Quantum approximate spectral decomposition

The linearity of quantum mechanics allows us to analyze dynamics via the spectral decomposition of the Hamiltonian. In many-body systems, however, the Hilbert space dimension is exponentially large in the number of degrees of freedom, and at finite energy densities the level separations are exponentially small. As a consequence, even fault-tolerant quantum computers cannot efficiently construct highly-excited eigenstates. This raises the question of whether the spectral decomposition remains a useful tool for the analysis of many-body dynamics. I will introduce a generalization of the spectral decomposition which allows for the approximate reconstruction of the dynamics of both observables and entanglement up to polynomial times, and present an algorithm allowing for its efficient extraction on quantum computers.