Quantum

Quantum Query Complexity and Span Programs from Pre-Geometry

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Briefings

This preprint establishes a matroidal framework for quantum query complexity, separating query dependence from span program structure. It derives exact bounds for $R_{10}$ closure, showing the natural program achieves cost 5 while another regular program is bounded by $\ ext{\sqrt{87/5}}$. The research provides explicit quantum algorithms with query complexities $O(N^{0.6500178\ldots})$ and establishes a lower bound of $\ ext{\Omega}(N^{0.7324867\ldots})$.

Key points

• The natural $R_{10}$ program achieves an exact cost of 5 for the $R_{10}$ closure problem, while a regular program has an upper bound of $\ ext{\sqrt{87/5}}$.
• Recursive self-composition yields a quantum algorithm with query complexity $O(N^{0.6500178\ldots})$ and a lower bound of $\\text{\Omega}(N^{0.7324867\ldots})$.

Context and limitations

Coverage is limited to fixed-weight objectives and specific $R_{10}$ closure instances; results rely on exact certificates and do not generalize to all query problems.

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