2 research outputs found
Superposition as memory: unlocking quantum automatic complexity
Imagine a lock with two states, "locked" and "unlocked", which may be
manipulated using two operations, called 0 and 1. Moreover, the only way to
(with certainty) unlock using four operations is to do them in the sequence
0011, i.e., where . In this scenario one might think that the
lock needs to be in certain further states after each operation, so that there
is some memory of what has been done so far. Here we show that this memory can
be entirely encoded in superpositions of the two basic states "locked" and
"unlocked", where, as dictated by quantum mechanics, the operations are given
by unitary matrices. Moreover, we show using the Jordan--Schur lemma that a
similar lock is not possible for .
We define the semi-classical quantum automatic complexity of a
word as the infimum in lexicographic order of those pairs of nonnegative
integers such that there is a subgroup of the projective unitary
group PU with and with such that, in terms of a
standard basis and with , we have
and for all with . We show that is
unbounded and not constant for strings of a given length. In particular, and
.Comment: Lecture Notes in Computer Science, UCNC (Unconventional Computation
and Natural Computation) 201