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Close the gap between classical lower and upper bounds for DXHOG

Close the gap between the current lower and upper bounds on the classical one-way communication required to achieve a target average linear cross-entropy benchmarking fidelity in the Distributed Linear Cross-Entropy Heavy Output Generation (DXHOG) task, where Alice’s n-qubit state is drawn from the Haar measure and Bob’s measurement is drawn from an ensemble such as random Clifford circuits, approximate unitary t-designs, or the Haar measure; accomplishing this likely requires sharper bounds on the sizes of high-dimensional spherical codes or improved classical protocol constructions.

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Background

The paper establishes exponential lower bounds on classical one-way communication needed to match the average linear cross-entropy benchmarking fidelity achieved by a quantum protocol for the DXHOG task, and also presents constructive classical upper bounds via randomized codebook protocols. While these bounds are within an order of magnitude for practical parameters, they do not match, leaving a quantitative gap.

The authors point out that improving the classical upper bound reduces to a high-dimensional packing problem: constructing arrangements of vectors that correlate well with most Haar-random states. They further note that obtaining sharper bounds on spherical codes—an area with longstanding unresolved questions—may be necessary to significantly tighten the upper bound. Tighter, preferably matching, bounds would clarify the exact classical memory resources required to replicate the demonstrated quantum advantage.

References

A natural question for future work is to close the gap between our lower and upper bounds. This seems challenging, as our upper bound on classical communication (\Cref{thm:classical_upper_bound} and \Cref{cor:classical_lower_bound}) is essentially solving a packing problem in high dimensions: we want to find an arrangement of vectors ${\ket{\varphi_x}}_{x \in {0,1}m}$ such that $\max_x {\braket{\varphi_x|\psi}2}$ is large for most $\ket{\psi}$. Improving the randomized construction might require obtaining sharper bounds on the sizes of high-dimensional spherical codes, which is a major unresolved question in mathematics.

— Demonstrating an unconditional separation between quantum and classical information resources (2509.07255 - Kretschmer et al., 8 Sep 2025) in Section 7, Summary of Classical Communication Complexity Bounds (Supplemental Material)