Dimension-Dependent Non-Adaptive Lower Bounds

Establish whether non-adaptive equivalence testing and independence testing in fidelity have an asymptotic advantage over trace-distance testing, or instead require dimension-dependent sample complexities of order approximately d^{3/2}/\varepsilon^2.

Background

The paper proves a non-adaptive lower bound of approximately 1/\varepsilon2 for fidelity equivalence testing even for qubits, demonstrating a separation from certification. However, it does not determine the general dependence on the Hilbert-space dimension.

The authors conjecture that non-adaptive fidelity testing may require the same dimension dependence as trace-distance testing, both for equivalence testing and, analogously, for independence testing.

References

Importantly, general lower bounds for non-adaptive algorithms including scaling in the dimension d remain open. It seems natural to conjecture that, analogous to tomography, non-adaptive protocols for equivalence testing in fidelity may not have an advantage over trace distance, such that $\widetilde{\Omega}(d{3/2}/2)$ samples are required, and analogously for independence testing.

— On the Power of Adaptivity in Testing Quantum States in Fidelity  (2609.08733 - Seyfried et al., 8 Sep 2026) in Introduction, subsection Discussion