Speaker
Description
A central open problem in quantum process tomography is whether access to quantum memory yields a learning separation for arbitrary quantum channels. Coherent protocols may preserve quantum information across channel uses, whereas incoherent protocols must measure after each use and proceed only with classical memory. The remaining question is whether classical adaptive protocols can compensate for the absence of quantum memory. In this work, we prove that any adaptive incoherent protocol that estimates every channel $\Phi:\mathcal{L}(\mathbb{C}^{d_{\mathrm{in}}})\to\mathcal{L}(\mathbb{C}^{d_{\mathrm{out}}})$ to diamond-norm error $\varepsilon$ with constant success probability requires
$$
\Omega(d^3_{\textrm{in}} d^3_{\textrm{out}}/\varepsilon^2)
$$
channel queries, even when each individual use may be ancilla-assisted.
This matches the known upper bound up to logarithmic factors and therefore establishes that incoherent quantum process tomography has optimal sample complexity $\widetilde{\Theta}(d^3_{\textrm{in}} d^3_{\textrm{out}}/\varepsilon^2)$. The key idea is a transcript-wise posterior anti-concentration argument for process testers, which handles the full adaptivity of classical strategies and shows that such strategies cannot concentrate around the true channel faster than the optimal incoherent rate. Since coherent tomography protocols using quantum memory achieve arbitrary-channel estimation with $\Theta(d^2_{\textrm{in}} d^2_{\textrm{out}}/\varepsilon^2)$ queries, our results establish a strict learning separation between process tomography with and without quantum memory.
| Topic | Theoretical foundations of quantum learning |
|---|---|
| Additional Topic | Quantum state and process tomography with learning-based approaches |