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The latter refers to a sudden change in a property of a state of n<\/mml:mi><\/mml:math> qubits, such as its entanglement entropy, depending on the rate at which individual qubits are measured. At the same time, quantum complexity emerged as a key quantity for the identification of complex behaviour in quantum many-body dynamics. In this work, we investigate the dynamics of the quantum state complexity in monitored random circuits, where n<\/mml:mi><\/mml:math> qubits evolve according to a random unitary circuit and are individually measured with a fixed probability at each time step. We find that the evolution of the exact quantum state complexity undergoes a phase transition when changing the measurement rate. Below a critical measurement rate, the complexity grows at least linearly in time until {saturating to a value e<\/mml:mi>&#x03A9;<\/mml:mi>(<\/mml:mo>n<\/mml:mi>)<\/mml:mo><\/mml:mrow><\/mml:msup><\/mml:math>.} Above, the complexity does not exceed poly<\/mml:mi>&#x2061;<\/mml:mo>(<\/mml:mo>n<\/mml:mi>)<\/mml:mo><\/mml:math>. In our proof, we make use of percolation theory to find paths along which an exponentially long quantum computation can be run below the critical rate, and to identify events where the state complexity is reset to zero above the critical rate. We lower bound the exact state complexity in the former regime using recently developed techniques from algebraic geometry. Our results combine quantum complexity growth, phase transitions, and computation with measurements to help understand the behavior of monitored random circuits and to make progress towards determining the computational power of measurements in many-body systems.<\/jats:p>","DOI":"10.22331\/q-2025-02-10-1627","type":"journal-article","created":{"date-parts":[[2025,2,10]],"date-time":"2025-02-10T15:25:18Z","timestamp":1739201118000},"page":"1627","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":3,"title":["Quantum complexity phase transitions in monitored random circuits"],"prefix":"10.22331","volume":"9","author":[{"given":"Ryotaro","family":"Suzuki","sequence":"first","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems, Freie Universit\u00e4t Berlin, Berlin 14195, Germany"}]},{"given":"Jonas","family":"Haferkamp","sequence":"additional","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems, Freie Universit\u00e4t Berlin, Berlin 14195, Germany"},{"name":"School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02318, USA"}]},{"given":"Jens","family":"Eisert","sequence":"additional","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems, Freie Universit\u00e4t Berlin, Berlin 14195, Germany"}]},{"given":"Philippe","family":"Faist","sequence":"additional","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems, Freie Universit\u00e4t Berlin, Berlin 14195, Germany"}]}],"member":"9598","published-online":{"date-parts":[[2025,2,10]]},"reference":[{"key":"0","unstructured":"Scott Aaronson ``The Complexity of Quantum States and Transformations: From Quantum Money to Black Holes'' (2016)."},{"key":"1","doi-asserted-by":"publisher","unstructured":"Leonard Susskind ``Entanglement is not enough'' Fortschritte Phy. 64, 49\u201371 (2016).","DOI":"10.1002\/prop.201500095"},{"key":"2","doi-asserted-by":"publisher","unstructured":"Jens Eisert ``Entangling power and quantum circuit complexity'' Phys. 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