Path Integral Optimization as Circuit Complexity

Camargo, Hugo A. (Max Planck Institute for Gravitational Physics (Albert Einstein Institute),Am Mühlenberg 1, 14476 Potsdam-Golm, Germany) (Department of Physics, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany) ; Heller, Michal P. (Max Planck Institute for Gravitational Physics (Albert Einstein Institute),Am Mühlenberg 1, 14476 Potsdam-Golm, Germany) ; Jefferson, Ro (Max Planck Institute for Gravitational Physics (Albert Einstein Institute),Am Mühlenberg 1, 14476 Potsdam-Golm, Germany) ; Knaute, Johannes (Max Planck Institute for Gravitational Physics (Albert Einstein Institute),Am Mühlenberg 1, 14476 Potsdam-Golm, Germany) (Department of Physics, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany)

01 July 2019

Abstract: Early efforts to understand complexity in field theory have primarily employed a geometric approach based on the concept of circuit complexity in quantum information theory. In a parallel vein, it has been proposed that certain deformations of the Euclidean path integral that prepare a given operator or state may provide an alternative definition, whose connection to the standard notion of complexity is less apparent. In this Letter, we bridge the gap between these two proposals in two-dimensional conformal field theories, by explicitly showing how the latter approach from path integral optimization may be given by a concrete realization within the standard gate counting framework. In particular, we show that, when the background geometry is deformed by a Weyl rescaling, a judicious gate counting allows one to recover the Liouville action as a particular choice within a more general class of cost functions.


Published in: Physical Review Letters 123 (2019)
Published by: APS
DOI: 10.1103/PhysRevLett.123.011601
License: CC-BY-4.0



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