Statistical properties of a quantum cellular automaton

We study a quantum cellular automaton (QCA) whose time evolution is defined using the global transition function of a classical cellular automaton (CA). In order to investigate natural transformations from CAs to QCAs, the present QCA includes the CA with Wolfram’s rules 150 and 105 as special cases. We first compute the time evolution of the QCA and examine its statistical properties. As a basic statistical value, the probability of finding an active cell averaged over spatial-temporal space is introduced, and the difference between the CA and QCA is considered. In addition, it is shown that statistical properties in QCAs are related to the classical trajectory in configuration space.

Figure 1

Time-evolution of a QCA based on a CA with Wolfram’s rule 150. The darkness is proportional to the probability of finding an active cell. The number of cells is five and the initial state is always $\{0,0,1,0,0\}$.

Figure 2

The change of the mean density of active cells in a QCA with $K=5$ and $\theta =0.35764$. The initial states are (a) ${\xi }_0=\{0,0,0,0,0\}$ and (b) ${\xi }_{11}=\{0,1,0,1,1\}$.

Figure 3

The distribution of time-averaged probability of finding the configuration ${\xi }_i$ in a QCA with size $K=5$,7,8,10. The parameter $\theta$ and the initial state are 0.7 and ${\delta }_{11}$ always. The sequences of numbers connected with arrows indicate changes of configurations in a CA.

Figure 4

(a) The time evolution of the probability of finding a configuration ${\xi }_i$. The darkness is proportion to the value of the probability. (b) The same simulation results as in (a), but the order of configurations is rearranged. The relation between the $i\mathrm{th}$ and the $(2^{K-1}+i)\mathrm{th}$ configurations is bit reversal.