A universal framework for the quantum simulation of Yang-Mills theory

TL;DR

The paper presents a universal framework for quantum simulation of SU($N$) Yang–Mills theories on fault-tolerant quantum computers using the orbifold lattice formulation, yielding a simple truncated Hamiltonian valid for arbitrary $N$, dimensions, and lattice sizes. By representing theories in a common form $\hat{H}=\tfrac{1}{2}\sum_a\hat{p}_a^2+V(\hat{x})$ and employing coordinate-basis truncation together with quantum Fourier transforms, the authors show that scalar QFT, matrix models, and orbifold YM share the same structure, enabling uniform circuit constructions built from CNOT and single-qubit gates. They provide detailed resource estimates for a single Suzuki–Trotter step across these models, highlighting the quartic interaction terms that dominate gate counts and outlining how diagonal kinetic terms can be efficiently implemented in the momentum basis. The work argues that the extended Hilbert space greatly simplifies practical quantum simulations of gauge theories, discusses gauge-symmetry truncation effects, and outlines potential hardware platforms and future directions, including inclusion of fermions. Overall, this framework offers a scalable, model-agnostic path toward first-principles quantum simulations of Yang–Mills dynamics on universal quantum computers.

Abstract

We provide a universal framework for the quantum simulation of SU(N) Yang--Mills theories on fault-tolerant digital quantum computers adopting the orbifold lattice formulation. As warm-up examples, we also consider simple models, including scalar field theory and the Yang--Mills matrix model, to illustrate the universality of our formulation, which shows up in the fact that the truncated Hamiltonian can be expressed in the same simple form for any N, any dimension, and any lattice size, in stark contrast to the popular approach based on the Kogut--Susskind formulation. In all these cases, the truncated Hamiltonian can be programmed on a quantum computer using only standard tools well-established in the field of quantum computation. As a concrete application of this universal framework, we consider Hamiltonian time evolution by Suzuki--Trotter decomposition. This turns out to be a straightforward task due to the simplicity of the truncated Hamiltonian. We also provide a simple circuit structure that contains only CNOT and one-qubit gates, independent of the details of the theory investigated.

Figures (7)

• Figure 1: Equation \ref{['eq:digitization']} is reproduced in this figure to highlight the schematic construction of the discretized and truncated Hilbert space $\mathcal{H}_a$ of a single boson $a$. The coordinate operator $\hat{x}_a$ can act on a limited number of $\Lambda$ different states $|n_a\rangle$ with discretized eigenvalues $x_{a, n_a}$, labeled by an integer $n_a$. A pictorial shaded profile represents a possible wavefunction realization for this single bosonic degree of freedom.
• Figure 2: Visual representation of $\hat{Z}_{j,\vec{n}} \hat{\bar{Z}}_{j,\vec{n}} \hat{Z}_{j,\vec{n}} \hat{\bar{Z}}_{j,\vec{n}}$ (left) and $\hat{Z}_{j,\vec{n}} \hat{\bar{Z}}_{j,\vec{n}} \hat{\bar{Z}}_{j,\vec{n}-\hat{j}}\hat{Z}_{j,\vec{n}-\hat{j}}$ (right). Red lines represent links.
• Figure 3: Visual representation of $\hat{Z}_{j,\vec{n}} \hat{\bar{Z}}_{j,\vec{n}} \hat{Z}_{k,\vec{n}} \hat{\bar{Z}}_{k,\vec{n}}$ (Top, Left), $\hat{Z}_{j,\vec{n}} \hat{\bar{Z}}_{j,\vec{n}} \hat{\bar{Z}}_{k,\vec{n}-\hat{k}}\hat{Z}_{k,\vec{n}-\hat{k}}$ (Top, Right), $\hat{\bar{Z}}_{j,\vec{n}-\hat{j}}\hat{Z}_{j,\vec{n}-\hat{j}}\hat{Z}_{k,\vec{n}} \hat{\bar{Z}}_{k,\vec{n}}$ (Bottom, Left), $\hat{\bar{Z}}_{j,\vec{n}-\hat{j}}\hat{Z}_{j,\vec{n}-\hat{j}} \hat{\bar{Z}}_{k,\vec{n}-\hat{k}}\hat{Z}_{k,\vec{n}-\hat{k}}$ (Bottom, Right). Red lines represent links.
• Figure 4: Circuit demonstrating $\exp\left(-\textrm{i}\theta\hat{\sigma}_z\cdots \hat{\sigma}_z\right)$. Here, $R_Z(\theta)=\exp(-\frac{\mathrm{i}\theta}{2}\hat{\sigma}_z)$.
• Figure 5: Microscopic Quantum Fourier Transform circuit