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Abstract
Random quantum circuits play an important role in the study of non-equilibrium properties of quantum many-body systems, such as entanglement and thermalization. In this thesis, we study the spreading of quantum information in two important examples of random circuits: a quantum circuit with charge and dipole conservation laws, and the random tree tensor network, which corresponds to the all-to-all circuit. We find that the dynamics of these two systems can fall into different phases depending on the spreading of quantum information in the system/Hilbert space.
We first study the time evolution of a one-dimensional fracton system with charge and dipole moment conservation using a random unitary circuit description. Previous work has shown that when the random unitary operators act on four or more sites, an arbitrary initial state eventually thermalizes via a universal subdiffusive dynamics. In contrast, a system evolving under three-site gates fails to thermalize due to strong ``fragmentation'' of the Hilbert space. Here we show that three-site gate dynamics causes a given initial state to evolve toward a highly nonthermal state on a time scale consistent with Brownian diffusion. Strikingly, the dynamics produces an effective attraction between isolated fractons or between a single fracton and the boundaries of the system, as in the Casimir effect of quantum electrodynamics. We show how this attraction can be understood by exact mapping to a simple classical statistical mechanics problem, which we solve exactly for the case of an initial state with either one or two fractons. What's more, we further find that in the limit of maximal single charge goes to infinity, the fragmentation of Hilbert space of different filling can be solved by mapping to a mathematical tournament problem. Using this mapping we can show that the system exhibits a transition from a non-thermalized to a thermalized phase by tuning the filling number, as observed in a previous work. Our technique allows us to get the exact critical point and scaling exponents which match well with the numerical results.
The second question we consider is the evolution of entanglement in monitored quantum tree system. Monitored many-body systems fall broadly into two dynamical phases, "entangling" or "disentan-
gling", separated by a transition as a function of the rate at which measurements are made on the
system. Producing an analytical theory of this measurement-induced transition is an outstanding
challenge. Recent work made progress in the context of tree tensor networks, which can be related
to all-to-all quantum circuit dynamics with forced (postselected) measurement outcomes. So far,
however, there are no exact solutions for dynamics of spin-1/2 degrees of freedom (qubits) with
"real" measurements, whose outcome probabilities are sampled according to the Born rule. Here
we define dynamical processes for qubits, with real measurements, that have a tree-like spacetime
interaction graph, either collapsing or expanding the system as a function of time. The former
case yields an exactly solvable measurement transition. We explore these processes analytically and
numerically, exploiting the recursive structure of the tree. We compare the case of "real" measurements with the case of "forced" measurements. Both cases show a transition at a nontrivial value of
the measurement strength, with the real measurement case exhibiting a smaller entangling phase.
Both exhibit exponential scaling of the entanglement near the transition, but they differ in the value
of a critical exponent. An intriguing difference between the two cases is that the real measurement
case lies at the boundary between two distinct types of critical scaling. On the basis of our results
we propose a protocol for realizing a measurement phase transition experimentally via an expansion
process.