The Second Law of Quantum Complexity

TL;DR

The paper proposes a thermodynamic framework for quantum complexity by mapping the growth of circuit complexity in a K-qubit system to the positional entropy of a classical auxiliary system with 2^K degrees of freedom. It develops a geometry of complexity on SU(2^K) with a right-invariant metric that penalizes non-k-local directions, derives negative sectional curvatures, and proposes an action-based relation between complexity and the auxiliary system's dynamics. A central claim is that ensemble-averaged quantum complexity equals the auxiliary entropy, with a dual role for Kolmogorov complexity as the kinetic entropy; together these yield a Second Law of Complexity and a resource theory around uncomplexity. The work also connects these ideas to black-hole physics, showing uncomplexity corresponds to spacetime behind horizons and that a single clean qubit can rejuvenate a horizon, linking computational and gravitational notions. Overall, the paper provides a conceptual bridge between quantum information, statistical mechanics, and holography, proposing testable links and rich avenues for further formalization.

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.

Figures (10)

• Figure 1: The conjectured evolution of quantum complexity of the operator $e^{-iHt}$, where $H$ is a generic time-independent $k$-local Hamiltonian. The complexity increases with rate $K$, and then saturates at a value exponential in $K$. It fluctuates around this value. Quantum recurrences occur on a timescale that is double-exponential in $K$; a very rare, very large fluctuation brings the complexity down to near zero. This figure would also describe the entropy of a classical chaotic system with $\exp[K]$ degrees of freedom.
• Figure 3: Two neighboring geodesics (in blue) leave the origin at $t=0$; this corresponds to evolution under two nearby $k$-local Hamiltonians. The two geodesics are connected by a Jacobi vector (in red); this corresponds to the (non-$k$-local) connecting operator $e^{\Lambda} = e^{-iHt} e^{i(H + \Delta d \theta) t }$. As $t$ increases the connecting Jacobi vector grows, and the acceleration of this growth rate gives the geodesic deviation. In the bi-invariant metric the geodesics always converge, but on the complexity metric the geodesics can diverge for $I_{3}>4/3$.
• Figure 4: The left panel shows a gas of $2N$ classical particles created with the particles paired. The entropy is the same as a gas of $N$ particles. In the right panel the gas has come to equilibrium and the particles become randomly distributed. The entropy in the right panel is twice the entropy in the left panel. No work can be extracted from a gas in equilibrium, but the gas of paired particles is far from equilibrium and so can be used to do work.
• Figure 5: The interior of the circles represent the space $CP(2^K-1)$ with the center point being the state $|0000\rangle.$ The blue regions are the target set. The left panel shows the evolution of a circuit programmed to get to a point on the target set. The trajectory is built from gates and each step increases the complexity.
• Figure 6: Left: adding an extra qubit doubles the maximum complexity (adding an annulus to the space of possible states) and replenishes the uncomplexity resource. Right: given the additional resource, what was previously a state of maximal complexity now has some uncomplexity and can be used to further computation; this is illustrated by showing how a target state can be reached from the original maximally complex configuration.