The minimal memory required to model a given stochastic process-known as the statistical complexity-is a widely adopted quantifier of structure in complexity science. Here, we ask if quantum mechanics can fundamentally change the qualitative behaviour of this measure. We study this question in the context of the classical Ising spin chain. In this system, the statistical complexity is known to grow monotonically with temperature. We evaluate the spin chain's quantum mechanical statistical complexity by explicitly constructing its provably simplest quantum model, and demonstrate that this measure exhibits drastically different behaviour: it rises to a maximum at some finite temperature then tends back towards zero for higher temperatures. This demonstrates how complexity, as captured by the amount of memory required to model a process, can exhibit radically different behaviour when quantum processing is allowed.Statistical complexity emerges from the scientific ideal that we understand reality through cause and effect -the more precisely we can isolate the causes of natural things, the greater our comprehension. In this context, the more causes one must postulate to fully replicate the behaviour of a process, the more complicated that process appears. This motivates the statistical complexity as a measure of the process's intrinsic structure and complexity, since it represents the minimal amount of causal information one must record about past observations of a phenomena to model the statistics of observations made at future times [1].Following these motivations, there has been significant work on developing a complete framework for statistical complexity. There are methods for evaluating the statistical complexity of general stationary stochastic processes [2,3], and for estimating its Whei Yeap Suen: wheiyeap@u.nus.edu Mile Gu: mgu@quantumcomplexity.org value directly from observational statistics [4,5,6]. Meanwhile, the corresponding optimal models can be systematically constructed. If a process has a statistical complexity of C, we can systematically replicate the process's statistical behaviour using a model that records only C bits of information about the past. This analytic tractability, combined with a clear operational motivation, has propelled the use of statistical complexity within complexity science as a key measure of structure. Its field of study -computational mechanics -has been applied to analyse structure in diverse settings [7,8,9,10], an early example being the Ising spin chain [11].Conventional studies only consider building classical models. Yet, recent advances show that even when modelling the same classical process, a quantum model may require less input information [12,13,14,15,16]. This motivates an important question: could statistical complexity exhibit very different qualitative behaviour in the quantum regime? If true, this would imply that many existing studies in statistical complexity could draw very different conclusions when taking quantum information processing into account....
