  In the decade since 2010, successes in artificial intelligence have been at
the forefront of computer science and technology, and vector space models have
solidified a position at the forefront of artificial intelligence. At the same
time, quantum computers have become much more powerful, and announcements of
major advances are frequently in the news.
  The mathematical techniques underlying both these areas have more in common
than is sometimes realized. Vector spaces took a position at the axiomatic
heart of quantum mechanics in the 1930s, and this adoption was a key motivation
for the derivation of logic and probability from the linear geometry of vector
spaces. Quantum interactions between particles are modelled using the tensor
product, which is also used to express objects and operations in artificial
neural networks.
  This paper describes some of these common mathematical areas, including
examples of how they are used in artificial intelligence (AI), particularly in
automated reasoning and natural language processing (NLP). Techniques discussed
include vector spaces, scalar products, subspaces and implication, orthogonal
projection and negation, dual vectors, density matrices, positive operators,
and tensor products. Application areas include information retrieval,
categorization and implication, modelling word-senses and disambiguation,
inference in knowledge bases, and semantic composition.
  Some of these approaches can potentially be implemented on quantum hardware.
Many of the practical steps in this implementation are in early stages, and
some are already realized. Explaining some of the common mathematical tools can
help researchers in both AI and quantum computing further exploit these
overlaps, recognizing and exploring new directions along the way.
