Enhancing Variational Quantum Eigensolvers for SU(2) Lattice Gauge Theory via Systematic State Preparation

Abstract

Computing the vacuum and energy spectrum in non-Abelian, interacting lattice gauge theories remains an open challenge, in part because approximating the continuum limit requires large lattices and huge Hilbert spaces. To address this difficulty with near-term quantum computing devices, we adapt the variational quantum eigensolver to non-Abelian gauge theories. We outline scaling advantages when using a spin-network basis to simulate the gauge-invariant Hilbert space and develop a systematic state preparation ansatz that creates gauge-invariant excitations while alleviating the barren plateau problem. We illustrate our method in the context of SU(2) Yang-Mills theory by testing it on a minimal toy model consisting of a single vertex in 3+1 dimensions. In this toy model, simulations allow us to investigate the impact of noise expected in current quantum devices.

Figures (15)

• Figure 1: (a) Schematic representation of an LGT system. A Gauss constraint $G_J$ enforces physical relations between the quantum numbers $j_e$ of edges $e$ meeting at vertex $v$. The electric part $H_E$ and the magnetic part $H_B$ of the Kogut-Susskind Hamiltonian act on edges and plaquettes, respectively. Physical states may feature multiple excitations $j_e\in\mathbb{N}/2$ on the same edge and can be created by exciting various (potentially overlapping) plaquettes of the lattice. (b) The usual enables short and dense gate layers but requires large optimization parameter count. (c) The physics-informed SSP ansatz preserves gauge-invariant excitations (shown for two-edge-plaquettes relevant for the toy model of \ref{['Sec5:ToyModel']}), thereby reducing the optimization parameter count at the expense of more complicated controlled multi-qubit gates.
• Figure 2: Graphical representation of the toy model. (a) A single vertex in 3D with periodic boundary conditions, i.e., in- and outgoing irreps in each direction carry the same label. Projecting to the gauge-invariant Hilbert space is accomplished by choosing a basis of intertwiners $(\pi_+,\pi_o,\pi_-)$ in (b). (c) We choose a discretization of the Hamiltonian approximating continuum Yang-Mills by attaching plaquettes such that $\pi_o$ is not changed by action of $H_B$, hence separating the graph into superselection sectors. (d) Focusing on the sector $\pi_o=0$ reduces the single-vertex graph to the $\Theta$-graph.
• Figure 3: Comparison of HEA (grey dots) and SSP (green crosses). Achieving a given infidelity $\bar{f}$ (averaged over all $\lambda$) requires deeper circuits with more parameters and implies more function evaluations $M_{\rm eval}$, i.e., calls to the quantum computer. For both ansätze, a fit (linear on the log-log scale) has been added as $M_{\rm eval}(\bar{f})=a\; \exp(b \bar{f})$ with fit parameters $(a,b)=(-0.5, 9)$ for HEA and $(a,b)=(-0.6,6.7)$ for SSP respectively.
• Figure 4: Best achieved minima for the energy expectation values for SSP3, a particular variant of the SSP circuit which creates $n=3$ gauge-invariant plaquette excitations. The has been emulated in an ideal environment (green) and in the presence of noise from a 0.5% 2-qubit gate error (red). Upon adding error mitigation (EM) protocols (orange), the accuracy comes close to the exact energies (blue) and falls within the uncertainty given by the finite cut-off $j_{max}=3/2$ (shaded region).
• Figure 5: 2-point correlator evaluated on $\psi(\vec{\theta}^*)$ prepared with the best optimization parameters found in \ref{['fig:energy_exp_val']}. Green triangles show ideal values, and orange points show error-mitigated and post-selected results. Data points indicate median and error bars indicate standard deviation centered around the mean. In blue is the exact correlator shown with a shaded uncertainty region due to the finite cut-off $j_{max}=3/2$.