The quantum query complexity of certification

TL;DR

This paper determines the quantum query complexity for certifying values in balanced NAND formulas, revealing tight bounds that depend on the formula's degree and levels, advancing understanding of quantum query efficiency.

Contribution

It establishes tight bounds for quantum query complexity of certification in balanced NAND formulas, showing the complexity depends on degree and levels, and demonstrates the quantum adversary method's direct sum property.

Findings

Quantum query complexity for 0-certificates is Theta(d^{(k+1)/2})

Quantum query complexity for 1-certificates is Theta(d^{k/2})

Zero-error quantum evaluation complexity is O(d^{(k+1)/2})

Abstract

We study the quantum query complexity of finding a certificate for a d-regular, k-level balanced NAND formula. Up to logarithmic factors, we show that the query complexity is Theta(d^{(k+1)/2}) for 0-certificates, and Theta(d^{k/2}) for 1-certificates. In particular, this shows that the zero-error quantum query complexity of evaluating such formulas is O(d^{(k+1)/2}) (again neglecting a logarithmic factor). Our lower bound relies on the fact that the quantum adversary method obeys a direct sum theorem.