Spacetime as a quantum circuit

Abstract

We propose that finite cutoff regions of holographic spacetimes represent quantum circuits that map between boundary states at different times and Wilsonian cutoffs, and that the complexity of those quantum circuits is given by the gravitational action. The optimal circuit minimizes the gravitational action. This is a generalization of both the “complexity equals volume” conjecture to unoptimized circuits, and path integral optimization to finite cutoffs. Using tools from holographic $$ T\overline{T} $$ T T ¯ , we find that surfaces of constant scalar curvature play a special role in optimizing quantum circuits. We also find an interesting connection of our proposal to kinematic space, and discuss possible circuit representations and gate counting interpretations of the gravitational action.

Exact Gravity Duals for Simple Quantum Circuits

“…Furthermore, another interesting question is whether any of the previously studied cost functions in [8][9][10][11][12] can be mapped to geometric quantities in the bulk. Conversely, it would be interesting to understand better bulk candidates for costs considered in [33][34][35][36]. To make 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 energymomentum tensor.…”

Q-curvature and Path Integral Complexity

“…matter) properties of the surface B could be incorporated in the holographic path integral proposal. In a precise sense, the surface B can be understood as a time-dependent cut-off [30,52] and e.g. adding counter-terms-like higher-derivative on B may be a natural step.…”

“…Given the interpretation that the holographic path integral optimization should be thought of as the boundary action plus finite cut-off terms, it is tempting to speculate that, in holographic settings, T T deformations could be used as a tool to introduce such finite cut-off corrections (see [30,52]). Making this precise is beyond the scope of this work however we discuss below how these two approaches may be mutually consistent.…”

“…On the other hand, finding B from extremizing an on-shell gravity action with counter-terms computed up to a finite-cutoff region of the bulk is natural in the T T context. We discuss this procedure below and point the main difference with the TN ideas based on T T deformations [30,52].…”

Circuit Complexity in Topological Quantum Field Theory

“…Quantum circuit complexity is roughly the number of basic quantum operations, or gates, needed to construct a quantum operator or to prepare a quantum state from a simple reference state.2 A category is a mathematical structure that consists of objects (e.g., sets, groups, vector spaces) along with "arrows" or maps between them (e.g., functions, homomorphisms, linear operators). A functor is a homomorphism of categories.3 Our perspective, while independent of holographic considerations, bears some conceptual similarities to tensor network toy models of holography[31][32][33][34][35][36][37][38], in which the geometry of a spatial slice is understood as a quantum circuit and thereby provides a natural notion of complexity for boundary states.…”