Cost of holographic path integrals

Abstract

We consider proposals for the cost of holographic path integrals. Gravitational path integrals within finite radial cutoff surfaces have a precise map to path integrals in T T deformed holographic CFTs. In Nielsen's geometric formulation cost is the length of a not-necessarily-geodesic path in a metric space of operators. Our cost proposals differ from holographic state complexity proposals in that (1) the boundary dual is cost, a quantity that can be 'optimised' to state complexity, (2) the set of proposals is large: all functions on all bulk subregions of any co-dimension which satisfy the physical properties of cost, and (3) the proposals are by construction UV-finite. The optimal path integral that prepares a given state is that with minimal cost, and cost proposals which reduce to the CV and CV2.0 complexity conjectures when the path integral is optimised are found, while bounded cost proposals based on gravitational action are not found. Related to our analysis of gravitational action-based proposals, we study bulk hypersurfaces with a constant intrinsic curvature of a specific value and give a Lorentzian version of the Gauss-Bonnet theorem valid in the presence of conical singularities.

Exact Gravity Duals for Simple Quantum Circuits

“…For instance, in [8] it was demonstrated that the Fubini-Study distance on the space of circuits starting from scalar primary states is encoded in the maximal and minimal perpendicular distances between infinitesimally close timelike geodesics in AdS. Conversely, it would be interesting to understand better bulk candidates for costs considered in [35][36][37][38]. To make further progress in this direction, cost functions on the boundary have to be determined in terms of the bulk metric or conversely bulk observables in terms of conformal field theory quantities like the boundary energy-momentum tensor.…”

“…Note especially the inclusion of the square in equation (38). This means that compared to the problem of geodesic motion in curved spaces, the functional that we are working with is more similar to the kinetic energy than the length functional.…”

“…Also, as shown in [5,6], the complexity (38) matches the change of volume under infinitesimal local conformal transformations to the leading nontrivial order. We will hence further study this cost further in this section, and allowing ourselves a little imprecision of nomenclature, we will refer to (38) as Fubini-Study complexity.…”