Observing the ambiguity of simplicity via quantum simulations of an Ising spin chain

Stochastic processes are used to model a large number of different phenomena where the quantity of interest varies discretely or continuously through time in a non-predictable fashion [1]. The simulation of stochastic... [ view full abstract ]

Stochastic processes are used to model a large number of different phenomena where the quantity of interest varies discretely or continuously through time in a non-predictable fashion [1]. The simulation of stochastic processes is crucial for understanding natural and social phenomena such as weather patterns and traffic congestion. However, most cases are very complicated, and it is almost impossible to follow all internal interactions and variables if simulators have limited memory. Therefore, realizing a memory-efficient mathematical model is important for simulating large systems.

A key metric for this is the minimum amount of memory the model (and its corresponding physical simulator) must store about the past to simulate the process' future [2,3]. In complexity science, this quantity has been formalized as the statistical complexity C_c [2], and it is regarded as a key quantifier of structure. Remarkably, quantum mechanics allows the simulation of many processes with drastically reduced memory C_q < C_c [4], providing the potential for significantly more memory-efficient simulators. 

In this work [5], we use a photonic quantum information processor (Figure 1) to demonstrate a quantum simulator capable of efficiently simulating a key problem in statistical physics, the one-dimensional Ising spin chain (Figure 2). This system is complex enough for two important questions to arise. The first is how to address imperfections in a real, necessarily non-ideal simulator, allowing us to go ahead with the simulation even in the presence of small deviations from design. The second is whether there is a universal notion of simplicity between systems [6]. Specifically, we experimentally answer the question of whether two models, simulating the same Ising system at two temperatures, T_A and T_B, give the same answer to the question, ''Is configuration A or B more complex?'' (Figure 2). The answer to this question carries fundamental significance for our understanding of the nature of complexity.

[1] J. Duan, An Introduction To Stochastic Dynamics, Vol. 51 (Cambridge University Press, 2015).

[2] Int. J. Theor. Phys. 25, 907 (1986).

[3] Phys. Rev. Lett. 63, 105 (1989).

[4] Nat. Commun. 3, 762 (2012).

[5] arXiv:1711.03661.

[6] Phys. Lett. A. 381, 14 (2017), Pages 1223-1227.