Quasi-polynomial Time Approximation of Output Probabilities of Geometrically-local, Shallow Quantum Circuits

@article{Coble2022QuasipolynomialTA,
  title={Quasi-polynomial Time Approximation of Output Probabilities of Geometrically-local, Shallow Quantum Circuits},
  author={Nolan J. Coble and Matthew Coudron},
  journal={2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)},
  year={2022},
  pages={598-609}
}
  • Nolan J. Coble, Matthew Coudron
  • Published 10 December 2020
  • Computer Science
  • 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)
We present a classical algorithm that, for any 3D geometrically-local, polylogarithmic-depth quantum circuit $C$, and any bit string $x\in\{0,1\}^{n}$, can compute the quantity $\vert \langle x\vert C\vert 0^{\otimes n}\rangle\vert ^{2}$ to within any inverse-polynomial additive error in quasi-polynomial time. It is known that it is $\# P$-hard to compute this same quantity to within $2^{-n^{2}}$ additive error [1], [2], and worst-case hardness results for this task date back to [3]. The… 
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