Axiomatic complexity in quantum field theory and its applications

Abstract

Based on three general axioms for complexity, inspired by circuit complexity, and two physical assumptions, we show that the complexity of the SU($n$) group operator is given by the length of a geodesic in bi-invariant Finsler manifolds, of which Finsler metric is uniquely determined. We also provide a new interpretation to Schr\{o}dinger's equation for isolated systems - the quantum state evolves by the process of minimizing computational cost. Building on the complexity of the operator, we propose the complexity between states. For pure states, our proposal shows an interesting connection to the complexity of cMERA (continuous multi-scale entanglement renormalization ansatz) tensor networks and implies the results obtained by both the Fubini-Study metric approach and the path-integral approach. Furthermore, our proposal also reproduces the holographic complexity for the thermofield double states by the CV (complexity = volume) and CA (complexity = action) conjectures.