Quantum Computation as Gravity

Quantum state complexity meets many-body scars

Scar eigenstates in a many-body system refers to a small subset of non-thermal finite energy density eigenstates embedded into an otherwise thermal spectrum. This novel non-thermal behaviour has been seen in recent experiments simulating a one-dimensional PXP model with a kinetically-constrained local Hilbert space realised by a chain of Rydberg atoms. We probe these small sets of special eigenstates starting from particular initial states by computing the spread complexity associated to time evolution of the PXP hamiltonian. Since the scar subspace in this model is embedded only loosely, the scar states form a weakly broken representation of the Lie algebra. We demonstrate why a careful usage of the forward scattering approximation (FSA), instead of any other method, is required to extract the most appropriate set of Lanczos coefficients in this case as the consequence of this approximate symmetry. Only such a method leads to a well defined notion of a closed Krylov subspace and consequently, that of spread complexity. We show this using three separate initial states, namely|Z2⟩,|Z3⟩and the vacuum state, due to the disparate classes of scar states hosted by these sectors. We also discuss systematic methods of remedying the imperfections in the FSA setup stemming from these approximate symmetries.

Circuit Complexity in Z2 EEFT

Motivated by recent studies of circuit complexity in weakly interacting scalar field theory, we explore the computation of circuit complexity in Z2 Even Effective Field Theories (Z2 EEFTs). We consider a massive free field theory with higher-order Wilsonian operators such as ϕ4, ϕ6, and ϕ8. To facilitate our computation, we regularize the theory by putting it on a lattice. First, we consider a simple case of two oscillators and later generalize the results to N oscillators. This study was carried out for nearly Gaussian states. In our computation, the reference state is an approximately Gaussian unentangled state, and the corresponding target state, calculated from our theory, is an approximately Gaussian entangled state. We compute the complexity using the geometric approach developed by Nielsen, parameterizing the path-ordered unitary transformation and minimizing the geodesic in the space of unitaries. The contribution of higher-order operators to the circuit complexity in our theory is discussed. We also explore the dependency of complexity on other parameters in our theory for various cases.

Complexity for Conformal Field Theories in General Dimensions

We study circuit complexity for conformal field theory states in an arbitrary number of dimensions. Our circuits start from a primary state and move along a unitary representation of the Lorentzian conformal group. Different choices of distance functions can be understood in terms of the geometry of coadjoint orbits of the conformal group. We explicitly relate our circuits to timelike geodesics in anti-de Sitter space and the complexity metric to distances between these geodesics. We extend our method to circuits in other symmetry groups using a group theoretic generalization of the notion of coherent states.

Fundamental Physics and Computation: The Computer-Theoretic Framework

The central goal of this manuscript is to survey the relationships between fundamental physics and computer science. We begin by providing a short historical review of how different concepts of computer science have entered the field of fundamental physics, highlighting the claim that the universe is a computer. Following the review, we explain why computational concepts have been embraced to interpret and describe physical phenomena. We then discuss seven arguments against the claim that the universe is a computational system and show that those arguments are wrong because of a misunderstanding of the extension of the concept of computation. Afterwards, we address a proposal to solve Hempel’s dilemma using the computability theory but conclude that it is incorrect. After that, we discuss the relationship between the proposals that the universe is a computational system and that our minds are a simulation. Analysing these issues leads us to proposing a new physical principle, called the principle of computability, which claims that the universe is a computational system (not restricted to digital computers) and that computational power and the computational complexity hierarchy are two fundamental physical constants. On the basis of this new principle, a scientific paradigm emerges to develop fundamental theories of physics: the computer-theoretic framework (CTF). The CTF brings to light different ideas already implicit in the work of several researchers and provides a new view on the universe based on computer theoretic concepts that expands the current view. We address different issues regarding the development of fundamental theories of physics in the new paradigm. Additionally, we discuss how the CTF brings new perspectives to different issues, such as the unreasonable effectiveness of mathematics and the foundations of cognitive science.

Lorentzian Threads as Gatelines and Holographic Complexity

The continuous min flow-max cut principle is used to reformulate the "complexity=volume" conjecture using Lorentzian flows-divergenceless norm-bounded timelike vector fields whose minimum flux through a boundary subregion is equal to the volume of the homologous maximal bulk Cauchy slice. The nesting property is used to show the rate of complexity is bounded below by "conditional complexity," describing a multistep optimization with intermediate and final target states. Conceptually, discretized Lorentzian flows are interpreted in terms of threads or gatelines such that complexity is equal to the minimum number of gatelines used to prepare a conformal field theory (CFT) state by an optimal tensor network (TN) discretizing the state. We propose a refined measure of complexity, capturing the role of suboptimal TNs, as an ensemble average. The bulk symplectic potential provides a "canonical" thread configuration characterizing perturbations around arbitrary CFT states. Its consistency requires the bulk to obey linearized Einstein's equations, which are shown to be equivalent to the holographic first law of complexity, thereby advocating a notion of "spacetime complexity."

Circuit Complexity from Cosmological Islands

Recently, in various theoretical works, path-breaking progress has been made in recovering the well-known page curve of an evaporating black hole with quantum extremal islands, proposed to solve the long-standing black hole information loss problem related to the unitarity issue. Motivated by this concept, in this paper, we study cosmological circuit complexity in the presence (or absence) of quantum extremal islands in negative (or positive) cosmological constant with radiation in the background of Friedmann-Lemai^tre-Robertson-Walker (FLRW) space-time, i.e., the presence and absence of islands in anti de Sitter and the de Sitter space-time having SO(2, 3) and SO(1, 4) isometries, respectively. Without using any explicit details of any gravity model, we study the behavior of the circuit complexity function with respect to the dynamical cosmological solution for the scale factors for the above mentioned two situations in FLRW space-time using squeezed state formalism. By studying the cosmological circuit complexity, out-of-time ordered correlators, and entanglement entropy of the modes of the squeezed state, in different parameter space, we conclude the non-universality of these measures. Their remarkably different features in the different parameter space suggests their dependence on the parameters of the model under consideration.

Conformal field theory complexity from Euler-Arnold equations

Defining complexity in quantum field theory is a difficult task, and the main challenge concerns going beyond free models and associated Gaussian states and operations. One take on this issue is to consider conformal field theories in 1+1 dimensions and our work is a comprehensive study of state and operator complexity in the universal sector of their energy-momentum tensor. The unifying conceptual ideas are Euler-Arnold equations and their integro-differential generalization, which guarantee well-posedness of the optimization problem between two generic states or transformations of interest. The present work provides an in-depth discussion of the results reported in arXiv:2005.02415 and techniques used in their derivation. Among the most important topics we cover are usage of differential regularization, solution of the integro-differential equation describing Fubini-Study state complexity and probing the underlying geometry.

Circuit complexity across a topological phase transition

We use Nielsen's geometric approach to quantify the circuit complexity in a one-dimensional Kitaev chain across a topological phase transition. We find that the circuit complexities of both the ground states and nonequilibrium steady states of the Kitaev model exhibit nonanalytical behaviors at the critical points, and thus can be used to detect both equilibrium and dynamical topological phase transitions. Moreover, we show that the locality property of the real-space optimal Hamiltonian connecting two different ground states depends crucially on whether the two states belong to the same or different phases. This provides a concrete example of classifying different gapped phases using Nielsen's circuit complexity. We further generalize our results to a Kitaev chain with long-range pairing, and we discuss generalizations to higher dimensions. Our result opens up an avenue for using circuit complexity as a tool to understand quantum many-body systems.

First Law of Holographic Complexity

We investigate the variation of holographic complexity for two nearby target states. Based on Nielsen's geometric approach, we find the variation only depends on the end point of the optimal trajectory, a result which we designate the first law of complexity. As an example, we examine the complexity=action conjecture when the anti-de Sitter vacuum is perturbed by a scalar field excitation, which corresponds to a coherent state. Remarkably, the gravitational contributions completely cancel and the final variation reduces to a boundary term coming entirely from the scalar field action. Hence, the null boundary of Wheeler-DeWitt patch appears to act like the "end of the quantum circuit".

Path Integral Optimization as Circuit Complexity

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.