A new definition of local complexity in quantum field theories

ANTON Z. CAPRI

Neil J. Cornish

Jincheng Zhang

In conventional quantum field theory (QFT) the local observables most frequently considered are energy density, charge density, and correlation functions. These quantities encode the dynamical and kinematic properties of fields but do not directly capture the structural intricacy of the quantum state at each point in space–time. In this work we propose a new local observable, the Local Complexity Density of Quantum Fields (LCDQF), denoted ρ_C(x), which quantifies the minimal computational resources required to prepare the quantum state in an infinitesimal neighbourhood of the space–time point x from a fixed reference state. The definition of ρ_C(x) is grounded in the circuit complexity framework of quantum information theory and is constructed to be a local scalar density, invariant under spatial diffeomorphisms and Lorentz transformations. We derive a set of general properties for ρ_C(x), including positivity, monotonicity under coarse‑graining, and covariance under field redefinitions. In addition, we establish a formal relationship between ρ_C(x) and the energy density ρ_E(x) by showing that, for a wide class of free and weakly interacting fields, the two densities are linked through a differential operator that captures the local scaling of the field modes. Explicit calculations for the free scalar, the φ^4 theory, and Maxwell electrodynamics illustrate how ρ_C(x) behaves in different regimes and highlight its sensitivity to ultraviolet cutoffs, renormalisation group flows, and topological excitations. We discuss the implications of LCDQF for the AdS/CFT correspondence, arguing that the local complexity density may serve as a bridge between bulk geometric quantities such as the volume of extremal slices and boundary field-theoretic complexity. Finally, we outline prospective avenues for experimental probing of ρ_C(x) via quantum simulators that can implement local state preparation protocols, suggesting that LCDQF may become a measurable characteristic in near‑term quantum devices. This new local observable opens a novel window into the internal structure of quantum fields and promises to enrich both theoretical and experimental studies of quantum many‑body systems.

Гарий Владимирович Ефимов

Andrew J. Hanson, Gerardo Ortíz, Amr Sabry +1 more

We explore finite-field frameworks for quantum theory and quantum computation. The simplest theory, defined over unrestricted finite fields, is unnaturally strong. A second framework employs only finite fields with no solution to x^2+1=0, and thus permits an elegant complex representation of the extended field by adjoining i=\sqrt{-1}. Quantum theories over these fields recover much of the structure of conventional quantum theory except for the condition that vanishing inner products arise only from null states; unnaturally strong computational power may still occur. Finally, we are led to consider one more framework, with further restrictions on the finite fields, that recovers a local transitive order and a locally-consistent notion of inner product with a new notion of cardinal probability. In this framework, conventional quantum mechanics and quantum computation emerge locally (though not globally) as the size of the underlying field increases. Interestingly, the framework allows one to choose separate finite fields for system description and for measurement: the size of the first field quantifies the resources needed to describe the system and the size of the second quantifies the resources used by the observer. This resource-based perspective potentially provides insights into quantitative measures for actual computational power, the complexity of quantum system definition and evolution, and the independent question of the cost of the measurement process.

Jincheng Zhang

Quantum-Inspired Constraint Field (QICF) is a novel theoretical framework designed to address the challenges of understanding and predicting the evolution of dynamical systems, particularly those exhibiting complex and stable configurations. This work proposes a method based on the application of quantum field theory concepts, specifically entanglement and tunneling, to define constraints within dynamical systems. The core mechanism involves utilizing these concepts to create a "quantum-inspired" set of constraints, allowing for the prediction of stable and complex configurations – a significant departure from traditional approaches. This framework holds the potential to unlock new insights into control theory and provide a powerful tool for designing robust and predictable systems. The proposed method offers a fundamentally different perspective on constraint definition, potentially leading to significant advances in the modeling and control of complex phenomena.