The evolution of entanglement entropy in quantum circuits composed of Haar-random gates and projective measurements shows versatile behavior, with connections to phase transitions and complexity theory. We reformulate the problem in terms of a classical Markov process for the dynamics of bipartition purities and establish a probabilistic cellular-automaton algorithm to compute entanglement entropy in monitored random circuits on arbitrary graphs. In one dimension, we further relate the evolution of the entropy to a simple classical spin model that naturally generalizes a two-dimensional lattice percolation problem. We also establish a Markov model for the evolution of the zeroth Rényi entropy and demonstrate that, in one dimension and in the limit of large local dimension, it coincides with the corresponding second-Rényi-entropy model. Finally, we extend the Markovian description to a more general setting that incorporates continuous-time dynamics, defined by stochastic Hamiltonians and weak local measurements continuously monitoring the system.

1 aShtanko, Oles1 aKharkov, Yaroslav, A.1 aGarcía-Pintos, Luis, Pedro1 aGorshkov, Alexey, V. uhttps://arxiv.org/abs/2004.0673601291nas a2200157 4500008004100000245004000041210003800081260001400119520085000133100001900983700002401002700002601026700002501052700001901077856003701096 2020 eng d00aA minimal model for fast scrambling0 aminimal model for fast scrambling c5/18/20203 aWe study quantum information scrambling in spin models with both long-range all-to-all and short-range interactions. We argue that a simple global, spatially homogeneous interaction together with local chaotic dynamics is sufficient to give rise to fast scrambling, which describes the spread of quantum information over the entire system in a time that is logarithmic in the system size. This is illustrated in two exactly solvable models: (1) a random circuit with Haar random local unitaries and a global interaction and (2) a classical model of globally coupled non-linear oscillators. We use exact numerics to provide further evidence by studying the time evolution of an out-of-time-order correlator and entanglement entropy in spin chains of intermediate sizes. Our results can be verified with state-of-the-art quantum simulators.

1 aBelyansky, Ron1 aBienias, Przemyslaw1 aKharkov, Yaroslav, A.1 aGorshkov, Alexey, V.1 aSwingle, Brian uhttps://arxiv.org/abs/2005.0536201529nas a2200145 4500008004100000245007400041210006900115260001500184490000800199520104100207100002601248700001701274700002201291856007001313 2020 eng d00aQuantum Lifshitz criticality in a frustrated two-dimensional XY model0 aQuantum Lifshitz criticality in a frustrated twodimensional XY m c01/09/20200 v1013 aAntiferromagnetic quantum spin systems can exhibit a transition between collinear and spiral ground states,

driven by frustration. Classically this is a smooth crossover and the crossover point is termed a Lifshitz point.

Quantum fluctuations change the nature of the transition. In particular, it has been argued previously that in the two-dimensional (2D) case a spin liquid (SL) state is developed in the vicinity of the Lifshitz point, termed a Lifshitz SL. In the present work, using a field theory approach, we solve the Lifshitz quantum phase transition problem for the 2D frustrated XY model. Specifically, we show that, unlike the SU (2) symmetric Lifshitz case, in the XY model, the SL exists only at the critical point. At zero temperature we calculate nonuniversal critical exponents in the Néel and in the spin spiral state and relate these to properties of the SL. We also solve the transition problem at a finite temperature and discuss the role of topological excitations.