Shivan Mittal Final Defense

Random quantum circuits are analytically tractable models capturing universal aspects of generic quantum many-body physics. In this thesis, we explore two facets of such circuits. First, we study local random quantum circuits on arbitrary architectures, constructed by applying Haar random 2-qudit gates to pairs of qudits by uniformly sampling edges of their connectivity graph. We characterize the influence of the qudit connectivity on the convergence of such circuits to Haar random unitaries. We give the first proof that local random quantum circuits on arbitrary connected architectures form approximate unitary $k$-designs in at most quasi-polynomial in system size gates. We also identify a large class of connectivity graphs such that local random quantum circuits on those form approximate unitary designs in as many gates as they do on 1D graphs, up to constant multiplicative factor. To obtain these results, (1) we generalize Knabe's finite size threshold method of lower bounding spectral gap to arbitrary connectivity graphs and (2) we develop a novel approach to relate spectral gaps of any spanning tree graph to those for 1D graph by recursive applications of Detectability Lemma and the Quantum Union Bound. Second, we investigate the dynamics of quantum state complexity of subsystems of output states of random quantum circuits. We formulate and confirm several features of holographic subsystem complexity in certain random quantum circuits. In particular, we prove a sharp subsystem complexity transition for contiguous subsystems: complexity of smaller-than-half subsystems grows to superlinear in subsystem size and then crashes to linear complexity, whereas complexity of larger-than-half subsystems grows linearly for exponential in subsystem size time. Furthermore, we prove that when smaller-than-half subsystems are fragmented into $m$ equal-size disjoint pieces, then the complexity crash time scale reduces by a multiplicative factor of $m$. To obtain these results, we generalize the method in (http://arxiv.org/abs/1912.04297) PRX Quantum 2 No. 3, (2021) 030316 to subsystem complexity by incorporating subsystem rank-dynamics. We also develop a domain-wall counting scheme for random circuit purity dynamics in terms of non-intersecting chords on an annulus.