Quantum Correlated Systems in the Modern Computing Era

147 pagesIn this dissertation, we employ state-of-art computational approaches to gain insights into long-standing problems in strongly correlated quantum systems. In the first part, we tackle the mysterious strange metal state, marked by a linear-in-temperature resistivity over an extended range of temperature. T-linear resistivity, which is ubiquiteously observed in strongly correlated metals, has long puzzled researchers, as such a temperature dependence is inaccessible from Fermi liquid theory. We begin with an investigation of T-linear resistivity down to very low temperature. We consider a lattice model of itinerant spin-1/2 fermions interacting via on-site Hubbard interaction and random infinite-ranged Heisenberg interaction. We show that the quantum critical point (QCP) associated with the melting of the spin-glass phase displays non-Fermi liquid behaviour. More precisely, the quantum critical regime above the QCP recovers the quantum spin liquid dynamics previously established in the large-M limit of SU(M) symmetric models and features T-linear resistivity driven by a "Planckian" T-linear scattering rate. Next, we consider T-linear resistivity at intermediate and high temperatures. We are particularly motivated by the observation that the slope of T-linear resistivity is often seen to stay constant through crossovers between different temperature regimes: a phenomenon we dub "slope invariance". We consider two strongly correlated models with local self-energies that demonstrate T-linearity: a lattice of coupled Sachdev-Ye-Kitaev (SYK) models and the Hubbard model in single-site dynamical mean-field theory (DMFT). The two models achieve T-linearity through distinct mechanisms at intermediate temperatures, but converge to an identical form at high temperatures. Surprisingly, we find that both models exhibit "slope invariance" across the two temperature regimes. We then turn our attention to the recently discovered strange metallic state in Dirac systems. The metallic "normal state" above the melting temperature of the correlated insulator in integer-filled magic-angle twisted bilayer graphene (MATBG) remains mysterious, with its excessively high entropy and linear-in-temperature resistivity. We focus on the effects of fluctuations of the order-parameters describing correlated insulating states at integer fillings of the low-energy flat bands on charge transport. More precisely, we capture the heterogeneity in the order-parameter landscape observed in MATBG samples with frustrating extended range interactions in an effective Ising model of the order-parameters. The gapless fluctuations of these heterogeneous order-parameter configurations are found to be responsible for the linear-in-temperature resistivity as well as the enhanced low-temperature entropy. We finally conclude the chapter with a section that develops the relevance of the effective Ising model to MATBG in greater detail. We observe that the Kramers intervalley-coherent state, the quantum Hall ferromagnetic ground state of MATBG at various integer fillings, features a 2-state degeneracy from spontaneous breaking of time-reversal symmetry. We find that these degenerate states on the scale of the moiré lattice play the role of spin in the effective Ising model. In the second part, we consider characterization of quantum states realized in quantum devices. With rapid progress across platforms for quantum systems, the problem of many-body quantum state reconstruction for noisy quantum states becomes an important challenge. There has been a growing interest in approaching the problem of quantum state reconstruction using generative neural network models. We begin by proposing the "Attention-based Quantum Tomography" (AQT). AQT is a quantum state characterization method that adapts an attention mechanism-based generative network, originally developed for natural language processing tasks, to learn the mixed state density matrix of a noisy quantum state. We demonstrate not only that AQT outperforms earlier neural-network-based quantum state reconstruction on identical tasks but that it can accurately reconstruct the density matrix associated with a noisy quantum state experimentally realized in an IBMQ quantum computer. We speculate the success of the AQT stems from its ability to model quantum correlations across the entire system much as the attention model for natural language processing captures the correlations among words in a sentence. We close the chapter with an investigation of "Machine-learning Augmented Shadow Tomography" (MAST), a hybrid method of quantum state characterization that augments physical measurement data with AQT generated data to improve the accuracy of shadow tomography. Although efficient algorithms of shadow tomography like classical shadows have garnered much attention in the community, their performance accuracy continues to be bottlenecked by the limited availability of physical measurement data. The motivation behind MAST is to improve accuracy of shadow tomography at large system sizes by incorporating information about the quantum state learned by AQT. We present results and discuss their implications for future studies of machine-learning augmentation