Quantum information has emerged as a unifying force in our understanding of large-scale quantum mechanical systems. In quantum technologies, the precise dispersal of quantum information enables computational advantages beyond those of any classical computer. In quantum materials, spatial patterns in the quantum information of electrons give rise to topological phases, with exotic properties when these patterns are "cut" by a material boundary. In more remarkable systems still, quantum information appears central to the unification of quantum mechanics and gravity, one of the frontier open questions in our understanding of the universe. At the same time, the emergence of quantum technologies and accompanying theoretical developments have motivated a widespread interest in dynamical quantum phenomena. In this thesis, I explore several questions at the interface of quantum information and many-body quantum dynamics. Can dynamical many-body quantum systems realize stable coherent quantum phenomena---for example, topological phases---beyond those of static systems? How does quantum information move under many-body dynamics? How can we characterize this movement, and what are its physical implications? And finally, what are the best strategies to learn properties of unknown quantum dynamics from measurements involving them? These inquiries are united by their pursuit to realize and understand large-scale quantum mechanical phenomena in the world around us.

The eternal quest to reconcile quantum mechanics and general relativity has recently led to new insights at the interface between gravitational physics, condensed matter, and quantum information science. Much of this progress owes to the development of the holographic principle, a conjectured duality between gravitational systems and ordinary quantum mechanical systems. This opens up the possibility of translating gravitational phenomena into the language of microscopic quantum physics, and vice versa. This thesis explores several surprising connections that have emerged from a sharper understanding of this duality, with a particular emphasis on questions regarding the dynamics of quantum information. What general lessons can we draw from the physics of black holes about the propagation of quantum information in quantum many-body systems? How can we characterize this propagation and leverage it as a resource for performing tasks on large-scale quantum technologies? From a complementary perspective, how can we utilize quantum simulators to probe features of quantum gravity? By investigating these questions, this thesis sheds light on a rich variety of long-standing topics—including quantum chaos, teleportation, and metrology. More broadly, it demonstrates holography as a powerful theoretical framework for guiding our exploration of quantum many-body physics.

Tuning pressure using diamond anvil cells (DACs) provides an interesting playground for exploring novel condensed phases and understanding important geophysical processes. However, the extension of conventional probes for materials characterization to extreme pressure environments is often constrained by large stress gradients and limited signal strengths. Here, I outline a novel platform for high pressure metrology in DACs based on the Nitrogen Vacancy (NV) color center in diamond. By integrating the NV center, a robust and versatile quantum sensor, directly into the diamond culet, we demonstrate in situ vector magnetometry and tensorial stress imaging with high sensitivity and diffraction-limited sub-micron spatial resolution. Proof of principle measurements demonstrate the platform’s versatility by imaging the dipole strength of Iron (Fe) across the α ↔ ε structural transition, mapping the full stress tensor across the culet surface, and exploring the pressure-temperature phase diagram of gadolinium (Gd). Applying this technique in synergy with piezoelectric displacement sensing and X-ray diffraction (XRD) to study the amorphization of chrysotile Serpentine, we observe phase transformational nanoseismicity driven by shear stress - a possible mechanism underpinning deep earthquakes. Furthermore, we explore surprising qualitative changes in the behavior of the NV center up to megabar pressures, motivating further studies to understand the spin-orbit, and spin-spin coupling in the excited electronic state and the inter-system crossing mechanism. We demonstrate the viability of using the NV center as a robust pressure calibrant capable of easily measuring spatial gradients in multiple stress components. By playing with different crystal cuts of diamond anvils, we extend continuous wave and pulsed NV sensing to pressures nearing ∼ 150 GPa achieving excellent signal contrast and sensitivity. Finally, we use simultaneous transport measurements and NV magnetometry to probe the dual hallmarks superconductivity in a superhydride material at megabar pressures. Our work constitutes the first spatially resolved measurement of the Meissner effect and provides clear evidence of flux trapping in these novel compounds.

Nitrogen-Vacancy (NV) centers in diamond have been established as exceptionally versatile probes for quantum technologies from sensing to simulation.Owing to the versatility of diamond fabrication, NVs can be integrated into probes with nanoscale resolution and superb sensitivity. However, because the diamond is host to many other types of defects, the local environment of the NV is unique for each NV, which poses challenges for any devices instrumenting NVs. This thesis contributes to the future of NV technologies along two essential directions. First, we work towards a more complete understanding of the local charge and strain environments of the NV. In addition to permitting the calibration of these effects for NV probes, we suggest potential uses for the local charges and strain that had previously been considered hindrances. Second, we develop new technologies for NV sensing and simulation. Our work integrating NVs into diamond anvil cells opens the door to detecting high pressure phenomena using the NVs as all-in-one sensors. Finally, our study of the coherence regimes of the local defect environment of the NV is a step towards versatile quantum simulation in dimensions lower than three.

In this disseration we explore the structure and construction of certain quantum many-body states, as well as operator dynamics in quantum many-body systems. First, we detail the implementation of a new tensor network ansatz for many-body ground states based on adiabatically evolving a state to repeatedly double in size. Next, we extend the notion of `strong zero modes, ' edge-localized operators stable at infinite temperature, to operators localized to boundaries between systems. We then discuss efforts to numerically characterize scrambling, a strong form of thermalization, in the Sachdev--Ye--Kitaev model. Finally, we show that a states with large momentum splittings can be created via symmetric Bloch oscillations in two optical lattices being accelerated in opposite directions.

For many years after the dawn of computing machines, it seemed to be the case that the dynamics of any physical system (including all computers, which of course are physical systems themselves) could be efficiently simulated by a simple model of a computer called the Turing machine. A consequence was that computational problems that were found to be hard for Turing machines---those requiring a runtime superpolynomial in the size of the input---remained hard no matter what machine was built to solve them. But in the latter part of the 20th century, an intriguing counterexample emerged: quantum mechanics. The simulation of straightforward quantum systems seemed to have an exponential computational cost. This led to a provocative question: what if one were to build a computer from quantum mechanical components? Could that machine outperform the Turing machine, and efficiently simulate arbitrary quantum processes? And are there other hard problems that such a machine could efficiently solve?

In this dissertation we explore several questions stemming from those ideas. First, classical simulation of quantum many-body physics may be hard, but modern supercomputers are extremely powerful---with cutting-edge innovations in both hardware and algorithms, what quantum simulations can be achieved, and what physics can we learn from them? Second, while there has been astounding progress in the development of quantum computers, they are still small and noisy---what can we do on these near-term devices, that cannot be done with the powerful classical supercomputers just described? Furthermore, if we do a quantum mechanical computation that seems to be infeasible for even the fastest classical machines, how do we check that the result is actually correct? Answering these questions requires deeply exploring the physical nature of computing; at heart, it comes down to the beautiful puzzle of organizing the physical world around us to process information.

Magnetic impurities in a superconductor offer a promising platform for realizing topologicalsuperconductivity. The presence of both classical and quantum impurities in a superconduc- tor can induce a localized subgap state called the Yu-Shiba-Rusinov (YSR) state, which can mediate the effective interaction between magnetic impurities. In this thesis, we explore the effective interaction between classical and quantum impurity spins. We analytically obtain the second-order effective Hamiltonian for the classical impurity spins in the presence of microwave radiation, demonstrating that the effective Hamiltonian exhibits a long-range in- teraction. Additionally, we propose an experimental implementation that enables the control and readout of the impurity spin state ensemble embedded on a Josephson junction from the junction’s integrated conductivity. Extending on the previous result, we also study the effective interaction between quantum impurity spins using a well-controlled perturbation theory. We discover that at the third-order correction, the transverse magnetic field and the quantum spin fluctuation can induce a scalar chirality interaction, which is absent for classical spins. To this end, we investigate the effect of the scalar chirality interaction on the quantum impurity spin order on a two-dimensional lattice using large-scale Density Matrix Renormalization Group (DMRG) calculations. Finally, we study the effects of impurities and disorder in a different context: the bosonic quantum Hall effect. Using quantum Monte Carlo, we uncover a disorder-induced compressible bosonic integer quantum Hall state and characterize its properties.