Polynomial round complexity for non-i.i.d. state tomography

Determine whether non-i.i.d. state tomography for quantum sources with description length at most s and registers of n qubits can be achieved with round complexity polynomial in s, ϵ^{-1}, and δ^{-1}, independently of the register size n and the preparation-time bound t.

Background

The paper constructs an information-theoretic non-i.i.d. state tomography algorithm with round complexity O(s min{2s, 2n} ϵ{-2}δ{-1}). The learner receives a single-copy prefix of an unknown, potentially correlated or entangled quantum source, performs measurements on the preceding registers, and outputs a classical description of the conditional state of the next register.

Although the paper proves that linear dependence on the source description length s is necessary, it does not establish whether the exponential factor min{2s, 2n} is intrinsic. The unresolved question is whether a substantially more efficient tomography procedure can obtain a bound polynomial in s, ϵ{-1}, and δ{-1} without dependence on n or the preparation time t.

References

However, we do not know whether the exponential factor min{2s, 2n} is necessary. It remains open whether one can achieve round complexity polynomial in s, ϵ−1, and δ−1, independently of n and t.

— Universal Inductive Inference of Quantum States  (2609.36912 - Hiroka et al., 29 Sep 2026) in Section 1.1, Non-i.i.d. state tomography (page 6)

De~Palma, Fanizza, Mowry and O'Donnell leave open whether the average state \rho_{\mathrm{avg}=\frac1n\sum_i\sigma_i of a product input \omega=\sigma_1\otimes\cdots\otimes\sigma_n can be learned with the i.i.d. copy complexity.

— Robustness of quantum spectrum estimation: weak Schur sampling under noisy inputs  (2610.03582 - Visnevskyi et al., 2 Oct 2026) in Applications subsection, “Learning the average of non-identical states”; Connections to existing work subsection, “Learning from non-i.i.d. sources”