From estimation of quantum probabilities to simulation of quantum circuits

Hakop Pashayan; Stephen D. Bartlett; David Gross

doi:10.22331/q-2020-01-13-223

From estimation of quantum probabilities to simulation of quantum circuits

Hakop Pashayan¹, Stephen D. Bartlett¹, and David Gross²

¹Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, Sydney, NSW 2006, Australia
²Institute for Theoretical Physics, University of Cologne, Germany

Published:	2020-01-13, volume 4, page 223
Eprint:	arXiv:1712.02806v3
Doi:	https://doi.org/10.22331/q-2020-01-13-223
Citation:	Quantum 4, 223 (2020).

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Abstract

Investigating the classical simulability of quantum circuits provides a promising avenue towards understanding the computational power of quantum systems. Whether a class of quantum circuits can be efficiently simulated with a probabilistic classical computer, or is provably hard to simulate, depends quite critically on the precise notion of ``classical simulation'' and in particular on the required accuracy. We argue that a notion of classical simulation, which we call EPSILON-simulation (or $\epsilon$-simulation for short), captures the essence of possessing ``equivalent computational power'' as the quantum system it simulates: It is statistically impossible to distinguish an agent with access to an $\epsilon$-simulator from one possessing the simulated quantum system. We relate $\epsilon$-simulation to various alternative notions of simulation predominantly focusing on a simulator we call a $\textit{poly-box}$. A poly-box outputs $1/poly$ precision additive estimates of Born probabilities and marginals. This notion of simulation has gained prominence through a number of recent simulability results. Accepting some plausible computational theoretic assumptions, we show that $\epsilon$-simulation is strictly stronger than a poly-box by showing that IQP circuits and unconditioned magic-state injected Clifford circuits are both hard to $\epsilon$-simulate and yet admit a poly-box. In contrast, we also show that these two notions are equivalent under an additional assumption on the sparsity of the output distribution ($\textit{poly-sparsity}$).

► BibTeX data

@article{Pashayan2020fromestimationof,
  doi = {10.22331/q-2020-01-13-223},
  url = {https://doi.org/10.22331/q-2020-01-13-223},
  title = {From estimation of quantum probabilities to simulation of quantum circuits},
  author = {Pashayan, Hakop and Bartlett, Stephen D. and Gross, David},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {4},
  pages = {223},
  month = jan,
  year = {2020}
}

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