Quantum algorithms for the ground state energy

Estimating the ground state energy of a quantum Hamiltonian is a fundamental task in quantum chemistry and condensed matter physics. Quantum algorithms offer a promising path towards getting highly precise estimates for quantum systems beyond the capability of classical algorithms. As quantum computers become more robust to noise with the improvement of quantum error correction capability, we may need different algorithms for this task at different stages of this development. I will first talk about quantum algorithms for this task that are designed for early fault-tolerant quantum computers, where besides the runtime we also want to minimize the number of qubits and the circuit depth. For quantum computers with sufficient error correcting power, I will present a near-optimal algorithm that optimizes the total runtime.