Geometric Speed Limit of Accessible ManyBody State Preparation
Abstract
We analyze state preparation within a restricted space of local control parameters between adiabatically connected states of control Hamiltonians. We formulate a conjecture that the time integral of energy fluctuations over the protocol duration is bounded from below by the geodesic length set by the quantum geometric tensor. The conjecture implies a geometric lower bound for the quantum speed limit (QSL). We prove the conjecture for arbitrary, sufficiently slow protocols using adiabatic perturbation theory and show that the bound is saturated by geodesic protocols, which keep the energy variance constant along the trajectory. Our conjecture implies that any optimal unitfidelity protocol, even those that drive the system far from equilibrium, are fundamentally constrained by the quantum geometry of adiabatic evolution. When the control space includes all possible couplings, spanning the full Hilbert space, we recover the wellknown MandelstamTamm bound. However, using only accessible local controls to anneal in complex models such as glasses or to target individual excited states in quantum chaotic systems, the geometric bound for the quantum speed limit can be exponentially large in the system size due to a diverging geodesic length. We validate our conjecture both analytically by constructing counterdiabatic and fastforward protocols for a threelevel system, and numerically in nonintegrable spin chains and a nonlocal SYK model.
 Publication:

Physical Review X
 Pub Date:
 January 2019
 DOI:
 10.1103/PhysRevX.9.011034
 arXiv:
 arXiv:1804.05399
 Bibcode:
 2019PhRvX...9a1034B
 Keywords:

 Quantum Physics;
 Condensed Matter  Other Condensed Matter;
 Condensed Matter  Quantum Gases
 EPrint:
 Phys. Rev. X 9, 011034 (2019)