Abstract
Classical stochastic processes can be generated by quantum simulators instead of the more standard classical ones, such as hidden Markov models. One reason for using quantum simulators has recently come to the fore: they generally require less memory than their classical counterparts. Here, we examine this quantum advantage for strongly coupled spin systems—in particular, the Dyson one-dimensional Ising spin chain with variable interaction length. We find that the advantage scales with both interaction range and temperature, growing without bound as interaction range increases. In particular, simulating Dyson’s original spin chain with the most memory-efficient classical algorithm known requires infinite memory, while a quantum simulator requires only finite memory. Thus, quantum systems can very efficiently simulate strongly coupled one-dimensional classical spin systems.
Highlights
The idea of a quantum computer, often attributed to Feynman[1], recognizes that while simulating quantum many-body systems is difficult, it is apparently something that the physical quantum system to be simulated itself accomplishes with ease
There is a variety of classical systems that can be simulated quantally with advantage[14] including thermal states[15], fluid flows[16, 17], electromagnetic fields[18], diffusion processes[19, 20], Burger’s equation[21], and molecular dynamics[22]
Our focus here is the problem of simultaneous generation, the potential quantum advantage therein, and the separation in classical-quantum scaling. [Quantum algorithms for sequential generation have been studied recently[34,35,36]
Summary
It is notoriously hard to find quantum advantage and even harder to prove[79]. We found such an advantage in the realm of stochastic process simulation. We analyzed the N-nearest neighbor Ising spin system and demonstrated that its quantum advantage displays a generic scaling behavior—quadratic in temperature and linear in interaction range. 80 showed that the 2D Ising model with external fields is universal in this sense This suggests that the quantum advantage described here may not be limited to the particular spin system we consider, but might be universal. We studied the cost of exact simulation of stochastic processes Both classical and quantum costs, though, can be very different when approximation is allowed. The price we pay to go from approximate to exact quantum simulation is relatively small
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