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.
We apply this measure to the Ising spin chain – a series of magnetically interacting spins that can each be aligned in one of two directions. The classical statistical complexity of an Ising spin chain only ever increases with temperature. On the other hand, the quantum statistical complexity rises to a maximum at some finite temperature then tends back towards zero for higher temperatures. This difference in the qualitative behaviour of complexity measures shows us that when we also consider quantum perspectives, our conclusions about “What is complex?” may be drastically changed.
 C. R. Shalizi et al., Ph.D. thesis, University of Wisconsin-Madison (2001).
 C. R. Shalizi and K. L. Shalizi, in Proceedings of the 20th conference on Uncertainty in artificial intelligence (AUAI Press, 2004), pp. 504-511.
 D. P. Feldman, Ph.D. thesis, University of California, Davis (1998).
 J. M. Yeomans, Statistical mechanics of phase transitions (Clarendon Press, 1992).
 M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information (Cambridge university press, 2010).
 C. Aghamohammadi, J. R. Mahoney, and J. P. Crutchfield, "Extreme Quantum Advantage when Simulating Strongly Coupled Classical Systems", arXiv:1609.03650 (2016).
 Cina Aghamohammadi, John R. Mahoney, and James P. Crutchfield, "The Ambiguity of Simplicity", arXiv:1602.08646 (2016).
 Andrew J. P. Garner, Jayne Thompson, Vlatko Vedral, and Mile Gu, "Thermodynamics of complexity and pattern manipulation", Physical Review E 95 4, 042140 (2017).
 J. R. Mahoney, C. Aghamohammadi, and J. P. Crutchfield, "Classical and Quantum Factors of Channels", arXiv:1709.08101 (2017).
 Andrew J. P. Garner, Jayne Thompson, Vlatko Vedral, and Mile Gu, "Thermodynamics of complexity and pattern manipulation", arXiv:1510.00010 (2015).
 Cina Aghamohammadi, John R. Mahoney, and James P. Crutchfield, "The ambiguity of simplicity in quantum and classical simulation", Physics Letters A 381 14, 1223 (2017).
 Jayne Thompson, Andrew J. P. Garner, Vlatko Vedral, and Mile Gu, "Using quantum theory to simplify input-output processes", npj Quantum Information 3, 6 (2017).
 Paul M. Riechers, John R. Mahoney, Cina Aghamohammadi, and James P. Crutchfield, "Minimized state complexity of quantum-encoded cryptic processes", Physical Review A 93 5, 052317 (2016).
 Thomas J. Elliott and Mile Gu, "Superior memory efficiency of quantum devices for the simulation of continuous-time stochastic processes", npj Quantum Information 4 1, 18 (2018).
 Thomas J Elliott, Andrew J P Garner, and Mile Gu, "Memory-efficient tracking of complex temporal and symbolic dynamics with quantum simulators", New Journal of Physics 21 1, 013021 (2019).
 Jayne Thompson, Andrew J. P. Garner, John R. Mahoney, James P. Crutchfield, Vlatko Vedral, and Mile Gu, "Causal Asymmetry in a Quantum World", Physical Review X 8 3, 031013 (2018).
 Andrew J P Garner, Qing Liu, Jayne Thompson, Vlatko Vedral, and mile Gu, "Provably unbounded memory advantage in stochastic simulation using quantum mechanics", New Journal of Physics 19 10, 103009 (2017).
 Felix C. Binder, Jayne Thompson, and Mile Gu, "Practical Unitary Simulator for Non-Markovian Complex Processes", Physical Review Letters 120 24, 240502 (2018).
 Chengran Yang, Felix C. Binder, Varun Narasimhachar, and Mile Gu, "Matrix Product States for Quantum Stochastic Modeling", Physical Review Letters 121 26, 260602 (2018).
The above citations are from Crossref's cited-by service (last updated 2019-02-20 14:52:11) and SAO/NASA ADS (last updated 2019-02-20 14:52:12). The list may be incomplete as not all publishers provide suitable and complete citation data.
This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.