Quantum Logical Depth and Shallowness of Streaming Data by One-Way Quantum Finite-State Transducers (Preliminary Report)

Abstract

The logical depth of a piece of data has served as a complexity measure to describe the amount of “usefulness” and “non-randomness” of information stored in the data itself, which is originated from Kolmogorov complexity. This notion of logical depth has been further expanded to various models of computation in the past literature to accommodate the needs for handling different computational circumstances. We focus on streaming data and streaming algorithms. With the use of one-way (or real-time) quantum finite-state automata equipped with write-once output tapes (called transducers) for recovering the desired information from the incoming compressed data sets, we introduce the notions of quantum finite-state depth and shallowness to capture the usefulness of the streaming data sets. We first layout a general setting of decompression of streaming data and, using its fundamental properties, we then argue the existence of deep and shallow data sets.