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Measurement-induced topological entanglement transitions in symmetric random quantum circuits


Random quantum circuits, in which an array of qubits is subjected to a series of randomly chosen unitary operations, have provided key insights into the dynamics of many-body quantum entanglement. Recent work has shown that interleaving the unitary operations with single-qubit measurements can drive a transition between high- and low-entanglement phases. Here we study a class of symmetric random quantum circuits with two competing types of measurement in addition to unitary dynamics. We find a rich phase diagram involving robust symmetry-protected topological, trivial and volume law entangled phases, where the transitions are hidden to expectation values of any operator and are only apparent by averaging the entanglement entropy over quantum trajectories. In the absence of unitary dynamics, we find a purely measurement-induced critical point, which maps exactly to two copies of a classical two-dimensional percolation problem. Numerical simulations indicate that this transition is a tricritical point that splits into two critical lines in the presence of arbitrarily sparse unitary dynamics with an intervening volume law entangled phase. Our results show that measurements alone are sufficient to induce criticality and logarithmic entanglement scaling, and arbitrarily sparse unitary dynamics can be sufficient to stabilize volume law entangled phases in the presence of rapid, yet competing, measurements.

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Fig. 1: Schematic of the circuit and its corresponding phase diagram.
Fig. 2: Chain cuts used to define the generalized topological entanglement entropy.
Fig. 3: The numerical results pertaining to the tricritical point at pu = 0.
Fig. 4: The phase transitions across the pu = 0.3 line.

Data availability

The data plotted in the figures of this Article that support the findings of this study are available via Zenodo at

Code availability

The source codes used to run the simulations of the symmetric random quantum circuit studied in this Article are available via Zenodo at


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We thank M. Hafezi, H. Dehghani and A. Nahum for helpful comments, and M. Gullans and D. Huse for suggesting the modified ancilla order parameter and for discussions regarding its saturation value in the topological phase. We acknowledge the University of Maryland supercomputing resources ( made available for conducting the research reported in this paper. A.L. and M.B. are supported by NSF CAREER (grant number DMR- 1753240), the Alfred P. Sloan Research Fellowship and JQI-PFC-UMD. Y.A. is supported by National Science Foundation grant number NSF DMR1555135 and JQI-NSF-PFC.

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Correspondence to Ali Lavasani.

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Peer review information Nature Physics thanks Xiao Chen, Masaki Oshikawa and Brayden Ware for their contribution to the peer review of this work.

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Lavasani, A., Alavirad, Y. & Barkeshli, M. Measurement-induced topological entanglement transitions in symmetric random quantum circuits. Nat. Phys. 17, 342–347 (2021).

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