^{1}

^{2}

^{3}.

We formulate Nielsen’s geometric approach to circuit complexity in the context of two-dimensional conformal field theories, where series of conformal transformations are interpreted as “unitary circuits” built from energy-momentum tensor gates. We show that the complexity functional in this setup can be written as the Polyakov action of two-dimensional gravity or, equivalently, as the geometric action on the coadjoint orbits of the Virasoro group. This way, we argue that gravity sets the rules for optimal quantum computation in conformal field theories.

Information, be it classical or quantum, is intimately related to geometry. This is manifest in the field of computational complexity, which, as shown by Nielsen

On the other hand, the Anti–de Sitter (AdS)/conformal field theories (CFT) correspondence

The motivation of our work is to bring these developments closer to each other and build bridge(s) between them by developing Nielsen’s geometric approach within 2D CFT. In particular, we focus on a subset of quantum symmetry gates (generally discussed in Ref.

The universality of our setup allows us to express the complexity action only in terms of the CFT’s central charge

Since the Polyakov and Liouville actions are directly related, our results neatly connect to the recent proposal for path integral complexity

We believe that our approach reintroduces, in a conceptually simple way, important aspects of CFTs which can set the holographic complexity discussion on a firmer ground. Crucially, all our building blocks (the energy-momentum tensors and costs) can be defined on both sides of the duality, so holographic complexity proposals should rest on algebraic or geometric relations between them. Last but not least, our considerations certainly open a new unexplored path in the field of quantum complexity in continuous field theories.

Before we begin, we would like to convey a general “take home” message from our study. The concept of circuit complexity comes with a quantum computational package such as universal sets of gates, cost functions etc. that seem hopeless to define in interacting continuous QFTs. On the other hand, motivated by

We start by reviewing the geometric approach to complexity based on Refs.

Nielsen’s attractive approach to estimate the associate costs is via metrics on the (group) manifold of unitaries. In this context, complexity becomes the length of the shortest geodesic between

More precisely, we can decompose any circuit into infinitesimal gates

Before defining

Next, we need to define the cost function

Finally, the complexity action (the length) is computed by Eq.

Now we consider the geometric approach to complexity in arguably the simplest setup relevant for holography, namely 2D CFTs (see Ref.

The Virasoro group is the central extension of the group of diffeomorphisms of the circle

To be precise, Virasoro elements are pairs

Following the previous sections, we consider a unitary CFT circuit build from the Virasoro symmetry gates

The circuit is defined as

To compute the cost function, we need to relate

As a reference state

This way, our large-

Finally, performing analogous steps for the second copy of the Virasoro group, the full CFT complexity action reads

We remark again that the cost is written in terms of Lie algebra elements. In

The connection of the 2D-CFT complexity functional

This way, we have, e.g., the one-point function

Let us elaborate more on this important connection and bring a new light that allows for generalizations toward complexity in quantum field theories.

The Polyakov action evaluated on metric

Let us point that for two Virasoro copies we get two Polyakov or complexity actions. It is well known

Finally, as shown in Ref.

As we will discuss in the last section, the fact that we can write our complexity as a geometric action gives a hint and a definite route for constructing complexity (based on “generalized symmetry gates”) in arbitrary quantum field theories.

The most promising generalization relies on the observed connection between complexity and geometric actions. Such geometric actions appear ubiquitously through the notion of generalized coherent states; see Refs.

In

Related generalizations within 2D CFTs arise when allowing instantaneous gates to include other primaries or symmetry currents (like Kac-Moody or higher spin W), or when including supersymmetry. Indeed, geometric actions for the Kac-Moody symmetry are well known and the approach of Alexeev and Shatashvili was also generalized to

We conclude that the present approach provides a novel starting point to derive gravity from complexity. In our setup, the CFT complexity corresponds to the Virasoro geometric action for each left and right sector. References

We wish to thank Hugo Camargo, Horacio Casini, Sumit Das, Jan de Boer, Nilay Kundu, Hugo Marrochio, Gautam Mandal, Rob Myers, Masamichi Miyaji, Tokiro Numasawa, Blagoje Oblak, Onkar Parrikar, Fernando Pastawski, Massimo Poratti, Guifre Vidal, and especially Joan Simon and Tadashi Takayanagi for guidance and important suggestions. Our work is supported by the Simons Foundation through the “It from Qubit” collaboration and work of P. C. was also supported by the JSPS starting Grant No. KAKENHI 17H06787. We thank the Galileo Galilei Institute for Theoretical Physics for the hospitality and the INFN for partial support during the completion of this work.