Quantum Simulation of non-Abelian Lattice Gauge Theories: a variational approach to $\mathbb{D}_8$

Abstract

In this work, we address the problem of a resource-efficient formulation of non-Abelian LGTs by focusing on the difficulty of simulating fermionic degrees of freedom and the Hilbert space redundancy. First, we show a procedure that removes the matter and improves the efficiency of the hardware resources. We demonstrate it for the simplest non-Abelian group addressable with this procedure, $\mathbb{D}_8$, both in the cases of one (1D) and two (2D) spatial dimensions. Then, with the objective of running a variational quantum simulation on real quantum hardware, we map the $\mathbb{D}_8$ lattice gauge theory onto qudit systems with local interactions. We propose a variational scheme for the qudit system with a local Hamiltonian, which can be implemented on a universal qudit quantum device as the one developed in $\href{https://doi.org/10.1038/s41567-022-01658-0}{[Nat. Phys. 18, 1053 (2022)]}$. Our results show the effectiveness of the matter-removing procedure, solving the redundancy problem and reducing the amount of quantum resources. This can serve as a way of simulating lattice gauge theories in high spatial dimensions, with non-Abelian gauge groups, and including dynamical fermions.

Figures (11)

• Figure 1: Illustration of gauge-invariant operators in the $d=2$ Hamiltonian formulation of LGT. Squares represent matter sites while circles are the gauge fields on the links. The orange cross depicts a local gauge transformation as described in Eq. \ref{['eqn:Gauss_D8']}. The red circle is an example of the electric Hamiltonian term we have on all the links; this come from $H_E$. The blue plaquette stands for the magnetic Hamiltonian term $H_B$ and the green square is an example of the local mass term $H_M$. The purple horizontal line represents a hopping interaction $H_{GM}$.
• Figure 2: Decomposition of the link hilbert space into a qubit and a qudit.
• Figure 3: 1D model, on the top the original model, where the gauge field is represented in blue, and the matter field in black. The lower part shows the transformed model, now featuring only the links, denoted as $c$, $l$, and $r$, corresponding to the center, left, and right, respectively. These labels will be used in subsequent references to indicate the specific link under consideration.
• Figure 4: Individual layer of the parametrized quantum circuit for the 1D system: Each layer of the parametrized quantum circuit used to find the approximate ground state of the 1D system with non-Abelian $\mathbb{D}_8$-symmetry consists of two-level single-qudit rotations on ququarts, entangling two-body MS gates between neighbouring qubits, non-local MS gates between a qubit and a ququart, and single-qubit Pauli-$X$ and Pauli-$Y$ rotations. Each gate in the variational circuit is parametrized by a variational parameter $\theta$, chosen to be different in each layer. The software used for drawing the quantum circuits is taken from drawio_library.
• Figure 5: VarQITE for the 1D system: (a) The energy of the variational state $\ket{\psi(\boldsymbol{\theta)}}$ as a function of the imaginary time for $N = 1,2,3,4,5$ number of layers. The dashed black line shows the numerical value of the true ground state, whereas the cyan line the one of the numerical first excited state. Note that the latter is not gauge invariant and depends on the coefficient $\lambda$ of a penalty term introduced to the Hamiltonian, see Section \ref{['sec:simulations_1D']} (here, $\lambda = 0.1$). After initial fast convergence to low energies, the secondary slow optimization is primarily due to the penalty for Gauss' law violation. From the inset, one can deduce that after $\tau = 5$, the optimization practically stops, yielding a relative error $\sim1\%$ for $N = 5$ layers. (b) The fidelity of the variational state w.r.t. the true ground state from exact diagonalization as a function of the imaginary time $\tau$. As for the energy, the optimization practically stops after $\tau = 5$, leading to a fidelity of $\sim99\%$ for $N = 5$ layers.