Geometric Speed Limit of Accessible Many-Body State Preparation

TL;DR

This paper establishes a geometric lower bound on the speed of many-body quantum state preparation based on quantum geometry, showing that local control constraints can significantly slow down the process in complex systems.

Contribution

It formulates and proves a conjecture linking energy fluctuations and quantum geometric tensor to the quantum speed limit, especially under local control restrictions.

Findings

Geometric lower bound for quantum speed limit derived

Bound is saturated by geodesic protocols with constant energy variance

In complex models, the bound can grow exponentially with system size

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 unit-fidelity 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 well-known Mandelstam-Tamm 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 counter-diabatic and fast-forward protocols for a three-level system, and numerically in nonintegrable spin chains and a nonlocal SYK model.