General relationship between quantum learnability and classical simulability

Establish whether efficient quantum learnability and efficient classical simulability are generally equivalent, and, for tensor-network states, determine the optimal graph-dependent exponents for learning and simulation, prove corresponding lower bounds for learning, and identify graph families that are efficiently learnable but hard to simulate or efficiently simulable but hard to learn.

Background

The paper compares the graph parameters governing tensor-network-state learning with those governing classical tensor-network simulation. Its direct tomography algorithm is bounded using contraction complexity, which also controls classical simulation through its relation to treewidth. The authors note that these upper bounds suggest a possible connection between learnability and simulability but do not establish one in general.

The unresolved problem has several parts: determine whether a general learnability–simulability connection exists, sharpen the graph-dependent exponents for both tasks, establish lower bounds for learning, and separate the two notions by finding graph families with asymmetric learning and simulation complexity.

References

While these separate results can be viewed as evidence for a connection between learnability and simulability, whether such a connection can be established in general remains an important open question in quantum learning theory and quantum machine learning . Our results strongly hint at the potential of parameterised complexity in investigating such a connection for TNSs by considering which graph parameters govern the complexities of learning and simulations of TNs, respectively. Our results relate the two upper bounds through contraction complexity. This parameter determines the cost of classical simulation through the treewidth relation of and, by \Cref{cor:lc-upper-bound}, bounds the exponent of our direct learner. It remains open to determine the optimal graph-dependent exponents for both tasks, to prove corresponding lower bounds for learning, and to identify graph families that are efficiently learnable but hard to simulate, or vice versa.

— Parameterised graph theory for tensor networks: entanglement rerouting, structural simplification, and agnostic tomography  (2609.04165 - Caro et al., 3 Sep 2026) in Section 1, subsection “Discussion and outlook,” paragraph “Learning versus simulation”

For which graph families and parameter regimes do we have \begin{equation}

\lc_{d,\chi}(G)

\Theta\left( \max\left{ 1, \left\lceil \CC(G)\log_d\chi \right\rceil \right} \right)? \end{equation} More generally, how large can the gap be between optimal learning complexity and the upper bound based on contraction, and which graph properties determine it?

— Parameterised graph theory for tensor networks: entanglement rerouting, structural simplification, and agnostic tomography  (2609.04165 - Caro et al., 3 Sep 2026) in Question “Tightness of the contraction-complexity bound,” subsection “Learning and contraction complexity”

What is the optimal copy complexity of TNS tomography as a function of the underlying graph? In particular: \begin{enumerate} \item Is there a family of graphs $G_n$ and states in $\mathcal S_d(G_n,\chi)$ for which every tomography algorithm requires $d{\Omega(\lc_{d,\chi}(G_n))}$ copies? \item Can an adaptive learner, which re-optimises the remaining learning sequence after each step using the information obtained so far, achieve copy complexity below $d{\lc_{d,\chi}(G)}$ for some graph family or some inputs? \end{enumerate}

— Parameterised graph theory for tensor networks: entanglement rerouting, structural simplification, and agnostic tomography  (2609.04165 - Caro et al., 3 Sep 2026) in Question “Optimal graph-dependent complexity of TNS tomography,” subsection “Learning and contraction complexity”