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arXiv:2603.29323

On the Entanglement Entropy Distribution of a Hybrid Quantum Circuit

Jeonghyeok Park, Hyukjoon Kwon, Hyunseok Jeong

quant-phSubmitted Mar 31, 2026• Announced Apr 1, 2026• 0 citations• Revised Apr 13, 2026 (v2)
v1 · Mar 31, 2026v2 · Apr 13, 2026
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Abstract

We investigate the distribution of entanglement entropy in hybrid quantum circuits consisting of random unitary gates and local measurements applied at a finite rate. We demonstrate that higher moments of the entanglement entropy distribution, such as the ratio between the variance and the mean and the skewness, capture nontrivial features of the measurement-induced dynamics that are invisible to the mean entropy alone. We demonstrate that these quantities exhibit distinct and robust behaviors across the volume-law and area-law phases, and can serve as effective diagnostics of measurement-induced entanglement transitions. We propose a phenomenological model describing the effect of measurements in the area-law regime, which, when combined with the directed polymer in a random environment description of the volume-law phase, well matches numerical simulations across the entire phase diagram.

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We investigate the distribution of entanglement entropy in hybrid quantum circuits consisting of random unitary gates and local measurements applied at a finite rate. We demonstrate that higher moments of the entanglement entropy distribution, such as the ratio between the variance and the mean and the skewness, capture nontrivial features of the measurement-induced dynamics that are invisible to the mean entropy alone. We demonstrate that these quantities exhibit distinct and robust behaviors across the volume-law and area-law phases, and can serve as effective diagnostics of measurement-induced entanglement transitions. We propose a phenomenological model describing the effect of measurements in the area-law regime, which, when combined with the directed polymer in a random environment description of the volume-law phase, well matches numerical simulations across the entire phase diagram.
We introduce and explore a one-dimensional ``hybrid'' quantum circuit model consisting of both unitary gates and projective measurements. While the unitary gates are drawn from a random distribution and act uniformly in the circuit, the measurements are made at random positions and times throughout the system. By varying the measurement rate we can tune between the volume law entangled phase for the random unitary circuit model (no measurements) and a ``quantum Zeno phase'' where strong measurements suppress the entanglement growth to saturate in an area law. Extensive numerical simulations of the quantum trajectories of the many-particle wave functions (exploiting Clifford circuitry to access systems up to 512 qubits) provide evidence for a stable ``weak measurement phase'' that exhibits volume-law entanglement entropy, with a coefficient decreasing with increasing measurement rate. We also present evidence for a continuous quantum dynamical phase transition between the ``weak measurement phase'' and the ``quantum Zeno phase,'' driven by a competition between the entangling tendencies of unitary evolution and the disentangling tendencies of projective measurements. Detailed steady-state and dynamic critical properties of this quantum entanglement transition are accessed.
Quantum information tends to spread out in interacting systems, leading to thermalization as characterized by the volume-law entropy of entanglement, while local measurements that extract classical information of the state tend to interrupt the propagation of information and reduce the entanglement entropy. Can entanglement sustain repeated measurements? The authors address this question in a toy hybrid-circuit model, where qubits undergo both unitary time evolution and sporadic local measurements. The phase diagram features a stable volume-law phase of entanglement for weak measurements and an area-law phase for strong measurements. The two domains in the diagram are separated by a single second-order phase transition. Characterization of both phases as well as of the critical point are presented.
When subject to a nonlocal unitary evolution, qubits in a quantum circuit become increasingly entangled. Conversely, measurements applied to individual qubits lead to their disentanglement from the collective system. The extent of entanglement reduction depends on the frequency of local projective measurements. A delicate balance emerges between unitary evolution, which enhances entanglement, and measurements which diminish it. In the thermodynamic limit, there is a phase transition from volume law entanglement to area law entanglement at a critical value of measurement frequency. This phenomenon, occurring in hybrid quantum circuits with both unitary gates and measurements, is termed as measurement-induced phase transition (MIPT). We study the behavior of MIPT in circuits comprising of two-qubit unitary gates parameterized by Cartan decomposition. We show that the entangling power and gate typicality of the two-qubit local unitaries employed in the circuit can be used to explain the behavior of global bipartite entanglement the circuit can sustain. When the two-qubit gate throughout the circuit is the identity and measurements are the sole driver of the entanglement behavior, we obtain an analytical estimate for the entanglement entropy that shows remarkable agreement with numerical simulations. We also find that the entangling power and gate typicality enable the classification of the two-qubit unitaries by different universality classes of phase transitions that can occur in the hybrid circuit. For all unitaries in a particular universality class, the transition from volume to area law of entanglement occurs with the same exponent that characterizes the phase transition.
We investigate the dynamics of two-dimensional quantum spin systems under the combined effect of random unitary gates and local projective measurements. When considering steady states, a measurement-induced transition occurs between two distinct dynamical phases, one characterized by a volume-law scaling of entanglement entropy, the other by an area law. Employing stabilizer states and Clifford random unitary gates, we numerically investigate square lattices of linear dimension up to L=48L=48L=48 for two distinct measurement protocols. For both protocols, we observe a transition point where the dominant contribution in the entanglement entropy displays multiplicative logarithmic violations to the area law. We obtain estimates of the correlation length critical exponent at the percent level; these estimates suggest universal behavior and are incompatible with the universality class of 3D percolation.
We introduce a class of hybrid quantum circuits, with random unitaries and projective measurements, which host long-range order in the area-law entanglement phase of the steady state.Our primary example is circuits with unitaries respecting a global Ising symmetry and two competing types of measurements.The phase diagram has an area-law phase with spin-glass order, which undergoes a direct transition to a paramagnetic phase with volume-law entanglement, as well as a critical regime.Using mutual information diagnostics, we find that such entanglement transitions preserving a global symmetry are in universality classes different from those without symmetry.We analyze generalizations of such hybrid circuits to higher dimensions, which allow for coexistence of order and volume-law entanglement, as well as topological order without any symmetry restrictions.
The presence of quantum noises inherent to real physical systems can strongly impact the physics in hybrid quantum circuits with local random unitaries and midcircuit measurements. The quantum noises with a size-independent occurring probability can lead to the disappearance of a measurement-induced entanglement phase transition and the emergence of a single area-law phase. In this work, we investigate the effects of quantum noises with size-dependent probabilities $q=p/{L}^{\ensuremath{\alpha}}$, where $\ensuremath{\alpha}$ represents the scaling exponent. We have identified a noise-induced entanglement phase transition from a volume law to a power (area) law in the presence (absence) of measurements as ppp increases when $\ensuremath{\alpha}=1$. With the help of an effective statistical model, we reveal that the phase transition is of first order arising from the competition between two types of spin configurations and shares the same analytical understanding as the noise-induced coding transition. This unified picture further deepens the understanding of the connection between entanglement behavior and the capacity of information protection. When $\ensuremath{\alpha}\ensuremath{\ne}1$, one spin configuration always dominates regardless of ppp and thus the phase transition disappears. Moreover, we highlight the difference between the effects of size-dependent bulk noise and boundary noises. We validate our analytical predictions with extensive numerical results from stabilizer circuit simulations.