# BQP-completeness of scattering in scalar quantum field theory

Stephen P. Jordan1,2, 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

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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.

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