The proposal by Peter Shor introduced a quantum algorithm to factorize large integers into their prime factors in almost linear time complexity. It was later realized that Shor’s algorithm was an example of a general class of mathematical problems called the “hidden subgroup” with important practical applications. Complemented by Richard Feynman’s insightful prediction that “there’s plenty of room at the bottom” and the discovery of atomic clock transitions currently used in the global positioning system, quantum information science was born and has presented us with the promise of efficiently solving exponentially complex problems in a variety of domains. As a result, the development of quantum computing hardware, and its associated quantum software, is a rapidly evolving research frontier in multiple areas of science and engineering. Multiple quantum computing technologies, such as ion-traps, superconducting coils, Bosonic processors with photons, solid-state devices and quantum dots inside cavities, and Rydberg atoms, have emerged as potential alternative computational platforms to address complex computational challenges. However, it is now accepted that universal, fully fault-tolerant, quantum computers are a rather distant dream, and new hybrid quantum-classical frontiers and Noisy Intermediate Scale Quantum (NISQ) systems have recently emerged. Interleaving computations by alternating between NISQ machines and classical computers has emerged as a paradigm for effectively exploiting NISQ machines in ways that go beyond the power of classical computers. In this talk we will present recent advances from our group towards the development of “hybrid” quantum-classical computing algorithms designed by solving chemistry problems such as correlated electronic structure and nuclear dynamics. The talk showcases a strong collaboration between theoretical chemists, atomic physicists and theoretical computer scientists. In the first half of my talk, I will consider the problem of quantum nuclear dynamics (or quantum molecular dynamics, where the nuclei and electrons are treated quantum mechanically), and discuss two approaches for the development of quantum algorithms. These can loosely be characterized using the familiar analogue and digital descriptions, but now in the quantum regime. Additionally, our algorithms have been implemented on state-of-the-art quantum platforms in IonQ, Inc. and Sandia National Lab’s QSCOUT ion-trap quantum computing systems. We have constructed quantum wavepacket dynamics studies on these platforms for the first time, and by studying the time-evolution of such quantum wavepackets, using a variant of the phase estimation algorithm, we obtained accurate vibrational properties for hydrogen bonded systems. As the dimensionality of nuclear degrees of freedom increases, we also show that using tensor networks, one can construct new, parallel quantum computing algorithms. Indeed, for the first time we have recently demonstrated parallel quantum computing on IonQ’s 11-qubit Harmony ion-trap systems. In the second half of this talk, we consider accurate post-Hartree Fock electronic structure. There have been several important algorithms introduced in the literature for this purpose, but in all cases, implementation of these algorithms leads to the quantum circuit depth problem. Here, as the quantum circuits become deeper and larger, the fidelity of the quantum hardware deeply affects the error obtained from quantum propagation. While many groups are working on better error correction strategies, our group has introduced a graph-theoretic procedure to drastically reduce the quantum circuit depth by several orders of magnitude. We have implemented our graph-theory based algorithms on IBM’s quantum simulators to study medium-to-large, protonated water clusters and obtained coupled cluster accuracy. If time permits, we will also discuss how the methods above lead to more efficient machine learning platforms. Indeed, our group remains strongly focused on the interface of quantum computing and machine learning.