Holographic complexity measures show universal linear growth followed by late-time saturation, proven necessary and sufficient via pole structures in the energy basis using the residue theorem, arising from random matrix statistics.
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Second law of quantum complexity
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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.
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Non-invertible symmetries define quantum gates with generalized complexity distances, and simple objects in symmetry categories turn out to be computationally complex in concrete 4D and 2D QFT examples.
Quantum complexity measures applied to the Schwinger model reveal nonlocal correlations along the string and show that entanglement and magic give complementary views of string formation and breaking.
Spectral functions of SYK, p-spin, and SU(M) Heisenberg models show exponential tails in spin-glass phases and quasiparticle families in spin-liquid phases, with a proof that exponential decay blocks detection of bulk causal structure.
Generalized Krylov complexity predicts the minimum time to realize target operations in analog quantum simulators such as Rydberg atom arrays.
Holographic Schwinger pair creation generates nonlocal magic for spacetime dimensions d>2, as shown by a non-flat entanglement spectrum that can be read from the probe brane free energy.
A timelike quantum focusing conjecture implies a complexity-based quantum strong energy condition and a complexity bound analogous to the covariant entropy bound for suitable codimension-0 field theory complexity measures.
Krylov complexity is a canonical, parameter-independent measure of operator spreading that probes chaotic dynamics to late times and admits a geometric interpretation in holographic duals.
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Bridging Krylov Complexity and Universal Analog Quantum Simulator
Generalized Krylov complexity predicts the minimum time to realize target operations in analog quantum simulators such as Rydberg atom arrays.