Quantum simulation of massive Thirring and Gross--Neveu models for arbitrary number of flavors

Abstract

The study of fermionic quantum field theories is an important problem for realizing the standard model of particle physics on a quantum computer. As a step towards this goal, we consider the massive Thirring and Gross--Neveu models with arbitrary number of fermion flavors, $N_f$, discretized on a spatial one-dimensional lattice of size $L$ in the Hamiltonian formulation. We compute the gate complexity using the higher-order product formula and using block-encoding/qubitization and quantum singular value transformations in the limit of large $N_f$ and $L$. We also prepare the ground states of both models with excellent fidelity for system sizes up to 20 qubits with $N_f = 1,2,3,4$ using the adaptive-variational quantum imaginary time algorithm. In addition, we also classify the dynamical Lie algebras of these relativistic fermionic models and show that they belong to the same isomorphism class. Our work is a concrete step towards the quantum simulation of real-time dynamics of large $N_f$ fermionic quantum field theories models relevant for chiral symmetry breaking, understanding dimensional transmutation, and exploring the conformal window of field theories on near-term and early fault-tolerant quantum computers.

Figures (5)

• Figure 1: Five-site lattice ($L=5$) with three flavors ($N_f =3$) of fermions (different colors) at each physical lattice site. Each blob (i.e., Dirac spinor) is represented by two qubits. The lattice spacing is denoted as $a$.
• Figure 2: Convergence of the AVQITE algorithm compared to the exact ground state energy (dashed lines) and fidelity (inset) for two representative examples from the set of simulations detailed in Table \ref{['tab:table_AVQITE']}.
• Figure 3: Normalized correlator defined by Eq. \ref{['eq:Cr_definition']} computed using the AVQITE-prepared ground state $\vert \psi_{\text{AVQITE}}\rangle$ (solid markers), compared to exact results computed using $\vert \psi_{\text{Exact}}\rangle$ (dashed-dotted lines) for the same datasets as considered in Fig. \ref{['fig:AVQITE_results1']}.
• Figure 4: Comparison of asymptotic simulation costs for a $p$-th order product-formula (Trotter) method and QSVT.
• Figure A1: Overlap of the initial (Néel) state $\ket{\Psi_0}=\ket{01}^{\otimes (N_fL)}$ with respect to the exact ground state $\ket{g}$ vs. number of qubits for the (a) Gross-Neveu (GN) and (b) Thirring models. The data points include all systems considered in Table \ref{['tab:table_AVQITE']}, with some additional points to better showcase the scaling trend.