# BQP-completeness of Scattering in Scalar Quantum Field Theory

## TL;DR

This paper proves that estimating the vacuum-to-vacuum transition amplitude in a scalar quantum field theory is BQP-complete, indicating it is as hard as the hardest problems solvable efficiently by quantum computers, and cannot be efficiently approximated classically.

## Contribution

It establishes the BQP-completeness of a fundamental quantum field theory problem, linking quantum computational complexity with quantum field theory.

## Key findings

- The problem is BQP-hard, implying classical algorithms cannot efficiently estimate it.
- The decision version of the problem is BQP-complete, solvable efficiently by quantum computers.
- This provides an idealized model for a universal quantum computer using scalar field theory.

## 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.