Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 67 / 2 / e / Solution

For a Boolean function $f$, its block sensitivity at $x$ is the maximum number of pairwise disjoint nonempty index sets $B_j$ such that flipping all bits in each $B_j$ individually changes $f(x)$. Maximize over $x$ to obtain $\mathrm{bs}(f)$.

Choose $x=(110)(110)\cdots (110)$, so every three-bit block has majority one and the conjunction is one. In each block, flipping either of its two one bits alone changes that block's majority to zero and therefore changes the conjunction to zero. These $2n$ singleton index sets are all disjoint, soThe supplied bounded-error quantum query complexity lower bound now givesHere $Q_2$ denotes a fixed two-sided error bound below $1/2$, such as $1/3$.

In fact the block sensitivity is exactly $2n$. At a one-input, selecting two one bits from each triple gives a size-$2n$ certificate: any block flip changing the output must touch it, so disjoint sensitive sets number at most $2n$. At a zero-input, one failing triple contains at least two zero bits; fixing those two zeros certifies zero, so at most two disjoint sensitive sets can change the output. The lower-bound witness above attains $2n$. The certificate complexity of a Boolean function supplies these upper bounds, though only the witness is needed for the requested result.