Anisotropic Quantum Annealing vs Trit Annealing

TL;DR

This work studies quantum annealing with spin-1 degrees of freedom augmented by a quadratic anisotropy term $D\sum_i (S_i^z)^2$, investigating whether higher-spin processors can outperform classical methods on ternary optimization problems. Using a one-dimensional open chain of length $N=5$, the authors compare anisotropic quantum annealing (AQA) to Trit Annealing (TA) under matched budgets, with the problem Hamiltonian $H_P=-J\sum_{\langle i,j\rangle}S_i^z S_j^z - h\sum_i S_i^z + D\sum_i (S_i^z)^2$ and driver $H_D(t)=-g(t)\sum_i S_i^x$, simulated via a fourth-order Suzuki–Trotter evolution. They find a finite-time quantum advantage for larger transverse scales $e$ and schedules that spend more time at small $g$, particularly in the easy-plane regime ($D>0$) near order-change boundaries, where coherent pathways through multiple avoided crossings can outperform classical relaxation. The results imply that higher-spin annealers can offer intrinsic robustness and flexibility for multivalued optimization, guiding schedule and anisotropy design, though the observed advantage is finite-size and finite-time and warrants scaling studies on larger graphs and topologies.

Abstract

Quantum annealing offers a promising strategy for solving complex optimization problems by encoding the solution into the ground state of a problem Hamiltonian. While most implementations rely on spin-$1/2$ systems, we explore the performance of quantum annealing on a spin-$1$ system where the problem Hamiltonian includes a single ion anisotropy term of the form $D\sum (S^z)^2$. Our results reveal that for a suitable range of the anisotropy strength $D$, the spin-$1$ annealer reaches the ground state with higher fidelity. We attribute this performance to the presence of the intermediate spin level and the tunable anisotropy, which together enable the algorithm to traverse the energy landscape through smaller, incremental steps instead of a single large spin flip. This mechanism effectively lowers barriers in the configuration space and stabilizes the evolution. These findings suggest that higher spin annealers offer intrinsic advantages for robust and flexible quantum optimization, especially for problems naturally formulated with ternary decision variables.

Figures (5)

• Figure 1: AQA versus TA across the coupling--anisotropy plane. Each panel displays the difference $P_{\mathrm{AQA}}-P_{\mathrm{TA}}$ over the $(J,D)$ plane for a fixed transverse field scale $e$ and annealing schedule $g(t)$, at longitudinal field $h=0.2$. Columns (left to right) correspond to schedules with late-time decay $g(t)\propto 1/\log(1{+}t)$, $1/\sqrt{t}$, $1/t$, and $1/t^2$, while rows (top to bottom) correspond to $e=1,5,10,20$. Red (blue) indicates parameter regions where AQA yields higher (lower) success probability than TA, and yellow markers indicate cases with $P_{\mathrm{AQA}}>0.9$. The dashed gray lines at $J=0$ and $D=0$ delineate the easy-plane ($D>0$) and easy-axis ($D<0$) sectors and assist in identifying symmetry-related structures.
• Figure 2: Number of basins versus $D$ along an antiferromagnetic cut ($J=-2$). Stepwise decreases reflect qualitative reorganizations of the set of minima as the anisotropy is varied.
• Figure 3: Dominance of the largest basin versus $D$ along the cut ($J=-2$). Growth of this fraction signals an increasingly funnel-like landscape; values near unity indicate that almost all configurations drain to a single minimum.
• Figure 4: Easy-plane energy landscape along an antiferromagnetic cut. Classical energy $H(s)$ versus the fraction $f(s)$ of spins in the $\pm1$ levels for $J=-2$ and $D=+2$. Light blue points show all configurations; the solid line indicates, for each $f$, the minimum energy over configurations with that $f$; star markers denote one-step local minima (basin roots). The local minima separate into distinct clusters at different $f$, reflecting competing orders and a multi-basin structure in this order-parameter coordinate.
• Figure 5: Easy-axis energy landscape along an antiferromagnetic cut. Classical energy $H(s)$ versus fraction $f(s)$ for $J=-2$ and $D=-2$, with one-step local minima marked by stars. In contrast to the easy-plane case, low energy states and local minima concentrate near $f= 1$ and are less clearly separated in $f$, indicating a comparatively less partitioned structure in this coordinate.