Multi-query quantum sums

TL;DR

This paper generalizes quantum query algorithms from parity to sum problems over rac{k}{n} elements, showing that fewer entangled quantum queries can compute sums with high probability, extending quantum advantage in query complexity.

Contribution

It introduces a new quantum algorithm for sum computation over rac{k}{n} elements, demonstrating reduced query complexity compared to classical methods.

Findings

Quantum algorithms can compute sums with fewer queries than classical methods.

The proposed algorithm uses entangled, sequential, and adaptive queries similar to Grover's approach.

The success probability depends on the ratio rac{n}{r} and the size of the alphabet rac{k}{n}.

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 $x\in\{0,1,...,n-1\}$ with the $x^{\rm th}$ 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 $\Z_k$ and the problem of determining $\sum f(x)$. By constructing an explicit algorithm, we show that $n-r$ ($n\ge r\in\N$) 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.