Lower Bounds for Hamiltonian Parameter Learning from Time Evolution
Ziyun Chen, Jerry Li
TL;DR
This work proves the first lower bounds on learning Hamiltonians from real-time evolution that scale with the number of parameters and locality. By reducing parameter-learning to planted-spike detection and leveraging Boolean Fourier analysis plus precise matrix-perturbation bounds, the authors show that learning an arbitrary $n$-qubit Hamiltonian to uniform accuracy $\beta$ requires at least $\Omega\left(2^{(1/2-o(1))n}/\beta\right)$ interaction rounds, and that $k$-local Hamiltonians with coefficients bounded by $1$ require $n^{\Omega(k)}$ rounds for constant error. These results persist under time-reversal and unlimited quantum memory, implying super-polynomial total evolution time for inverse-polynomial resolution. They also extend to average-case, local-spike settings, ruling out fixed-parameter tractable evolution for spike-detection-based hardness. Overall, the paper establishes fundamental, structure-sensitive limitations for Hamiltonian-learning protocols and clarifies when efficient, scalable Hamiltonian inference from dynamics is unlikely without additional structure.
Abstract
We consider the problem of learning Hamiltonians from time evolution: for a Hamiltonian $P = \sum_{P} α_P P$, where the sum is taken over all Pauli matrices, given the ability to apply $e^{-iHt}$, the goal is to approximately recover the coefficients $α_P$ of the Hamiltonian. This is a well-studied problem in quantum learning theory, with applications to quantum metrology, sensing, device benchmarking, and many-body physics. For this problem, we demonstrate the first lower bounds which scale with the number of parameters of the unknown Hamiltonian. When the unknown Hamiltonian is arbitrary, we show that learning the coefficients to entry-wise error $ε$ requires $2^{(1/2 - o(1))n} / ε$ rounds of interaction with the Hamiltonian. If the Hamiltonian is additionally assumed to be $k$-local, we show that learning the coefficients to entry-wise constant error requires $n^{Ω(k)}$ rounds of interaction with the Hamiltonian, resolving an open question of Bakshi, Liu, Moitra, and Tang. These bounds immediately imply that any learning algorithm with inverse polynomial time resolution requires super-polynomial total evolution time. Our lower bounds hold even for very simple planted spike detection problems, where the goal is to decide whether or not there is a single coefficient which is super-polynomially larger than the other coefficients of the Hamiltonian. Our lower bound also holds in natural average-case settings, suggesting that this hardness is likely unavoidable in many natural settings, without additional structure.