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Quantum Computation of Scattering in Scalar Quantum Field Theories

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

TL;DR

This work introduces a digital quantum algorithm to compute relativistic scattering amplitudes in massive $\phi^4$ theory on lattices in up to 4 dimensions, achieving polynomial-time scaling in system size, energy, and precision $\epsilon$ while remaining valid at both weak and strong coupling. The approach combines ground-state preparation for the free theory, adiabatic turn-on of interactions to reach the interacting theory, controlled wavepacket preparation, and robust measurement strategies, leveraging phase estimation, Suzuki–Trotter time evolution, and adiabatic state preparation. A thorough analysis addresses discretization errors, continuum and finite-volume limits via effective field theory, and localized detectors to extract observables, providing explicit scaling for weak and strong coupling regimes. The results indicate exponential classical speedups are possible for high-precision or strongly coupled scattering problems, suggesting a path toward quantum simulations of the Standard Model and broader quantum-field-theoretic dynamics, with clear directions for extending to fermions and gauge theories.

Abstract

Quantum field theory provides the framework for the most fundamental physical theories to be confirmed experimentally and has enabled predictions of unprecedented precision. However, calculations of physical observables often require great computational complexity and can generally be performed only when the interaction strength is weak. A full understanding of the foundations and rich consequences of quantum field theory remains an outstanding challenge. We develop a quantum algorithm to compute relativistic scattering amplitudes in massive phi-fourth theory in spacetime of four and fewer dimensions. The algorithm runs in a time that is polynomial in the number of particles, their energy, and the desired precision, and applies at both weak and strong coupling. Thus, it offers exponential speedup over existing classical methods at high precision or strong coupling.

Quantum Computation of Scattering in Scalar Quantum Field Theories

TL;DR

This work introduces a digital quantum algorithm to compute relativistic scattering amplitudes in massive theory on lattices in up to 4 dimensions, achieving polynomial-time scaling in system size, energy, and precision while remaining valid at both weak and strong coupling. The approach combines ground-state preparation for the free theory, adiabatic turn-on of interactions to reach the interacting theory, controlled wavepacket preparation, and robust measurement strategies, leveraging phase estimation, Suzuki–Trotter time evolution, and adiabatic state preparation. A thorough analysis addresses discretization errors, continuum and finite-volume limits via effective field theory, and localized detectors to extract observables, providing explicit scaling for weak and strong coupling regimes. The results indicate exponential classical speedups are possible for high-precision or strongly coupled scattering problems, suggesting a path toward quantum simulations of the Standard Model and broader quantum-field-theoretic dynamics, with clear directions for extending to fermions and gauge theories.

Abstract

Quantum field theory provides the framework for the most fundamental physical theories to be confirmed experimentally and has enabled predictions of unprecedented precision. However, calculations of physical observables often require great computational complexity and can generally be performed only when the interaction strength is weak. A full understanding of the foundations and rich consequences of quantum field theory remains an outstanding challenge. We develop a quantum algorithm to compute relativistic scattering amplitudes in massive phi-fourth theory in spacetime of four and fewer dimensions. The algorithm runs in a time that is polynomial in the number of particles, their energy, and the desired precision, and applies at both weak and strong coupling. Thus, it offers exponential speedup over existing classical methods at high precision or strong coupling.

Paper Structure

This paper contains 39 sections, 5 theorems, 234 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Quantum Field Theory
  4. Lattice $\phi^4$ Theory
  5. Continuum Limit
  6. Phase Transitions
  7. Quantum Computing
  8. Quantum Fourier Transform
  9. Reversible Circuits and Phase Kickback
  10. Phase Estimation
  11. Simulating Hamiltonian Time Evolution
  12. Adiabatic State Preparation
  13. Quantum Algorithm
  14. Description and Discussion
  15. Efficiency
  16. ...and 24 more sections

Key Result

Proposition 1

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

Figures (2)

  • 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.
  • Figure 2: The required number of qubits is shown for $2 \to 4$ scattering as a function of $1/\epsilon$. The insets display the interparticle separation in lattice units ($r/a$) and the number of qubits per site ($n_b$), each as a function of $1/\epsilon$. Our estimate $N$ for the total number of qubits is $6 \times (r/a) \times n_b$. The prefactor six is chosen (somewhat arbitrarily) to provide enough space for four outgoing particles to be well separated, with an extra factor of $1.5$ to allow for the possibility that they are not evenly spaced.

Theorems & Definitions (9)

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