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Quantum complexity in gravity, quantum field theory, and quantum information science (2503.10753v2)

Published 13 Mar 2025 in hep-th, cond-mat.stat-mech, cond-mat.str-el, gr-qc, and quant-ph

Abstract: Quantum complexity quantifies the difficulty of preparing a state or implementing a unitary transformation with limited resources. Applications range from quantum computation to condensed matter physics and quantum gravity. We seek to bridge the approaches of these fields, which define and study complexity using different frameworks and tools. We describe several definitions of complexity, along with their key properties. In quantum information theory, we focus on complexity growth in random quantum circuits. In quantum many-body systems and quantum field theory (QFT), we discuss a geometric definition of complexity in terms of geodesics on the unitary group. In dynamical systems, we explore a definition of complexity in terms of state or operator spreading, as well as concepts from tensor-networks. We also outline applications to simple quantum systems, quantum many-body models, and QFTs including conformal field theories (CFTs). Finally, we explain the proposed relationship between complexity and gravitational observables within the holographic anti-de Sitter (AdS)/CFT correspondence.

Summary

Overview of Quantum Complexity in Gravity, Quantum Field Theory, and Quantum Information Science

This paper offers a comprehensive review of quantum complexity, which is an interdisciplinary subject bridging gravity, quantum field theory (QFT), and quantum information science. Written by Stefano Baiguera, Vijay Balasubramanian, Pawel Caputa, Shira Chapman, Jonas Haferkamp, Michal P. Heller, and Nicole Yunger Halpern, the paper integrates perspectives from these diverse fields, providing a rich synthesis of contemporary understanding.

Definitions and Concepts

Quantum complexity refers to the minimum number of operations required to prepare a quantum state or implement a unitary transformation. This concept comes from quantum computation but has broad applications in areas like condensed matter physics and quantum gravity. The definition can vary significantly based on the context, with various properties and metrics being applied by different disciplines.

In quantum information science, for instance, complexity metrics are often derived from random quantum circuits, which help in understanding the dynamics of quantum states. In quantum many-body systems and QFT, complexity might be understood geometrically through path lengths on certain manifolds. These different conceptions highlight the necessity of bridging definitions between disciplines.

Research in Different Fields

Quantum Information Science

In quantum information science, complexity plays a critical role in describing computational tasks and the evolution of quantum algorithms. Here, random quantum circuits serve as a model for studying how complexity grows with time in chaotic systems. The paper discusses how complexity metrics derived from these circuits can stand as proxies for understanding quantum information scrambling.

Quantum Field Theory

For quantum field theories, complexity is often tied to geometric considerations. Complexity geometries can describe how quantum states evolve and thermalize, providing insights into operator growth and chaos. The review mentions approaches such as defining complexity via tensor networks, which provide efficient representations of quantum states and can be used to paper entanglement and other properties in quantum systems.

Quantum Gravity

In the domain of quantum gravity, complexity is linked to the anti-de Sitter space/conformal field theory (AdS/CFT) correspondence. Various conjectures propose that quantities such as the volume of the Einstein-Rosen bridge in a black hole’s geometry might correlate with quantum complexity. These proposals, though speculative, aim to provide an operational framework for understanding black hole interiors and spacetime geometry from a quantum mechanical perspective.

Results and Implications

The paper gathers evidence supporting these conjectures across various quantum systems. Particularly intriguing are findings related to the linear growth of complexity over time, complexity saturation, and phenomena like the switchback effect, where complexity exhibits delayed linear growth due to perturbations. These behaviors are relevant not only for theoretical models but also for potential experimental realizations in condensed matter systems or quantum simulators.

The implications are profound both theoretically and practically. They suggest new ways to quantify the resources required for quantum computation, improve understanding of chaos in quantum systems, and possibly interpret complex spacetime structures via quantum mechanics. Future work could explore these challenges further, with potential developments in both theoretical constructs and experimental techniques in quantum computing and quantum field experiments.

Conclusion

By synthesizing quantum complexity across gravitational, computational, and field-theoretic contexts, the paper reveals it as a versatile and valuable concept potentially paving new pathways in understanding both foundational physics and advanced computational processes. The interdisciplinary approach underlined by these authors is crucial for advancing research at the intersection of these fields.