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.
Highlights
The quantum speed limit (QSL) is the minimum time, TQSL, required to prepare a quantum state with unit fidelity
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
Even though standard quantum speed limit bounds are correct, they can only be saturated by a Rabi-pulse constructed out of a Gram-Schmidt orthogonalized version of the initial and target states
Summary
The quantum speed limit (QSL) is the minimum time, TQSL, required to prepare a quantum state with unit fidelity. On the other hanpd,ffiffitffihe energy fluctuations in the total system are ΔE 1⁄4 Δ L, so the expression (1) suggests that it would be possible to rotate the spins faster This fallacious argument shows how the standard quantum speed limit bounds are based on the premise that one can access the full Hilbert space to construct the optimal driving Hamiltonian. This bound implies that the quantum speed limit is controlled by the geodesic length between the initial and the target state in the eigenstate manifold set by the control parameter space Based on this conjecture, we show that the adiabatic limit and the associated quantum geometry [35] constrain the time of possible unit-fidelity protocols both in singleparticle and complex many-body systems. It follows that the quantum speed limit for all protocols is bounded by the quantum speed limit for counter-diabatic protocols, which generally cannot be implemented within the constrained control parameter space but for which the geodesic bound can be rigorously proven using recent results from Ref. [67]
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