This work proposes quantum circuit complexity—the minimal number of elementary operations needed to implement a quantum transformation—be established as a legitimate physical observable. We prove that circuit complexity satisfies all requirements for physical observables, including self-adjointness, gauge invariance, and a consistent measurement theory with well-defined uncertainty relations. We develop complete protocols for measuring complexity in quantum systems and demonstrate its connections to gauge theory and quantum gravity. Our results suggest that computational requirements may constitute physical laws as fundamental as energy conservation. This framework grants insights into the relationship between quantum information, gravity, and the emergence of spacetime geometry while offering practical methods for experimental verification. Our results indicate that the physical universe may be governed by both energetic and computational constraints, with profound implications for our understanding of fundamental physics.
This work proposes quantum circuit complexity—the minimal number of elementary operations needed to implement a quantum transformation—be established as a legitimate physical observable. We prove that circuit complexity satisfies all requirements for physical observables, including self-adjointness, gauge invariance, and a consistent measurement theory with well-defined uncertainty relations. We develop complete protocols for measuring complexity in quantum systems and demonstrate its connections to gauge theory and quantum gravity. Our results suggest that computational requirements may constitute physical laws as fundamental as energy conservation. This framework grants insights into the relationship between quantum information, gravity, and the emergence of spacetime geometry while offering practical methods for experimental verification. Our results indicate that the physical universe may be governed by both energetic and computational constraints, with profound implications for our understanding of fundamental physics.
We establish quantum circuit complexity as a fundamental physical observable and prove that it satisfies an uncertainty relation with energy, analogous to Heisenberg’s canonical uncertainty principle. Through rigorous operator theory, we demonstrate that the complexity operator meets all mathematical requirements for a legitimate quantum observable, including self-adjointness, gauge invariance, and proper spectral decomposition. This enables us to derive a fundamental bound that constrains how quickly complexity can increase in physical systems given available energy resources. We provide complete mathematical proofs of these results and demonstrate their far-reaching implications across quantum computation, black hole physics, and computational complexity theory. In particular, we show that this uncertainty relation imposes fundamental speed limits on quantum circuits, explains maximal complexity growth in black holes, and suggests that physical constraints may enforce an effective separation between complexity classes independent of their mathematical relationships. We outline explicit experimental protocols for testing these predictions using current quantum computing platforms and discuss the profound implications for our understanding of the relationship between computational complexity and fundamental physics. Our results indicate that computational requirements may be as basic to physics as energy conservation, suggesting a deep connection between the structure of physical law and fundamental limits on computation.
Observing the physical world is a foundational pursuit in science. In the quantum realm, however, observation necessitates a fundamental quantum-to-classical conversion: destructive measurements irreversibly project quantum states into classical data, inevitably incurring a loss of information. What physical principles govern this information loss, and how can we construct optimal measurements to maximize the readout? Here, we address these questions by establishing an intrinsic relationship between readout capability--quantified by the ratio of accessible classical Fisher information to the total quantum Fisher information (QFI), and measurement complexity--defined as the quantum circuit depth required prior to projection. Remarkably, we uncover a sudden emergence of observability: a sharp hidden-to-visible transition driven entirely by measurement complexity. We rigorously prove that below critical depth thresholds--Θ((logn)1/δ) for δ-dimensional architectures and Θ(loglogn) for all-to-all connectivity--readout capability decays exponentially with system size n, rendering the quantum information fundamentally inaccessible. Surprisingly, immediately above this threshold, the system enters a visible regime: we demonstrate that randomized measurements universally recover a constant fraction of the QFI using approximate unitary 3-designs, for which we explicitly develop optimal-depth circuit constructions tailored to finite-dimensional architectures. By unveiling the fundamental scaling laws and transitions that govern quantum observation, our results delineate definitive resource boundaries for quantum learning, state certification, and quantum metrology.
Quantum circuit complexity is a fundamental concept whose importance permeates quantum information, computation, many-body physics and high-energy physics. While extensively studied in closed systems, its characterization and behaviors in the widely important setting where the system is embedded within a larger one -- encompassing measurement-assisted state preparation -- lack systematic understanding. We introduce the notion of embedded complexity that characterizes the complexity of projected states and measurement operators in this general setting incorporating auxiliary systems and measurements. For random circuits and certain strongly interacting time-independent Hamiltonian dynamics, we show that the embedded complexity is lower-bounded by the circuit volume -- the total number of gates acting on both the subsystem and its complement. This strengthens the complexity linear growth theorems, enriches the understanding of deep thermalization, and indicates that measurement-assisted methods generically cannot yield significant advantages in state preparation cost, contrary to expectations. We further demonstrate a spacetime conversion of certain circuit models that concentrates circuit volume onto a subsystem, and showcase applications for random circuit sampling and shadow tomography. Our theory establishes a unified framework for space and time aspects of quantum circuit complexity, yielding profound new insights and applications across quantum information and physics.
Any computation is facilitated by some physical process, and the observable quantities of any physical process can be viewed as a computation. These close ties suggest that the study of what universal computers are capable of may lead to additional insight about the physical universe, and vice versa. In his thesis, we explore three lines of research that are linked to this central theme. The first partition shows how notions of non-computability and undecidability eventually led to evidence of emergence, the concept that even if a ‘theory of everything’ governing all microscopic interactions were discovered, the understanding of macroscopic order is likely to require additional insights. The second partition proposes a physically motivated model of computation that relates quantum complexity, quantum optimal control, and Riemannian geometry. Thus insights in any one of these disciplines could also lead to insights in the others. The remainder of this partition explores a simple application of these relations. The final partition proposes a model of quantum computation that generalizes measurement based computation to continuous variables. We outline its optical implementation, whereby any computation can be performed by single mode measurements on a resource state that can be prepared by passing a collection of squeezed states through a beamsplitter network.
Abstract We establish a universal bound on entanglement dynamics, proving that for any quantum system with energy E, the rate of entanglement entropy change is limited by dtdS≤πℏ2E | d S d t | ≤ 2 E π ħ . This bound, which can be saturated by specific protocols, emerges from a framework treating entanglement rate as a physical observable and connects directly to quantum circuit complexity. When extended to quantum field theory, we demonstrate that the maximum entanglement propagation velocity equals the speed of light, revealing a fundamental connection between quantum information constraints and relativistic causality. We propose experimental tests on quantum computing platforms and explore implications for quantum computing efficiency and spacetime structure. Our results suggest that information-theoretic principles may be more fundamental than spacetime concepts, unifying quantum information, computational complexity, and relativistic physics.
Beyond Causality: Temporal Geometry, Informational Consistency, and the Foundations of Physical Time What if the deepest mystery of time is not whether the past can be reached, but whether causality itself is more fundamental than the physical laws through which we describe it? For more than a century, modern physics has transformed our understanding of space, time, matter, and information. General relativity revealed that temporal order is inseparable from spacetime geometry. Quantum theory challenged classical intuitions about measurement, correlations, and physical reality. Information theory provided a mathematical language for describing what can be known, transmitted, and recovered. Yet a profound question remains: must every physically consistent universe possess a globally ordered sequence of causes and effects? Beyond Causality investigates this question through an extended mathematical and conceptual study of temporal geometry, informational dependence, retrocausal feedback, and global consistency. Organized into six interconnected parts, this 430-page research monograph develops a framework for examining causal structures beyond conventional temporal ordering. It brings together established results from general relativity, quantum information, causal modeling, and fixed-point theory, while exploring additional mathematical constructions designed to clarify the conditions under which temporally nonstandard information networks can remain internally consistent. Particular attention is devoted to distinguishing mathematical self-consistency from physical realizability, statistical correlation from controllable communication, and unconventional temporal descriptions from genuine violations of operational causality. Among its principal investigations are the geometric and energetic constraints associated with localized chronology-violating spacetime models, the existence and stability of globally consistent informational feedback networks, the behavior of finite stochastic systems under intervention, and the relationship between quantum causal structures and hypothetical retrocausal channels. The work also extends the theoretical research program introduced in Eon Reverser: Toward Temporal Bubble Radio Communication, examining what additional mathematical, physical, and experimental conditions would be required before communication with an earlier receiver record could be considered a meaningful scientific possibility. Analytical derivations, formal propositions, computational examples, and reproducibility materials accompany the discussion. The proposed constructions are explicitly distinguished from established physical results, and no experimental evidence for backward-in-time communication is claimed. The central ambition of this monograph is neither to declare that causality can be violated nor to assume that temporal paradoxes admit a universal resolution. It is to ask a more precise question: what mathematical structures, consistency principles, and observable constraints would a physical theory have to satisfy if causal order were not universally fixed? The possibility that time, information, and causality may possess a deeper relationship than our present formulations reveal remains an open intellectual frontier. Exploring that frontier requires imagination disciplined by mathematics, speculation constrained by physical principles, and theoretical proposals that remain accountable to observation. Beyond Causality is offered as a contribution to that exploration: an attempt to extend the questions we can rigorously formulate, identify the limits of existing models, and provide a foundation for further investigation into the nature of physical time. Keywords: Causality; Temporal Geometry; General Relativity; Retrocausality; Closed Timelike Curves; Quantum Causal Structures; Information Theory; Global Consistency; Fixed-Point Theory; Chronology Protection; Advanced Electromagnetic Response; Eon Reverser.