Problem
Research questions and friction points this paper is trying to address.
Hamiltonian property testing
Heisenberg limit
total evolution time lower bound
continuous-time query model
amplitude estimation
Hamiltonian locality testing and certification do not achieve the Heisenberg limit
🤖 AI Summary
This study addresses the problem of characterizing complexity lower bounds for testing and certifying the k-locality of Hamiltonians in restricted settings where only forward queries to the time evolution operator are permitted. To this end, it proposes a continuous-time adversary method that derives theoretical bounds by constructing a distinguishing task between the zero Hamiltonian and a random ensemble. The primary contribution is the first proof that such natural problems cannot achieve the Heisenberg limit, thereby ruling out 1/ε scaling and establishing an Ω(1/ε²) lower bound on the total evolution time. This result tightly matches existing upper bounds and recovers related conclusions within the amplitude estimation framework.
📝 Abstract
We establish lower bounds for Hamiltonian property testing with access to the time-evolution operator but not its inverse. Each experiment may query the time-evolution operator multiple times, and distances between Hamiltonians are measured in the normalized Frobenius norm.
In this model, we show that testing whether a Hamiltonian is $k$-local or $\varepsilon$-far from every $k$-local Hamiltonian requires $Ω(1/\varepsilon^2)$ total evolution time, matching the upper bound of Kallaugher and Liang (TQC'25). We also prove that testing whether an unknown Hamiltonian equals a target Hamiltonian or is $\varepsilon$-far from it requires $Ω(1/\varepsilon^2)$ total evolution time, matching the upper bound of Sinha and Tong (2025). These are the first lower bounds for natural problems in Hamiltonian learning and testing that rule out Heisenberg-limited scaling of $1/\varepsilon$.
As a third result, we show that amplitude estimation to precision $\varepsilon$ requires $Ω(1/\varepsilon^2)$ total time evolution, recovering the result of Tang and Wright (QIP'26) in the continuous-time query model. All three results follow from the hardness of distinguishing the zero Hamiltonian from a suitably chosen ensemble of random Hamiltonians. We establish this hardness by adapting the continuous-time adversary method to forward Hamiltonian evolution.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Hamiltonian locality testing
Heisenberg limit
continuous-time adversary method
total evolution time lower bound
amplitude estimation
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