Quantum complexity in gravity, quantum field theory, and quantum information science

TL;DR

The survey maps quantum complexity across quantum information, many-body physics, quantum field theory, and holographic gravity, outlining how Nielsen, Krylov/spread, and tensor-network-inspired notions capture time evolution, chaos, and entanglement. It highlights universal features such as linear growth and switchback effects in chaotic dynamics, and surveys rigorous results from random circuits and unitary designs. A central thread is the holographic conjecture that certain gravitational observables track boundary complexity, with precise matches in two-dimensional JT gravity/SYK duality and broader discussions of CV, CA, and CAny frameworks, including subregions and de Sitter spacetime. The work also emphasizes open problems, including proving superpolynomial lower bounds, connecting Nielsen and Krylov formalisms, and extending holographic matches to higher dimensions and dynamical settings. Overall, the review provides a consolidated roadmap for using quantum complexity as a diagnostic and bridge between information theory and quantum gravity.

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.

Figures (15)

• Figure 1: Structure of the review.
• Figure 2: Illustration of the circuit that models unitary evolution following from a generic 2-local Hamiltonian.
• Figure 3: Illustration of the time evolution of complexity for a generic 2-local fast-scrambling Hamiltonian.
• Figure 4: Circuit implementation of the precursor operator $W(t)$. Yellow gates are necessary for implementing $W(t)$. Gray gates are not necessary and cancel out, failing to appear in the optimal circuit.
• Figure 5: Precursor's expected complexity, as a function of time. The switchback effect is the delay, until $t \approx t_*$, in the complexity's linear growth.