Finding suitable characterizations of quantum chaos is a major challenge in many-body physics, with a central difficulty posed by the linearity of the Schrödinger equation. A possible solution for recovering non-linearity is to project the dynamics onto some variational manifold. The classical chaos induced via this procedure may be used as a signature of quantum chaos in the full Hilbert space. Here, we demonstrate analytically a previously heuristic connection between the Lyapunov spectrum from projection onto the matrix product state (MPS) manifold and the growth of entanglement. This growth occurs by squeezing a localized distribution on the variational manifold. The process qualitatively resembles the Cardy-Calabrese picture, where local perturbations to a moving MPS reference are interpreted as bosonic quasi-particles. Taking careful account of the number of distinct channels for these processes recovers the connection to the Lyapunov spectrum. Our results rigorously establish the physical significance of the projected Lyapunov spectrum, suggesting it as an alternative method of characterizing chaos in quantum many-body systems, one that is manifestly linked to classical chaos.
We calculate the frame potential for Brownian clusters of
N spins or fermions with time-dependent all-to-all interactions. In both cases the problem can be mapped to an effective statistical mechanics problem which we study using a path integral approach. We argue that the
kth frame potential comes within
ε of the Haar value after a time of order
t∼kN+klogk+logε−1. Using a bound on the diamond norm, this implies that such circuits are capable of coming very close to a unitary
k-design after a time of order
t∼kN. We also consider the same question for systems with a time-independent Hamiltonian and argue that a small amount of time-dependent randomness is sufficient to generate a
k-design in linear time provided the underlying Hamiltonian is quantum chaotic. These models provide explicit examples of linear complexity growth that are also analytically tractable.
We investigate work statistics in monitored transverse-field Ising chains subjected to both a quantum quench of the transverse field and either stochastic quantum jumps or controlled measurement sequences. For generalized measurements, we derive a trajectory-resolved generating function for work statistics in the two-point energy measurement scheme. Evaluating it using a fermionic Gaussian-state formalism, we show that, under stochastic jump dynamics, the work distribution crosses over from a comb-like structure to an essentially Gaussian form with shrinking sub-Gaussian tails, as the number of detection events grows. For controlled jump protocols, the energy added by each jump is constant when successive jumps are causally disconnected but decreases and then saturates when they lie within each other's light cone, leading to linear growth of average work in the former case and a transient sublinear regime followed by linear growth with a reduced slope in the latter. For monitored quenches, continuous observation washes out the fine structure of the isolated-quench distribution and again drives the statistics toward Gaussian behavior. Together, these results establish work statistics as a trajectory-resolved diagnostic of measurement-induced energy injection and of the emergence or breakdown of additivity in monitored many-body dynamics.