BQP-completeness of scattering in scalar quantum field theory

Stephen P. Jordan; Hari Krovi; Keith S. M. Lee; John Preskill

doi:10.22331/q-2018-01-08-44

BQP-completeness of scattering in scalar quantum field theory

Stephen P. Jordan^1,2, Hari Krovi³, Keith S. M. Lee⁴, and John Preskill⁵

¹National Institute of Standards and Technology, Gaithersburg, MD, USA
²Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, MD, USA
³Quantum Information Processing Group, Raytheon BBN Technologies, Cambridge, MA, USA
⁴Centre for Quantum Information & Quantum Control and Department of Physics, University of Toronto, Toronto, ON, Canada
⁵Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA, USA

Published:	2018-01-08, volume 2, page 44
Eprint:	arXiv:1703.00454v4
Doi:	https://doi.org/10.22331/q-2018-01-08-44
Citation:	Quantum 2, 44 (2018).

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Abstract

Recent work has shown that quantum computers can compute scattering probabilities in massive quantum field theories, with a run time that is polynomial in the number of particles, their energy, and the desired precision. Here we study a closely related quantum field-theoretical problem: estimating the vacuum-to-vacuum transition amplitude, in the presence of spacetime-dependent classical sources, for a massive scalar field theory in (1+1) dimensions. We show that this problem is BQP-hard; in other words, its solution enables one to solve any problem that is solvable in polynomial time by a quantum computer. Hence, the vacuum-to-vacuum amplitude cannot be accurately estimated by any efficient classical algorithm, even if the field theory is very weakly coupled, unless BQP=BPP. Furthermore, the corresponding decision problem can be solved by a quantum computer in a time scaling polynomially with the number of bits needed to specify the classical source fields, and this problem is therefore BQP-complete. Our construction can be regarded as an idealized architecture for a universal quantum computer in a laboratory system described by massive phi^4 theory coupled to classical spacetime-dependent sources.

► BibTeX data

@article{Jordan2018bqpcompletenessof,
  doi = {10.22331/q-2018-01-08-44},
  url = {https://doi.org/10.22331/q-2018-01-08-44},
  title = {{BQP}-completeness of scattering in scalar quantum field theory},
  author = {Jordan, Stephen P. and Krovi, Hari and Lee, Keith S. M. and Preskill, John},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {2},
  pages = {44},
  month = jan,
  year = {2018}
}

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