Quantum simulations of large lattice models and chemistry beyond the existence of exact solutions

In the last decade, variational algorithms have been the tool of choice for researchers and practitioners of quantum computing to tackle ground state problems on pre-fault-tolerant quantum processors. Currently, a number of practical and theoretical issues prevent scalability of variational algorithms to large system sizes. In this talk, I will discuss two quantum diagonalization methods, based on subspaces obtained from quantum computers, which overcome the scaling limitations of variational algorithms. The Krylov quantum diagonalization, which allowed us to perform quantum ground state calculations for lattice models of up to 50 spins, and a sample-based quantum diagonalization, which enabled realistic chemistry computations of up to 77 qubits on a quantum centric supercomputing architecture, using a Heron quantum processor and the supercomputer Fugaku.