Inquantum gravity andquantum complexity theory, thecomplexity equals action duality (CA-duality) is theconjecture that thegravitational action of anysemiclassical state with an asymptoticallyanti-de Sitter background corresponds to quantum computational complexity, and that black holes produce complexity at the fastest possible rate.[1] In technical terms, the complexity of aquantum state on a spacelike slice of theconformal field theory dual is proportional to theaction of theWheeler–DeWitt patch (WDW patch) of that spacelike slice in the bulk. The WDW patch is the union of all possible spacelike slices of the bulk with the CFT slice as its boundary.[2][3][4]
This conjecture has been tested against severalanti-de Sitterblack hole backgrounds with and withoutshock waves, and was found to pass all the tests.[3] The action for the WDW patch of a wormhole grows linearly in time for an exponentially long period. Dually, quantum circuit complexity has also been shown to grow linearly for an exponentially long time[5]
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