Superposition as Memory: Unlocking Quantum Automatic Complexity

Abstract

We define the semi-classical quantum automatic complexity $Q_{s}(x)$ of a word x as the infimum in lexicographic order of those pairs of nonnegative integers (n, q) such that there is a subgroup G of the projective unitary group ${{\mathrm{PU}}}(n)$ with $|G|\le q$ and with $U_0,U_1\in G$ such that, in terms of a standard basis $\{e_k\}$ and with $U_z=\prod _k U_{z(k)}$, we have $U_x e_1=e_2$ and $U_y e_1 \ne e_2$ for all $y\ne x$ with $|y|=|x|$. We show that $Q_s$ is unbounded and not constant for strings of a given length. In particular,

$$\begin{aligned} Q_{s}(0^21^2)\le (2,12) < (3,1) \le Q_{s}(0^{60}1^{60}) \end{aligned}$$

and $Q_s(0^{120})\le (2,121)$.