How Tensor Network Renormalization quantifies circuit complexity and why this is a problem of [considerable] gravity

          @misc{ pirsa_PIRSA:17040043,
            doi = {10.48660/17040043},
            url = {https://pirsa.org/17040043},
            author = {Czech, Bartek},
            keywords = {Condensed Matter, Quantum Fields and Strings, Quantum Gravity, Quantum Information},
            language = {en},
            title = {How Tensor Network Renormalization quantifies circuit complexity and why this is a problem of [considerable] gravity},
            publisher = {Perimeter Institute},
            year = {2017},
            month = {apr},
            note = {PIRSA:17040043 see, \url{https://pirsa.org}}
          }
          

Abstract

According to a recent proposal, in the AdS/CFT correspondence the circuit complexity of a CFT state is dual to the Einstein-Hilbert action of a certain region in the dual space-time. If the proposal is correct, it should be possible to derive Einstein's equations by varying the complexity in a class of circuits that prepare the requisite CFT state. This talk attempts such a derivation in very special settings: Virasoro descendants of the CFT2 ground state, which are dual to locally AdS3 geometries. By applying Tensor Network Renormalization to the discretized Euclidean path integral that prepares the CFT state, I will justify the recent suggestion by Caputa et al. that the complexity of a path integral is quantified by the Liouville action. The Liouville field specifies the conformal frame in which the path integral is evaluated; in the most efficient / least complexity frame, the Liouville field is closely related to entanglement entropies of CFT2 intervals. Assuming the Ryu-Takayanagi proposal, the said entanglement entropies are lengths of geodesics living in the dual space-time. The Liouville equation of motion satisfied by the minimal complexity Liouville field is a geodesic-wise rewriting of the non-linear vacuum Einstein's equations in 3d with a negative cosmological constant. I emphasize that this is very much work in progress; I hope the audience will help me to sharpen the arguments.