The classical-quantum divergence of complexity in modelling spin chains

Whei Yeap Suen; Jayne Thompson; Andrew J. P. Garner; Vlatko Vedral; Mile Gu

doi:10.22331/q-2017-08-11-25

The classical-quantum divergence of complexity in modelling spin chains

Whei Yeap Suen¹, Jayne Thompson¹, Andrew J. P. Garner¹, Vlatko Vedral^1,2,3, and Mile Gu^1,4,5

¹Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, 117543, Singapore
²Atomic and Laser Physics, University of Oxford, Clarendon Laboratory, Parks Road, Oxford, OX1 3PU, United Kingdom
³Department of Physics, National University of Singapore, 3 Science Drive 2, Singapore 117543
⁴School of Mathematical and Physical Scieces, Nanyang Technological University, Singapore
⁵Complexity Institute, Nanyang Technological University, Singapore

Published:	2017-08-11, volume 1, page 25
Eprint:	arXiv:1511.05738v4
Doi:	https://doi.org/10.22331/q-2017-08-11-25
Citation:	Quantum 1, 25 (2017).

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Abstract

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.

Featured image: Plot of classical and quantum statistical complexity for various temperature of the Ising spin chain. Whereas the classical statistical complexity monotonically increases with temperature, its quantum counterpart attains maximal value for finite T.

Popular summary

Can quantum information fundamentally change the way we perceive what is complex? Statistical complexity quantifies the minimal information we must store about a process to simulate its future behaviour using classical information processing. Here, we construct a quantum variant of this measure, which also allows for simulation using quantum mechanical systems. The resulting complexity measure – quantum statistical complexity – exhibits drastically different qualitative behaviour than its classical counterpart.

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.

► BibTeX data

@article{Suen2017classicalquantum,
  doi = {10.22331/q-2017-08-11-25},
  url = {https://doi.org/10.22331/q-2017-08-11-25},
  title = {The classical-quantum divergence of complexity in modelling spin chains},
  author = {Suen, Whei Yeap and Thompson, Jayne and Garner, Andrew J. P. and Vedral, Vlatko and Gu, Mile},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {1},
  pages = {25},
  month = aug,
  year = {2017}
}

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Cited by

[1] Thomas J. Elliott, Chengran Yang, Felix C. Binder, Andrew J. P. Garner, Jayne Thompson, and Mile Gu, "Extreme Dimensionality Reduction with Quantum Modeling", Physical Review Letters 125 26, 260501 (2020).

[2] 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).

[3] 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).

[4] Felix C. Binder, Jayne Thompson, and Mile Gu, "Practical Unitary Simulator for Non-Markovian Complex Processes", Physical Review Letters 120 24, 240502 (2018).

[5] Qing Liu, Thomas J. Elliott, Felix C. Binder, Carlo Di Franco, and Mile Gu, "Optimal stochastic modeling with unitary quantum dynamics", Physical Review A 99 6, 062110 (2019).

[6] Chengran Yang, Felix C. Binder, Varun Narasimhachar, and Mile Gu, "Matrix Product States for Quantum Stochastic Modeling", Physical Review Letters 121 26, 260602 (2018).

[7] Thomas J. Elliott, Mile Gu, Andrew J. P. Garner, and Jayne Thompson, "Quantum Adaptive Agents with Efficient Long-Term Memories", Physical Review X 12 1, 011007 (2022).

[8] Andrew J. P. Garner, "The fundamental thermodynamic bounds on finite models", Chaos: An Interdisciplinary Journal of Nonlinear Science 31 6, 063131 (2021).

[9] 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).

[10] 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).

[11] Whei Yeap Suen, Thomas J. Elliott, Jayne Thompson, Andrew J. P. Garner, John R. Mahoney, Vlatko Vedral, and Mile Gu, "Surveying Structural Complexity in Quantum Many-Body Systems", Journal of Statistical Physics 187 1, 4 (2022).

[12] Farzad Ghafari, Nora Tischler, Jayne Thompson, Mile Gu, Lynden K. Shalm, Varun B. Verma, Sae Woo Nam, Raj B. Patel, Howard M. Wiseman, and Geoff J. Pryde, "Dimensional Quantum Memory Advantage in the Simulation of Stochastic Processes", Physical Review X 9 4, 041013 (2019).

[13] Matthew Ho, Mile Gu, and Thomas J. Elliott, "Robust inference of memory structure for efficient quantum modeling of stochastic processes", Physical Review A 101 3, 032327 (2020).

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The above citations are from Crossref's cited-by service (last updated successfully 2023-06-05 13:16:01) and SAO/NASA ADS (last updated successfully 2023-06-05 13:16:02). The list may be incomplete as not all publishers provide suitable and complete citation data.

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