Complexity of operators generated by quantum mechanical Hamiltonians

Run-Qiu Yang, Keun-Young Kim

Published 2018-10-22Version 1

We propose how to compute the complexity of operators generated by Hamiltonians in quantum field theory(QFT) and quantum mechanics(QM). The Hamiltonians in QFT/QM and quantum circuit have a few essential differences, for which we introduce new principles and methods for complexity. We show that the complexity geometry corresponding to one-dimensional quadratic Hamiltonians is equivalent to AdS$_3$ spacetime. Here, the requirement that the Hamiltonian is bounded below corresponds to the speed of a particle is not superluminal. Our proposal proves the complexity of the operator generated by a free Hamiltonian is zero, as expected. We also show that the complexity can be used as an indicator of quantum phase transitions. By studying non-relativistic particles in compact Riemannian manifolds we find the complexity is given by the global geometric property of the space. In particular, we show that in low energy limit the critical spacetime dimension to insure nonnegative complexity is 3+1 dimension.