The Second Law of Quantum Complexity

Abstract: We give arguments for the existence of a thermodynamics of quantum complexity that includes a "Second Law of Complexity". To guide us, we derive a correspondence between the computational (circuit) complexity of a quantum system of $K$ qubits, and the positional entropy of a related classical system with $2^K$ degrees of freedom. We also argue that the kinetic entropy of the classical system is equivalent to the Kolmogorov complexity of the quantum Hamiltonian. We observe that the expected pattern of growth of the complexity of the quantum system parallels the growth of entropy of the classical system. We argue that the property of having less-than-maximal complexity (uncomplexity) is a resource that can be expended to perform directed quantum computation. Although this paper is not primarily about black holes, we find a surprising interpretation of the uncomplexity-resource as the accessible volume of spacetime behind a black hole horizon.

• The paper proposes a Second Law of Quantum Complexity by drawing an analogy between quantum circuit complexity and classical entropy growth.
• It develops a framework linking the complexity of a K-qubit system to the entropy of a 2^K degree system, emphasizing thermodynamic parallels.
• It introduces uncomplexity as a computational resource, with implications for optimizing quantum computation and understanding black hole interiors.

Insights into the Second Law of Quantum Complexity

The paper by Adam R. Brown and Leonard Susskind, titled "The Second Law of Quantum Complexity," embarks on an exploration into the thermodynamic-like behavior of quantum complexity and conjectures the existence of a "Second Law of Complexity" akin to the second law of thermodynamics. This work explores the intricacies of quantum complexity theory through the lens of statistical mechanics, bringing forth a compelling analogy between complexity growth in quantum systems and entropy growth in classical systems.

The authors begin by establishing a framework that parallels quantum computational complexity with the entropy of classical systems. Specifically, they propose a correspondence between the circuit complexity of a quantum system composed of $K$ qubits and the positional entropy of a classical system with $2^K$ degrees of freedom. This conceptual framework interlinks the fields of quantum computation and classical thermodynamics, enabling a rigorous discussion on the evolution and saturation of complexity in quantum systems.

The paper presents a clear conjecture regarding the evolution of quantum complexity, suggesting that after reaching a point of maximum entropy, systems continue to increase in complexity before eventually saturating at a maximum value. Computational complexity, they argue, resembles the entropy of classical systems but on a vastly expanded scale due to the exponential nature of quantum states. This analogy forms the bedrock of their argument for the existence of a Second Law of Complexity, which postulates that complexity tends to increase over time, similar to how the entropy of a closed system tends to grow.

A pivotal contribution of their work is the analysis of uncomplexity as a resource. The property of being less than maximally complex—a state referred to as 'uncomplexity'—is posited as a computational resource that can be harnessed for directed quantum computation. This redefines the role of complexity, not just as a measure of gate operations but as a thermodynamically pertinent quantity that can influence computational power and efficiency.

One of the most intriguing aspects of the discussion is the practical implications of uncomplexity and its relationship to black holes. While not central to their thesis, Brown and Susskind explore the implications of their findings on the understanding of black holes in the context of the holographic principle. They propose that the 'uncomplexity resource' could metaphorically represent the accessible volume behind a black hole's horizon, providing new angles to deciphering the enigma of black hole interiors in quantum gravity.

Despite the detailed theoretical exposition and the compelling insights it offers, the assertions in the paper rest on unproven conjectures. The parallels drawn between the growth and saturation of complexity and those of entropy invite further theoretical scrutiny and experimental validation within the framework of quantum mechanics.

This work contributes significantly to the dialogue on quantum complexity by proposing a thermodynamic interpretation, which could potentially open new paths in studying quantum systems, particularly in the context of quantum Shannon theory and complexity-based encryption. Future work should aim to flesh out the precise interplay between complexity and computational dynamics, sharpening the tools for probing classical-quantum boundaries and practical realization of complexity-related computations in quantum computing.

In summary, the insights into the Second Law of Quantum Complexity not only serve as a stepping stone for future theoretical and experimental undertakings but also invite deeper contemplation on the nature of complexity as a fundamental aspect of physics, with profound implications for our understanding of the universe.