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On leave from National Centre for Nuclear Research, Pasteura 7, 02-093 Warsaw, Poland.

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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.

One of the most interesting developments of the past decade is the confluence of ideas from quantum gravity, quantum field theory (QFT), and quantum information science. The impact of this research trend is perhaps most striking in holography (

These so-called holographic complexity proposals have not only contributed to ongoing research on the emergence of dynamical spacetimes from microscopic quantum mechanical degrees of freedom, but have also motivated the exploration of complexity in QFTs. The concept of complexity originates in quantum computing and is based on approximating a given (typically unitary) operator in terms of some fundamental building blocks. In the context of quantum circuit design, the latter correspond to the gates used in constructing circuits that realize the given operation, the complexity of which is defined as the minimum-length circuit that achieves this goal. Note that, by fixing a reference state, this same framework allows one to speak of the complexity of states in the theory by optimizing over all circuits (operators) that produce the desired target state. From this perspective, the complexity of operators is the more elementary notion, and it is this question on which we will focus in the present Letter.

In practice, performing the optimization over all possible constructions is a difficult task. To surmount this problem in the original context of quantum circuit design, Nielsen

A second main approach based on so-called path integral optimization has been proposed in

The aim of the present Letter is to explicitly demonstrate that path integral optimization in its best-explored setting of two-dimensional conformal field theories (CFTs) may be given a precise formulation in terms of circuit complexity.

Let us begin by briefly reviewing the geometric approach introduced in Refs.

We now turn to the path integral approach, in preparation to reframe the latter in the language of circuit complexity. We focus here on a specific example pioneered in Ref.

The matrix elements of

Euclidean strip over which we perform the path integral

In the present Letter, we will demonstrate that, up to an unimportant overall normalization,

We now wish to interpret the Euclidean path integral on the Weyl-rescaled geometry as a circuit in the manner of Eq.

The relevant statement is the following. If one considers a Euclidean path integral for a

We then observe that Eq.

To proceed, we must identify coordinate transformations from the

Before we proceed to specifying the cost function, it will be very convenient to introduce an angle

In counting gates, our point of departure is regarding Eq.

As a closing remark, we note that for

This Letter can be regarded as an attempt to define circuit complexity in QFT as a direct functional of sources appearing in the underlying Euclidean path integral. Furthermore, as opposed to the earlier works on gate complexity in QFT, we explicitly use only local gates. In this framework, we were able to show that the cost function appearing in the path integral optimization approach, namely, the Liouville action given by Eq.

One aspect to emphasize is that the covariance with respect to an underlying metric is lost for a genuine cost function, since these are defined with respect to some time foliation. So far we were only able to recover covariance approximately to next-to-leading order in the UV cutoff in the relevant expansion for obtaining the Liouville action, cf. Eq.

Another important item that we want to stress in the context of Liouville is that our cost functions are not sensitive to the central charge of the CFT. Of course, one has the freedom to introduce the central charge in the overall normalization (or in penalty factors), but the arguments above do not provide any physical justification for doing so, and consequently, we have only considered cost functions defined up to some overall prefactor.

There are two further implications of this Letter we wish to highlight. First, a proof of the holographic complexity proposals should occur through the equality of bulk and boundary Euclidean partition functions, and the interface between the two sides is governed by the boundary (QFT) sources. While there have already been several attempts in this general direction

Regarding direct extensions of this work, it would be very interesting to find a cost function that is fully covariant to all orders in

We wish to thank E. Brehm, P. Caputa, B. Czech, M. Flory, B. Freivogel, L. Hackl, D. Hofman, L. Hadasz, R. Janik, J.-L. Lehners, T. Osborne, V. Schomerus, V. Svensson, S. Theisen, T. Takayanagi, and H. Wang for helpful discussions and correspondence and, especially, J. Hung for her key insight inspired by DBI actions that allowed us to derive the Liouville action as an approximation to a cost function. We are also very grateful to D. Das and S. He for their input during earlier stages of the project. Our special thanks go also to S. Singh, M. Walter, and F. Witteveen, as well as J. de Boer, S. Chapman, and I. Reyes for their feedback and collaboration with some of us on related topics. The Gravity, Quantum Fields, and Information group at AEI is generously supported by the Alexander von Humboldt Foundation and the Federal Ministry for Education and Research through the Sofja Kovalevskaja Award. H. C. is partially supported by the Konrad-Adenauer-Stiftung through their Sponsorship Program for Foreign Students.