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On leave from National Centre for Nuclear Research, 00-681 Warsaw, Poland.

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We investigate notions of complexity of states in continuous many-body quantum systems. We focus on Gaussian states which include ground states of free quantum field theories and their approximations encountered in the context of the continuous version of the multiscale entanglement renormalization ansatz. Our proposal for quantifying state complexity is based on the Fubini-Study metric. It leads to counting the number of applications of each gate (infinitesimal generator) in the transformation, subject to a state-dependent metric. We minimize the defined complexity with respect to momentum-preserving quadratic generators which form

Applications of quantum information concepts to high-energy physics and gravity have recently led to many far-reaching developments. In particular, it has become apparent that special properties of entanglement in holographic

Quantum state complexity originates from the field of quantum computations, which are usually modeled in a finite Hilbert space as the application of a sequence of gates chosen from a discrete set. In this context, the complexity of a unitary

In the context of holography, the organization of discrete tensor networks (seen as a quantum circuit

The main challenges in providing a workable definition of complexity in the continuum are related to choosing (a) a reference state

While the FS prescription is quite general, our choices for (a), (b), and (d) render the necessary calculations tractable. Some of these choices are inspired by the continuous MERA (cMERA) approach to free QFTs

As a first step, we consider the two-mode squeezing operator for each pair of opposite momenta

While a full literature review is outside the scope of this Letter, there is a substantial body of important recent developments which include, e.g., Refs.

We are interested in considering unitary operators

If the path

We consider a theory of free relativistic bosons in (

A general translation-invariant

The ground state

A natural choice for a reference state

As our target state, we consider the approximate ground state

The target states

We start by evaluating our proposed complexity under the assumption that we allow for a single generator per pair of momenta

Suppose, on the other hand, that

Here, we extend our minimization to a larger set of generators

We will see that the manifold of states generated by each

Evaluating the FS line element

The Poincaré disk, parametrized by real (horizontal) and imaginary (vertical) components of

The geodesic connecting

There are two proposals for the gravity dual of complexity in terms of maximal codimension-1 volumes (CV

We proposed a definition of state complexity in QFTs, independent from a notion of unitary complexity. This measure is derived from the FS metric by restricting to directions, in the space of states, generated by exponentiating allowed generators

We could verify using our methods that cMERA circuits are optimal in the

We worked in momentum space and restricted the generators to be quadratic. In position space, our generators are bilocal, which suggests an analogy to the two-qubit operations of traditional quantum circuits. However, our gates are spread in position space, and it would be interesting to explore the implications of working with local gates. Future directions include evaluating the complexity for fermionic systems and studying the time evolution of thermofield double states. Finally, it would be interesting to understand what universal data can be extracted from complexity, whether complexity in QFTs can serve as an order parameter, and if it plays a role in the context of RG flows.

We express first our special gratitude to R. C. Myers for numerous illuminating discussions and suggestions that helped shape the ideas and results of this Letter and for sharing with us his preliminary results with R. Jefferson on defining complexity in QFTs using Nielsen’s approach. We are also particularly thankful to R. Jefferson for many illuminating discussions and for providing comments on the Supplemental Material comparing the two works, and to J. Eisert for providing numerous comments on this work. In addition, we thank G. Verdon-Akzam, J. de Boer, P. Caputa, M. Fleury, A. Franco, K. Hashimoto, Q. Hu, R. Janik, J. Jottar, T. Osborne, G. Policastro, K. Rejzner, D. Sarkar, V. Scholz, M. Spalinski, T. Takayanagi, K. Temme, J. Teschner, G. Vidal, F. Verstraete, and P. Witaszczyk for valuable comments and discussions. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation. S. C. acknowledges additional support from an Israeli Women in Science Fellowship from the Israeli Council of Higher Education. The research of M. P. H. is supported by the Alexander von Humboldt Foundation and the Federal Ministry for Education and Research through the Sofja Kovalevskaja Award. M. P. H. is also grateful to Perimeter Institute, ETH, and the University of Amsterdam for stimulating hospitality during the completion of this project and to Kyoto University where this work was presented for the first time during the workshop