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arXiv:1809.06678 · Journal of High Energy Physics

More on complexity of operators in quantum field theory

Run-Qiu Yang, Yu-Sen An, Chao Niu, Cheng-Yong Zhang, Keun-Young Kim

Published Mar 1, 2019• 47 citations
arXivHTMLPDF

Abstract

Abstract Recently it has been shown that the complexity of SU(n) operator is determined by the geodesic length in a bi-invariant Finsler geometry, which is constrained by some symmetries of quantum field theory. It is based on three axioms and one assumption regarding the complexity in continuous systems. By relaxing one axiom and an assumption, we find that the complexity formula is naturally generalized to the Schatten p-norm type. We also clarify the relation between our complexity and other works. First, we show that our results in a bi-invariant geometry are consistent with the ones in a right-invariant geometry such as k-local geometry. Here, a careful analysis of the sectional curvature is crucial. Second, we show that our complexity can concretely realize the conjectured pattern of the time-evolution of the complexity: the linear growth up to saturation time. The saturation time can be estimated by the relation between the topology and curvature of SU(n) groups.

DOI
10.1007/jhep03(2019)161

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Abstract Recently it has been shown that the complexity of SU(n) operator is determined by the geodesic length in a bi-invariant Finsler geometry, which is constrained by some symmetries of quantum field theory. It is based on three axioms and one assumption regarding the complexity in continuous systems. By relaxing one axiom and an assumption, we find that the complexity formula is naturally generalized to the Schatten p-norm type. We also clarify the relation between our complexity and other works. First, we show that our results in a bi-invariant geometry are consistent with the ones in a right-invariant geometry such as k-local geometry. Here, a careful analysis of the sectional curvature is crucial. Second, we show that our complexity can concretely realize the conjectured pattern of the time-evolution of the complexity: the linear growth up to saturation time. The saturation time can be estimated by the relation between the topology and curvature of SU(n) groups.
Based on general and minimal properties of the discrete circuit complexity, we define the complexity in continuous systems in a geometrical way. We first show that the Finsler metric naturally emerges in the geometry of the complexity in continuous systems. Due to fundamental symmetries of quantum field theories, the Finsler metric is more constrained and consequently, the complexity of SU( n ) operators is uniquely determined as a length of a geodesic in the Finsler geometry. Our Finsler metric is bi-invariant contrary to the right-invariance of discrete qubit systems. We clarify why the bi-invariance is relevant in quantum field theoretic systems. After comparing our results with discrete qubit systems we show most results in k -local right-invariant metric can also appear in our framework. Based on the bi-invariance of our formalism, we propose a new interpretation for the Schrödinger’s equation in isolated systems – the quantum state evolves by the process of minimizing “computational cost”.
Abstract We investigate the time evolution of the complexity of the operator by the Sachdev-Ye-Kitaev (SYK) model with N Majorana fermions. We follow Nielsen’s idea of complexity geometry and geodesics thereof. We show that it is possible that the bi- invariant complexity geometry can exhibit the conjectured time evolution of the complexity in chaotic systems: i) linear growth until t ∼ e N , ii) saturation and small fluctuations after then. We also show that the Lloyd’s bound is realized in this model. Interestingly, these characteristic features appear only if the complexity geometry is the most natural “non-Riemannian” Finsler geometry. This serves as a concrete example showing that the bi-invariant complexity may be a competitive candidate for the complexity in quantum mechanics/field theory (QM/QFT). We provide another argument showing a naturalness of bi-invariant complexity in QM/QFT. That is that the bi-invariance naturally implies the equivalence of the right-invariant complexity and left-invariant complexity, either of which may correspond to the complexity of a given operator. Without bi-invariance, one needs to answer why only right (left) invariant complexity corresponds to the “complexity”, instead of only left (right) invariant complexity.
Motivated by the recent rapid development of complexity theory applied to quantum mechanical processes we present the complete derivation of Nielsen's complexity of unitaries belonging to the representations of oscillator group. Our approach is based on the observation that the whole problem refers to the structure of the underlying group. The questions concerning the complexity of particular unitaries are solved by lifting the abstract structure to the operator level by considering the relevant unitary representation. For the class of right-invariant metrics obeying natural invariance condition we solve the geodesic equations on oscillator group. The solution is given explicitly in terms of elementary functions. Imposing the boundary conditions yield a transcendental equation and the length of the geodesic is given in terms of the solutions to the latter. Since the unitary irreducible representations of oscillator group are classified this allows us to compute, at least in principle, the complexity of any unitary operator belonging to the representation.
We present a collection of recent results concerning quantum information theory applied to quantum gravity. We first study the entanglement structure of Euclidean path integral states in SU(2) Chern-Simons theory, where we elucidate a connection between topological entanglement and quantum mechanical entanglement. We prove that the topology of certain three-manifolds controls the entanglement structure of the resulting quantum state, and conjecture a more general relationship for arbitrary three-manifolds. We then analyze the quantum circuit complexity of the time evolution operator in the Sachdev-Ye-Kitaev model, a theory of near-extremal black hole microstates. We find that this complexity grows linearly for a time exponential in the entropy, modulo a caveat concerning global obstructions to the growth of the distance function along geodesics, which we do not rule out. This constitutes a partial proof of Susskind’s conjecture about the complexity growth of black holes. Finally, we address the black hole information paradox in three dimensions by considering a toy model of black hole microstates in the form of an end-of-the-world brane. This brane carries a quantum theory which is itself holographic, and we compute entanglement entropies in the glued dual geometry by using a Ryu-Takayanagi formula where the minimal entropy surface can pass through the gluing surface.
We develop a geometric approach to operator growth and Krylov complexity in many-body quantum systems governed by symmetries. We start by showing a direct link between a unitary evolution with the Liouvillian and the displacement operator of appropriate generalized coherent states. This connection maps operator growth to a purely classical motion in phase space. The phase spaces are endowed with a natural information metric. We show that, in this geometry, operator growth is represented by geodesics, and Krylov complexity is proportional to a volume. This geometric perspective also provides two novel avenues toward computation of Lanczos coefficients, and it sheds new light on the origin of their maximal growth. We describe the general idea and analyze it in explicit examples, among which we reproduce known results from the Sachdev-Ye-Kitaev model, derive operator growth based on SU(2) and Heisenberg-Weyl symmetries, and generalize the discussion to conformal field theories. Finally, we use techniques from quantum optics to study operator evolution with quantum information tools such as entanglement and Renyi entropies, negativity, fidelity, relative entropy, and capacity of entanglement.
As time passes, once simple quantum states tend to become more complex. For strongly coupled kkk-local Hamiltonians, this growth of computational complexity has been conjectured to follow a distinctive and universal pattern. In this paper we show that the same pattern is exhibited by a much simpler system---classical geodesics on a compact two-dimensional geometry of uniform negative curvature. This striking parallel persists whether the system is allowed to evolve naturally or is perturbed from the outside.