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Quantum Algorithms for Quantum Field Theories

Stephen P. Jordan, Keith S. M. Lee, John Preskill

TL;DR

A quantum algorithm to compute relativistic scattering probabilities in a massive quantum field theory with quartic self-interactions in spacetime of four and fewer dimensions is developed and achieves exponential speedup over the fastest known classical algorithm.

Abstract

Quantum field theory reconciles quantum mechanics and special relativity, and plays a central role in many areas of physics. We develop a quantum algorithm to compute relativistic scattering probabilities in a massive quantum field theory with quartic self-interactions (phi-fourth theory) in spacetime of four and fewer dimensions. Its run time is polynomial in the number of particles, their energy, and the desired precision, and applies at both weak and strong coupling. In the strong-coupling and high-precision regimes, our quantum algorithm achieves exponential speedup over the fastest known classical algorithm.

Quantum Algorithms for Quantum Field Theories

TL;DR

A quantum algorithm to compute relativistic scattering probabilities in a massive quantum field theory with quartic self-interactions in spacetime of four and fewer dimensions is developed and achieves exponential speedup over the fastest known classical algorithm.

Abstract

Quantum field theory reconciles quantum mechanics and special relativity, and plays a central role in many areas of physics. We develop a quantum algorithm to compute relativistic scattering probabilities in a massive quantum field theory with quartic self-interactions (phi-fourth theory) in spacetime of four and fewer dimensions. Its run time is polynomial in the number of particles, their energy, and the desired precision, and applies at both weak and strong coupling. In the strong-coupling and high-precision regimes, our quantum algorithm achieves exponential speedup over the fastest known classical algorithm.

Paper Structure

This paper contains 9 sections, 5 theorems, 88 equations, 1 figure, 2 tables.

Table of Contents

  1. Supplementary Material
  2. Steps of Algorithm and Comments
  3. Efficiency
  4. Mass Renormalization
  5. Representation by Qubits
  6. Adiabatic Preparation of Interacting Wavepackets
  7. Weak Coupling
  8. Strong Coupling
  9. Suzuki-Trotter Formulae for Large Lattices

Key Result

Proposition 1

Let $\hat{p}$ and $\hat{q}$ be Hermitian operators on $L^2(\mathbb{R})$ obeying the canonical commutation relation $[\hat{p},\hat{q}]=i \mathds{1}$. Then the eigenbasis of $\hat{p}$ is the Fourier transform of the eigenbasis of $\hat{q}$.

Figures (1)

  • Figure 1: The dashed line illustrates schematically the location of a quantum phase transition of $\phi^4$ theory in two and three spacetime dimensions. A and B denote weakly and strongly coupled continuum-like theories, respectively. We prepare them adiabatically by following the arrows starting from the massive free theory ($m_0^2 >0$, $\lambda_0 = 0$). To maintain adiabaticity, the path must not cross the quantum phase transition.

Theorems & Definitions (5)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Theorem 1