical mechanics model by extending a replica-based method to random-basis measurement-only circuits, and show that a continuous-time limit yields an effective long-range XX hamiltonian in the steady state. This connection links the observed volume-law to sub-volume-law entanglement transition to the boundary between a continuous symmetry broken phase and a critical XY phase. Strikingly, in structured (single-basis) circuits we find a phase in which volume-law and long-range entanglement coexists with rapid, size-independent purification of an ancilla qubit, and the absence of scrambling, highlighting measurement-only circuits as a promising route to efficiently preparing highly entangled and technologically useful quantum states.
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An entanglement transition can occur without a corresponding transition in purification. We demonstrate this separation in spatially local monitored Clifford circuits with a fixed number of layers per Floquet period, where a spatially random pattern of Clifford gates and Pauli measurements is repeated exactly in time. Rare spatial structures persist under this repetition and break the correspondence between entanglement and purification found in conventional spacetime-random circuits. In one spatial dimension, the late-time entanglement produced with an initial product state remains area-law, while the system remains extensively mixed in the thermodynamic and long-time limits if the initial state is maximally mixed. In two or more dimensions, we find a transition between volume-law and area-law entanglement while the system remains mixed on both sides of this transition. An extensive amount of quantum information survives indefinitely in both phases and evolves unitarily despite the repeated measurements. We introduce the logical-support radius to characterize its spatial spreading and identify the volume-law and area-law phases with delocalized and localized logical dynamics, respectively. We further introduce the logical Krylov dimension, which counts the independent logical operators dynamically generated from an initially local operator. Its scaling resolves the transition and provides evidence for a new universality class of measurement-induced phase transitions.