Multi-query Quantum Sums

Abstract

Parity is the problem of determining the parity of a string $f$ of $n$ bits given access to an oracle that responds to a query $x\in \{0,1,\ldots ,n-1\}$ with the $x^\mathrm{th}$ bit of the string, $f(x)$. Classically, $n$ queries are required to succeed with probability greater than $1/2$ (assuming equal prior probabilities for all length $n$ bitstrings), but only $\lceil n/2\rceil$ quantum queries suffice to determine the parity with probability $1$. We consider a generalization to strings $f$ of $n$ elements of ${{\mathbb {Z}}}_k$ and the problem of determining $\sum f(x)$. By constructing an explicit algorithm, we show that $n-r$ ($n\ge r\in {\mathbb {N}}$) entangled quantum queries suffice to compute the sum correctly with worst case probability $\min \{\lfloor n/r\rfloor /k,1\}$. This quantum algorithm utilizes the $n-r$ queries sequentially and adaptively, like Grover’s algorithm, but in a different way that is not amplitude amplification.