This paper shows how to set up and count the quantum resources needed to simulate Quantum Chromodynamics (QCD) — the SU(3) gauge theory of quarks and gluons — in full 3+1 dimensions on a lattice by choosing the axial gauge. The axial gauge fixes one component of the gauge field so that the usual Gauss’s-law constraints do not have to be enforced at every lattice site. That simplification lets the authors write a Hamiltonian with only independent dynamical fields and to follow their real-time evolution in a way that is straightforward to implement on a quantum computer.

Concretely, the authors start from a lattice Hamiltonian in the axial gauge and solve the temporal component of the gauge field analytically in terms of the independent spatial components and a lattice-regulated Green’s function. They work in a local field basis for the gauge variables and show how to transform those fields into their canonical conjugate (momentum) basis by local quantum Fourier transforms. The fermion fields are mapped to qubits with a standard Jordan–Wigner transformation. The paper assumes a cubic lattice with Dirichlet boundary conditions and describes how the Green’s function singularity is regulated on that lattice.

One of the main technical results is a provable upper bound on the number of qubits needed to represent all states up to an energy E with accuracy ε on a lattice of volume V at bare coupling g. The bound is written as 16 n_A V + 12 n_f V, where n_A is the number of qubits used for each independent gauge field degree of freedom per site and n_f is the number of fermion flavors. The authors give an approximate expression for n_A that grows only logarithmically with combinations of E, V, ε and g, so the total qubit count rises only polynomially with those physical parameters.

They also give a constructive quantum algorithm for time evolution. It combines Trotterization (a standard way to break evolution into short steps), local quantum Fourier transforms, and the Jordan–Wigner mapping. The paper shows how to build the needed quantum circuits under an arbitrary truncation and digitization of the gauge fields. For the gate cost per Trotter step (counting CNOTs and single-qubit rotations), they find a scaling of order n_A^4 V^{4/3} plus a term of order V^{5/3}, for fixed fermion flavor number n_f ≤ 6. Taken together, these results say that the quantum resources scale polynomially with volume, energy, simulation time, target accuracy, and the bare Hamiltonian parameters.