Span Programs are Equivalent to Quantum Query Algorithms

Abstract

Span programs form a linear-algebraic model of computation that is studied to prove classical lower bounds. Quantum query complexity is a coherent generalization for quantum algorithms of decision-tree complexity. It is characterized by a semidefinite program known as the general adversary bound. We connect these classical and quantum models by proving that for any boolean function, the optimal “witness size” of a span program equals the general adversary bound. Therefore, span program witness size and quantum query complexity are equivalent measures. In particular, quantum algorithms can be designed based on span programs.