Beyond qubits: Quantum optimization and lattice gauge theories in a qudit framework

Luca Spagnoli, Alessandro Roggero, Nathan Wiebe

In this work we extend the connection between Quantum Error Correction (QEC) and Lattice Gauge Theories (LGTs) by showing that a $\mathbb{Z}_N$ gauge theory with prime dimension $N$ coupled to dynamical matter can be expressed as a qudit stabilizer code. Using the stabilizer formalism we show how to formulate an exact mapping of the encoded $\mathbb{Z}_N$ gauge theory onto two different bosonic models, uncovering a logical duality generated by error correction itself. From this perspective, quantum error correction provides a unifying language to expose dual descriptions of lattice gauge theories. In addition, we generalize earlier $\mathbb{Z}_2$ constructions on qubits to $\mathbb{Z}_N$ on $N$-level qudits and demonstrate how universal fault-tolerant gates can be implemented via state injection between compatible qudit codes.

Hersh Singh

Martin J. Savage A

Emanuele Gaz, Pavel P. Popov, Guy Pardo +3 more

Quantum simulation of lattice gauge theories (LGTs) provides a powerful framework for understanding nonperturbative phenomena. However, to realize the hope of answering challenging physics questions on near-term devices it is crucial to develop resource-efficient formulations, in particular for non-Abelian LGTs and beyond (1+1) dimensions. In this work, we address this issue by focusing on the difficulty of simulating fermionic degrees of freedom and on mitigating the Hilbert space redundancy, i.e., the presence of exponentially many nonphysical states that do not obey gauge invariance. First, we show a procedure that removes the matter and improves the hardware-resource efficiency. We demonstrate it for the simplest non-Abelian group addressable with this procedure, D 8 , in the cases of both one and two spatial dimensions. Then, with the objective of running a variational quantum simulation on real quantum hardware, we map the D 8 LGT 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. Our results show the effectiveness of the matter-removing procedure, solving the redundancy problem, and reducing the amount of quantum resources. This approach can serve as a way of simulating LGTs in high spatial dimensions, with non-Abelian gauge groups, and including dynamical fermions.

M. H. S. Amin, Neil G. Dickson, Peter Smith

Jacky Jiang, Natalie Klco, Olivia Di Matteo

Recent developments in mapping lattice gauge theories relevant to the Standard Model onto digital quantum computers identify scalable paths with well-defined quantum compilation challenges toward the continuum. As an entry point to these challenges, we address the simulation of SU(2) lattice gauge theory. Using qudit registers to encode the digitized gauge field, we provide quantum resource estimates, in terms of elementary qudit gates, for arbitrarily high local gauge field truncations. We then demonstrate an end-to-end simulation of real-time, qutrit-digitized SU(2) dynamics on a cube. Through optimizing the simulation, we improved circuit decompositions for uniformly controlled qudit rotations, an algorithmic primitive for general applications of quantum computing. The decompositions also apply to mixed-dimensional qudit systems, which we found advantageous for compiling lattice gauge theory simulations. Furthermore, we parallelize the evolution of opposite faces in anticipation of similar opportunities arising in three-dimensional lattice volumes. This work details an ambitious executable for future qudit hardware and attests to the value of codesign strategies between lattice gauge theory simulation and quantum compilation.