Quantum Computational Structure of $SU(N)$ Scattering

TL;DR

It is shown that for scattering between particles which transform in the fundamental or anti-fundamental representations, all 2-2 scattering amplitudes can be constructed from only three quantum gates, suggesting that scattering in this context is fundamentally governed by the action of ``bit flips'' on the internal quantum numbers.

Abstract

We study scattering of particles which obey an $SU(N)$ global symmetry through the lens of quantum computation and quantum algorithms. We show that for scattering between particles which transform in the fundamental or anti-fundamental representations, i.e. qudits, all 2-2 scattering amplitudes can be constructed from only three quantum gates. Further, for any $N$, all 2-2 scattering channels are shown to emerge from the span of a $\mathbb{Z}_{2}$ algebra, suggesting that scattering in this context is fundamentally governed by the action of ``bit flips'' on the internal quantum numbers. We frame these findings in terms of quantum algorithms constructed from Linear Combinations of Unitaries and block encoding.

Figures (2)

• Figure 1: Space of $SU(N)$-invariant 2-2 amplitudes in the two-operator subspace $\mathrm{span}\{\mathbf{S}_{\mathbb{I}},i\mathbb{Z}\}$. The shaded disk depicts the physically allowed region in any closed unitary sector (e.g. a fixed partial wave or helicity block); the boundary circle marks $|a_J|^2+|b_J |^2=1$. Coefficients $(a(s,\Omega),b(s,\Omega))$ for fixed $s$ may lie outside the disk prior to this projection. The dotted arrow illustrates a representative crossing transformation.
• Figure 2: Block encoding of 2-2 $SU(N)$-invariant scattering amplitudes.