BQP-completeness of scattering in scalar quantum field theory

TLDR

Quantum computers can compute scattering probabilities in massive quantum field theories in polynomial time. The study investigates estimating the vacuum‑to‑vacuum transition amplitude for a massive scalar field in (1+1) dimensions with spacetime‑dependent classical sources. The authors construct an idealized universal quantum computer architecture using massive φ⁴ theory coupled to classical spacetime‑dependent sources. They prove the vacuum‑to‑vacuum amplitude problem is BQP‑hard and BQP‑complete, implying no efficient classical algorithm can estimate it unless BQP=BPP.

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.