Complexity of operators generated by quantum mechanical Hamiltonians

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.

A revised version and gives more detailed examples on why "complexity'' in quantum mechanics and quantum field theory should be bi-invariant

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Cited by in corpus (16)

• Chaos and Complexity in Quantum Mechanics
• The Complexity Geometry of a Single Qubit
• Circuit Complexity across a Topological Phase Transition
• Time evolution of spread complexity in quenched Lipkin-Meshkov-Glick model
• WdW-patches in AdS${}_3$ and complexity change under conformal transformations II
• Spread complexity as classical dilaton solutions
• Time evolution of the complexity in chaotic systems: concrete examples
• More on complexity of operators in quantum field theory
• Geometry and Complexity of Path Integrals in Inhomogeneous CFTs
• Generalized Volume-Complexity For Two-Sided Hyperscaling Violating Black Branes
• Circuit Complexity of Knot States in Chern-Simons theory
• Nielsen complexity of coherent spin state operators
• Complexity and scaling in quantum quench in $1+1$ dimensional fermionic field theories
• Real space circuit complexity as a probe of phase diagrams
• New Topological Gauss-Bonnet Black Holes in Five Dimensions
• Time-dependent Hamiltonians and Geometry of Operators Generated by Them