Quantum Algorithms for Fermionic Quantum Field Theories

TL;DR

This work tackles the challenge of simulating fermionic quantum field theories on quantum computers by extending prior bosonic methods to the two-dimensional Gross-Neveu model. It presents a detailed digital quantum algorithm with six main steps—vacuum preparation, adiabatic turn-ons, wavepacket excitation, real-time evolution, and two measurement strategies—along with a rigorous complexity analysis showing polynomial scaling in precision and energy. Key contributions include a Bravyi-Kitaev fermion-encoding scheme, a Suzuki–Trotter-based time-evolution framework, and practical state-preparation and measurement protocols (including local-charge observables) that address lattice artifacts and mass renormalization. The results advance the prospect of efficiently simulating dynamical quantum field processes and move toward the broader goal of quantum-computational access to the Standard Model.

Abstract

Extending previous work on scalar field theories, we develop a quantum algorithm to compute relativistic scattering amplitudes in fermionic field theories, exemplified by the massive Gross-Neveu model, a theory in two spacetime dimensions with quartic interactions. The algorithm introduces new techniques to meet the additional challenges posed by the characteristics of fermionic fields, and its run time is polynomial in the desired precision and the energy. Thus, it constitutes further progress towards an efficient quantum algorithm for simulating the Standard Model of particle physics.

Figures (2)

• Figure 1: Vertices represent elements of $\{0,1\} \times \Omega$ two vertices are connected by an edge if $H_{\mathrm{nn}}$ couples these sites. (Different species are never coupled by $H_{\mathrm{nn}}$, so the full graph with vertices corresponding to elements of $\{1,\ldots,N\} \times \{0,1\} \times \Omega$ would consist of $N$ disconnected copies of the graph shown.) The edges can be colored with four colors such that each node has no more than one incident edge of each color. One can obtain the decomposition $H_{\mathrm{nn}} = H_1 + H_2 + H_3 + H_4$ by choosing $H_1$ to be the sum of all interaction terms along the edges labeled 1 (which are blue), $H_2$ to be the sum of all the interaction terms along edges labeled 2 (which are red), and so on.
• Figure 2: Our perturbative calculations of the physical mass in the massive Gross-Neveu model indicate a phase diagram with the qualitative features illustrated above. The phase above the dashed curve is accessible adiabatically from the free theory but the phase below is not. The arrow depicts our linear adiabatic path, described in (\ref{['deltam']}). Our perturbative analysis shows that the first two derivatives of the phase transition curve with respect to $g_0^2$ are both positive and diverge only as $\mathrm{poly}(\log(m_0 a))$ in the limit $a \to 0$.

Theorems & Definitions (1)

• Theorem 1