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#### Path integral optimization as circuit complexity

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1904.02713.pdf

(Preprint), 433KB

PhysRevLett.123.011601.pdf

(Publisher version), 224KB

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##### Citation

Camargo, H., Heller, M. P., Jefferson, R., & Knaute, J. (2019). Path integral optimization
as circuit complexity.* Physical Review Letters,* *123*(1):
011601. doi:10.1103/PhysRevLett.123.011601.

Cite as: https://hdl.handle.net/21.11116/0000-0003-5536-1

##### 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 prepares 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 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.

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 prepares 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 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.