Improved Hamiltonians for Quantum Simulations of Gauge Theories

Quantum simulations of lattice gauge theories for the foreseeable future will be hampered by limited resources. The historical success of improved lattice actions in classical simulations strongly suggests that Hamiltonians with improved discretization errors will reduce quantum resources, i.e., require $\gtrsim 2^d$ fewer qubits in quantum simulations for lattices with $d$-spatial dimensions. In this work, we consider $\mathcal{O}(a^2)$-improved Hamiltonians for pure gauge theories and design the corresponding quantum circuits for its real-time evolution in terms of primitive gates. An explicit demonstration for ${\mathbb{Z}}_2$ gauge theory is presented including exploratory tests using the ibm_perth device.

Figure 1

3D lattice with example contributions to $H_I$—the plaquette $P_{xy}$, rectangles $R_{yz}$ and $R_{zx}$, and the bent loop $C_{xyz}$—and the two links $U_1$ and $U_2$ used for $K_{2L}$.

Figure 2

Quantum circuits for the time evolution of ${\widehat{H}}_I$.

Figure 3

${\mathcal{U}}_{V_{\mathrm{rect}}}$ (a) and ${\mathcal{U}}_{K_{2L}}$ (b) for ${\mathbb{Z}}_2$ gauge theory. (c) Link-to-qubit map on ibm_perth.

Figure 4

${\mathcal{U}}_{\mathrm{circ}}^{|n⟩}$ for studying the errors of ${\mathcal{U}}_{V_{\mathrm{rect}}}$.

Figure 5

Probability of measuring Hamming weights for selected $|{\mathrm{\Psi }}_n⟩$ compared to the noise-dominated results. In the noiseless limit, $P(w_H)={\delta }_{w_H,0}$ for all $|{\mathrm{\Psi }}_n⟩$.