Time-optimal tomography for general rank-r mixed states

Determine whether there exists an algorithm for learning n-qubit rank-r mixed states to infidelity at most ε in O(rn2^n/ε) time.

Background

The paper establishes near-optimal sample complexity for sparse mixed-state tomography but does not obtain matching time complexity. Its reduction through the random purification channel is computationally expensive, motivating the broader problem of achieving the standard input-reading time scale for arbitrary rank-r mixed states. This question asks for an algorithm whose runtime is linear in the number of input qubits, the rank, and the ambient Hilbert-space dimension, up to the accuracy factor.

References

Does there exist an algorithm for learning $n$-qubit rank-$r$ mixed states in $O(rn2n/\eps)$ time to infidelity at most $\eps$?

— Learning Sparse Quantum States  (2609.12219 - Sen, 10 Sep 2026) in Section "Open Questions"

This also gives a route to answering the further question: Does there exist an algorithm for learning $n$-qubit $k$-sparse rank-$r$ mixed states in $O(nkr/\eps)$ time to infidelity at most $\eps$? We show in \cref{section:mixed-state} that the sample complexity scales with the support size $k$, could we additionally get the time complexity of the random purification to scale optimally with $k$ (or $r$)?

— Learning Sparse Quantum States  (2609.12219 - Sen, 10 Sep 2026) in Section "Open Questions"

Obtaining (near) optimal time complexity bounds, along with scaling in $k$, remains open.

— Learning Sparse Quantum States  (2609.12219 - Sen, 10 Sep 2026) in Section "Open Questions"