Lower bounds and complexity dependence on locality and sparsity

Establish lower bounds for Hamiltonian learning from real-time evolution that clarify how the complexity of learning all parameters scales with the locality of the Hamiltonian and with the effective sparsity; in particular, ascertain whether dependence on effective sparsity is necessary and determine the optimal dependence.

Background

A lower bound of (1/ε) log(1/δ) is known for estimating a single parameter, but the overall complexity for learning all parameters remains unclear.

Understanding whether and how the complexity must intrinsically depend on locality and effective sparsity is important for characterizing fundamental limits of Hamiltonian learning algorithms.

References

Can one prove lower bounds on Hamiltonian learning? A lower bound of \frac{1}{\eps}\log\frac{1}{\delta} is known for estimating one parameter. It is not known how the complexity of learning all parameters scales with underlying locality. Is a dependence on effective sparsity \sparse necessary? What is the optimal dependence?

— Structure learning of Hamiltonians from real-time evolution  (2405.00082 - Bakshi et al., 2024) in Discussion (Future directions), Item 1

Another question is whether the additional system-size factors in the single-qubit algorithms can be reduced.

— Hamiltonian Learning and Certification via Eigenphase Engineering  (2609.26596 - Shin et al., 22 Sep 2026) in Section Discussion and future work

Our learning lower bound leaves a factor \widetilde{\mathcal O}(s{1/(2k)}) in sparsity, so the optimal dependence on s also remains to be determined.

— Hamiltonian Learning and Certification via Eigenphase Engineering  (2609.26596 - Shin et al., 22 Sep 2026) in Section Discussion and future work

An important open question is therefore how the interaction geometry determines the optimal learning complexity.

— Learning Many-Body Hamiltonians Using a Local Probe  (2610.02157 - Liu et al., 1 Oct 2026) in Section Discussion and outlook

Can further adaptation of the input, allowed measurement basis, or evolution time based on previous outcomes reduce the overall sample complexity? Could it reduce the number of experimental settings, or the amplification of measurement errors?

— Out-of-control Hamiltonian Learning  (2610.06709 - Gong et al., 5 Oct 2026) in Section 6, Outlook, paragraph “The role of adaptivity”