Anti–de Sitter Space from Optimization of Path Integrals in Conformal Field Theories

We introduce a new optimization procedure for Euclidean path integrals, which compute wave functionals in conformal field theories (CFTs). We optimize the background metric in the space on which the path integration is performed. Equivalently, this is interpreted as a position-dependent UV cutoff. For two-dimensional CFT vacua, we find the optimized metric is given by that of a hyperbolic space, and we interpret this as a continuous limit of the conjectured relation between tensor networks and Anti-de Sitter (AdS)/conformal field theory (CFT) correspondence. We confirm our procedure for excited states, the thermofield double state, the Sachdev-Ye-Kitaev model, and discuss its extension to higher-dimensional CFTs. We also show that when applied to reduced density matrices, it reproduces entanglement wedges and holographic entanglement entropy. We suggest that our optimization prescription is analogous to the estimation of computational complexity.