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 ε {0, 1,., n-1} with the xth 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 [n/2] quantum queries suffice to determine the parity with probability 1. We consider a generalization to strings f of n elements of ℤk and the problem of determining Σf(x). By constructing an explicit algorithm, we show that n-r (n ≥ r ε N) entangled quantum queries suffice to compute the sum correctly with worst case probability min{[n/r]/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. © 2014 Springer-Verlag Berlin Heidelberg.