Complexity of operators generated by quantum mechanical Hamiltonians

TL;DR

The paper develops a framework to compute the complexity of operators generated by Hamiltonians in QFT/QM by extending Nielsen's complexity geometry to infinite-dimensional settings and imposing unitary (bi-invariant) symmetry. For a general one-dimensional quadratic Hamiltonian, the resulting complexity geometry is shown to be AdS$_3$ with AdS radius $\ell_{\text{AdS}} = \lambda_0$, and nonnegativity of complexity is tied to the Hamiltonian being lower-bounded and to subluminal propagation; the free Hamiltonian has zero complexity. In a compact Riemannian manifold, complexity is governed by global geometric properties and, in the low-energy 3+1 dimensional limit, nonnegative complexity is ensured, signaling a potential universal principle linking physical viability to nonnegative complexity. The work emphasizes that bi-invariance, reinforced by unitary invariance, provides a natural and necessary structure for complexity in QFT/QM and hints at deep connections between complexity, spacetime geometry, and renormalization in quantum theories.

Abstract

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 complexity is nonnegative corresponds to the fact that the Hamiltonian is lower bounded and 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. By studying a non-relativistic particle 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 ensure the "nonnegative" complexity is the 3+1 dimension.

Figures (1)

• Figure 1: Schematic diagram on the similarities between the complexity geometry of the quadratic Hamiltonian (left panel) and the spacetime geometry of AdS$_3$ (right panel).