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arXiv:2004.03619 · Journal of High Energy Physics

Complexity measures from geometric actions onVirasoro and Kac-Moody orbits

Johanna Erdmenger B, Marius Gerbershagen, Anna-Lena Weigel

Published Nov 1, 2020• 86 citations
arXivHTMLPDF

Abstract

Abstract We further advance the study of the notion of computational complexity for 2d CFTs based on a gate set built out of conformal symmetry transformations. Previously, it was shown that by choosing a suitable cost function, the resulting complexity functional is equivalent to geometric (group) actions on coadjoint orbits of the Virasoro group, up to a term that originates from the central extension. We show that this term can be recovered by modifying the cost function, making the equivalence exact. Moreover, we generalize our approach to Kac-Moody symmetry groups, finding again an exact equivalence between complexity functionals and geometric actions. We then determine the optimal circuits for these complexity measures and calculate the corresponding costs for several examples of optimal transformations. In the Virasoro case, we find that for all choices of reference state except for the vacuum state, the complexity only measures the cost associated to phase changes, while assigning zero cost to the non-phase changing part of the transformation. For Kac-Moody groups in contrast, there do exist non-trivial optimal transformations beyond phase changes that contribute to the complexity, yielding a finite gauge invariant result. Moreover, we also show that our Virasoro complexity proposal is equivalent to the on-shell value of the Liouville action, which is a complexity functional proposed in the context of path integral optimization. This equivalence provides an interpretation for the path integral optimization proposal in terms of a gate set and reference state. Finally, we further develop a new proposal for a complexity definition for the Virasoro group that measures the cost associated to non-trivial transformations beyond phase changes. This proposal is based on a cost function given by a metric on the Lie group of conformal transformations. The minimization of the corresponding complexity functional is achieved using the Euler-Arnold method yielding the Korteweg-de Vries equation as equation of motion.

DOI
10.1007/jhep11(2020)003

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Abstract We further advance the study of the notion of computational complexity for 2d CFTs based on a gate set built out of conformal symmetry transformations. Previously, it was shown that by choosing a suitable cost function, the resulting complexity functional is equivalent to geometric (group) actions on coadjoint orbits of the Virasoro group, up to a term that originates from the central extension. We show that this term can be recovered by modifying the cost function, making the equivalence exact. Moreover, we generalize our approach to Kac-Moody symmetry groups, finding again an exact equivalence between complexity functionals and geometric actions. We then determine the optimal circuits for these complexity measures and calculate the corresponding costs for several examples of optimal transformations. In the Virasoro case, we find that for all choices of reference state except for the vacuum state, the complexity only measures the cost associated to phase changes, while assigning zero cost to the non-phase changing part of the transformation. For Kac-Moody groups in contrast, there do exist non-trivial optimal transformations beyond phase changes that contribute to the complexity, yielding a finite gauge invariant result. Moreover, we also show that our Virasoro complexity proposal is equivalent to the on-shell value of the Liouville action, which is a complexity functional proposed in the context of path integral optimization. This equivalence provides an interpretation for the path integral optimization proposal in terms of a gate set and reference state. Finally, we further develop a new proposal for a complexity definition for the Virasoro group that measures the cost associated to non-trivial transformations beyond phase changes. This proposal is based on a cost function given by a metric on the Lie group of conformal transformations. The minimization of the corresponding complexity functional is achieved using the Euler-Arnold method yielding the Korteweg-de Vries equation as equation of motion.
Recently, the notion of circuit complexity defined in symmetry group manifolds has been related to geometric actions which generally arise in the coadjoint orbit method in representation theory and play an important role in geometric quantization. On the other hand, it is known that there exists a precise relation between geometric actions and Berry phases defined in group representations. Motivated by these connections, we elaborate on a relation between circuit complexity and the group theoretic Berry phase. As the simplest setup relevant for holography, we discuss the case of two dimensional conformal field theories. In the large central charge limit, we identify the computational cost function with the Berry connection in the unitary representation of the Virasoro group. We then use the latter identification to express the Berry phase in terms of the Virasoro circuit complexity. The former can be seen as the holonomy of the Berry connection along the path in the group manifold which defines the protocol. In addition, we derive a proportionality relation between Virasoro circuit complexity and the logarithm of the inner product between a particularly chosen reference state and the prepared target state. In this sense, the logarithmic formula turns out to be approximating the complexity up to some additive constant if the building blocks of the circuit are taken to be the underlying symmetry gates. Predictions based on this formula have recently been shown to coincide with the holographic complexity proposals and the path integral optimization procedure. The found connections may therefore help to better understand such coincidences. We also discuss that our findings, put together with earlier observations, may suggest a connection between the Virasoro Berry phase and the complexity measure in the path integral optimization proposal.
A bstract Quantum complexity of conformal field theory (CFT) states has recently gained significant attention, both as a diagnostic tool in condensed matter systems and in connection with holographic observables probing black hole interiors. Previous studies have primarily focused on cases where all generators of the conformal group contribute equally to the cost of building a circuit. In this work, we present a general framework for studying the complexity of circuits in generic Lie groups, where penalty factors assign relative weights to different generators. Our approach constructs a metric on the coset space of quantum states, induced from a (pseudo-)Riemannian norm on the space of unitary circuits. The geodesics of this metric are interpreted as optimal circuits. The method builds on the formalism of (pseudo-)Riemannian submersions and connects naturally to other prescriptions in the literature, including cost function minimization along stabilizer directions and constructions based on coadjoint orbits. As a concrete application, we compute state complexity for states in one- and two-dimensional CFTs. For specific choices of penalty factors, our prescription yields a positive-definite metric with a viable interpretation as complexity; in other cases, the resulting metric is indefinite. In the viable regime, we derive analytic results when a specific penalty factor is turned off, develop perturbative expansions for small values of the penalty factors, and provide numerical results in the general case. We comment on the relation of our measure of complexity to holography.
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.
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.
Abstract We study the conditions under which, given a generic quantum system, complexity metrics provide actual lower bounds to the circuit complexity associated to a set of quantum gates. Inhomogeneous cost functions — many examples of which have been recently proposed in the literature — are ruled out by our analysis. Such measures are shown to be unrelated to circuit complexity in general and to produce severe violations of Lloyd’s bound in simple situations. Among the metrics which do provide lower bounds, the idea is to select those which produce the tightest possible ones. This establishes a hierarchy of cost functions and considerably reduces the list of candidate complexity measures. In particular, the criterion suggests a canonical way of dealing with penalties, consisting in assigning infinite costs to directions not belonging to the gate set. We discuss how this can be implemented through the use of Lagrange multipliers. We argue that one of the surviving cost functions defines a particularly canonical notion in the sense that: i) it straightforwardly follows from the standard Hermitian metric in Hilbert space; ii) its associated complexity functional is closely related to Kirillov’s coadjoint orbit action, providing an explicit realization of the “complexity equals action” idea; iii) it arises from a Hamilton-Jacobi analysis of the “quantum action” describing quantum dynamics in the phase space canonically associated to every Hilbert space. Finally, we explain how these structures provide a natural framework for characterizing chaos in classical and quantum systems on an equal footing, find the minimal geodesic connecting two nearby trajectories, and describe how complexity measures are sensitive to Lyapunov exponents.
We study the conditions under which, given a generic quantum system, complexity metrics provide actual lower bounds to the circuit complexity associated to a set of quantum gates. Inhomogeneous cost functions ---many examples of which have been recently proposed in the literature--- are ruled out by our analysis. Such measures are shown to be unrelated to circuit complexity in general and to produce severe violations of Lloyd's bound in simple situations. Among the metrics which do provide lower bounds, the idea is to select those which produce the tightest possible ones. This establishes a hierarchy of cost functions and considerably reduces the list of candidate complexity measures. In particular, the criterion suggests a canonical way of dealing with penalties, consisting in assigning infinite costs to directions not belonging to the gate set. We discuss how this can be implemented through the use of Lagrange multipliers. We argue that one of the surviving cost functions defines a particularly canonical notion in the sense that: i) it straightforwardly follows from the standard Hermitian metric in Hilbert space; ii) its associated complexity functional is closely related to Kirillov's coadjoint orbit action, providing an explicit realization of the ``complexity equals action'' idea; iii) it arises from a Hamilton-Jacobi analysis of the ``quantum action'' describing quantum dynamics in the phase space canonically associated to every Hilbert space. Finally, we explain how these structures provide a natural framework for characterizing chaos in classical and quantum systems on an equal footing, find the minimal geodesic connecting two nearby trajectories, and describe how complexity measures are sensitive to Lyapunov exponents.