Physicality oracle for SU(3) Loop-String-Hadron dynamics: a digital quantum circuit

TL;DR

The paper tackles real-time simulation of SU$(3)$ lattice gauge theory with fermions by adopting the Loop-String-Hadron (LSH) formulation in 1+1D, which encodes gauge-invariant degrees of freedom on-site and automatically satisfies the non-Abelian Gauss laws. A quantum oracle is proposed to enforce the remaining Abelian Gauss law between neighboring sites, using adder and comparator circuits to compute flux balance and flipping a flag qubit when the law holds, with full uncomputation to preserve the quantum state. The authors compare the qubit-resource demands of the KS irrep basis with the LSH basis, showing a substantial reduction in qubits per site to $2N+3$ for LSH (with flux cutoff $\, abla=2^N$), and provide detailed gate-count and depth analyses for the oracle, including an explicit depth formula of $30N-38$ for $N\ge3$. They also discuss extensions to qudit architectures and higher dimensions, arguing that LSH offers a practical, scalable path toward quantum simulations of lattice QCD and related gauge theories, with potential applications in error mitigation and quantum algorithms for gauge invariance preservation.

Abstract

Within the aim of understanding quantum chromodynamics through simulation, an increasingly studied approach is that of quantum computation and simulation. Challenges exist in encoding the minimal and physical degrees of freedom for a non-Abelian gauge theory and maintaining physical or gauge-invariant dynamics in a simulation. In this work, the Loop-String-Hadron (LSH) formulation of the 1+1-dimensional SU(3) lattice gauge theory is used to define an efficient mapping of SU(3) invariant degrees of freedom onto qubits. It is shown that the required number of qubits in the LSH basis is significantly reduced compared to its IRREP basis counterpart. While the non-Abelian Gauss laws of the SU(3) theory are automatically satisfied by the usage of LSH variables, the remnant constraints on the consistency of the flux numbers still exist. During time evolution, the noise can accumulate and take the state out of the sector of the Hilbert space where the constraint is satisfied. With this motivation, an oracle algorithm is constructed to be applied to the qubits for checking the local constraint at a given link. Costs in terms of qubit and gate number, and circuit depth, are found.

Figures (10)

• Figure 1: Kogut-Susskind variables for a 1-dimensional lattice
• Figure 2: Kogut-Susskind SU(2) rigid rotator variables
• Figure 3: SU(2) Schwinger bosons - variables are now only on the link ends, with none on the entire link itself.
• Figure 4: SU(3) Schwinger bosons. There are two triplets at each site as there is a fundamental and an anti-fundamental representation, each of dimension three, at every link end. These are not the proper IRREP degrees of freedom, but their appropriate combinations provide them.
• Figure 5: Illustration of how the two kinds of flux, $P$ (arrow to the right) and $Q$ (arrows to the left), can be generated on either side of the site by certain combinations of fermionic numbers $\nu_{\underline 1}$, $\nu_0$ and $\nu_1$. The bosonic quantum numbers $n_P$ and $n_Q$ define the number of flux lines that pass through the site and get added to the ones generated at the site by the fermions. As an example they are given as 2 and 3 here.