Path Integral Optimization as Circuit Complexity

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 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.

Aspects of the first law of complexity

“…This approach was first applied to a concrete quantum field theory calculation in [57], where the authors adapted Nielsen's approach to evaluate the complexity of the vacuum state of a free scalar field theory. These calculations have been extended in a number of interesting ways in the past few years, e.g., [58][59][60][61][62][63][64][65][66][67][68][69][70][71][72][73][74][75][76], but we will be particularly interested in [65] where the same techniques were applied to explore the complexity of coherent states in the same QFT.…”

Complexity and emergence of warped AdS3 space-time from chiral Liouville action

“…for the case of zero angle σ where t = −ṫ tan σ has been proposed in [59]. The dual action of DBI case for the warped case could be considered.…”

“…Before doing that, we should remind that, from the perspective of [59], the Liouville action is a particular cost function. The chiral Liouville action then could be considered as a particular cost function for the non-local and Lorentz-breaking theory of warped CFTs.…”

“…Other ideas such as quantum error correction could also help to get a better picture. In [59], where the connections between path integral complexity and circuit complexity has been outlined, the authors showed that the Liouville action which in general is expressed in terms of Weyl parameter, could be considered as a general cost function in the gate counting setup. In terms of quantum error correcting codes (QECCs), the dual of this physical CFT anomaly, have been conjectured to be a logical bulk gate.…”

“…In fact, the cost function D [59], would be related to the dynamics of the string, and the way the end-points of these strings would be coupled to any boundary or D-brane. For instance, taking the L 2 -norm D 2 = λ 0 dκ IJ η IJ Y I Y J for calculating complexity would correspond to taking the Nambu-Goto action describing the dynamics of the strings.…”