  Fingerprinting techniques are widely used for localization because of their
accuracy, especially in the presence of wireless channel noise. However, the
fingerprinting techniques require significant storage and running time, which
is a concern when implementing such systems on a global worldwide scale.
  In this paper, we propose an efficient quantum Euclidean similarity algorithm
for wireless localization systems. The proposed quantum algorithm offers
exponentially improved complexity compared to its classical counterpart and
even the state-of-the-art quantum localization systems, in terms of both
storage space and running time. The basic idea is to entangle the test received
signal strength (RSS) vector with the fingerprint vectors at different
locations and perform the similarity calculation in parallel to all fingerprint
locations. We give the details of how to construct the quantum fingerprint, how
to encode the RSS measurements in quantum particles, and finally; present the
quantum algorithm for calculating the Euclidean similarity between the online
RSS measurements and the fingerprint ones.
  Implementation and evaluation of our algorithm in a real testbed using a real
IBM quantum machine as well as a simulation for a larger testbed confirm its
ability to correctly obtain the estimated location with an exponential
enhancement in both time and space compared to the traditional classical
fingerprinting techniques and the state-of-the-art quantum localization
techniques.
