Gluon field digitization for quantum computers

Simulations of gauge theories on quantum computers require the digitization of continuous field variables. Digitization schemes that use the minimum amount of qubits are desirable. We present a practical scheme for digitizing $SU(3)$ gauge theories via its discrete subgroup $S(1080)$. The $S(1080)$ standard Wilson action cannot be used since a phase transition occurs as the coupling is decreased, well before the scaling regime. We propose a modified action that allows simulations in the scaling window and carry out classical Monte Carlo calculations down to lattice spacings of order $a\approx 0.08\mathrm{fm}$. We compute a set of observables with subpercent precision at multiple lattice spacings and show that the continuum extrapolated value agrees with the full $SU(3)$ results. This suggests that this digitization scheme provides sufficient precision for noisy intermediate-scale quantum era QCD simulations.

Figure 1

Average energy per plaquette, $⟨E_0⟩=1-\mathrm{Re}⟨\mathrm{Tr}{U}_p⟩/3$, vs ${\beta }_0$ on $2^4$ lattice for (filled black square) $SU(3)$ and $S(1080)$ with (filled red circle) ${\beta }_1=0$ and with (filled blue up-pointing triangle) ${\beta }_1=-0.6$. (Inset) ${\chi }_0$ vs ${\beta }_0$ for ${\beta }_1=0$.

Figure 2

Phase diagram in the $({\beta }_0,{\beta }_1)$ plane for a $2^4$ lattice where $⟨E_1⟩=1-\mathrm{Re}⟨\mathrm{Tr}{U}_p^2⟩/3$ is an additional order parameter. (filled blue square) and (filled red circle) correspond to peaks in ${\chi }_0$ and (filled green down-pointing triangle) to peaks in ${\chi }_1$. Also shown as a dashed line is the trajectory used for exploring larger lattices.

Figure 3

(Top) $T_c\sqrt{t_0}$ vs $a^2/t_0$. Our results (blue square) compared to $SU(3)$ results from [61] using Wilson (black up-pointing triangle) and Wilson-improved (red circle) energies are reproduced for comparison. Our extrapolated value (filled blue square) is compared to $SU(3)$ results of [61] (filled red circle) and [62] (filled green down-pointing triangle). (Bottom) $T_c\sqrt{t_{0.2}}$ vs $a^2/t_{0.2}$ and the extrapolation compared with the results of [63] with same symbols.