Imaginary Hamiltonian variational <i>Ansatz</i> for combinatorial optimization problems

Obtaining exact solutions to combinatorial optimization problems using classical computing is computationally expensive. The current tenet in the field is that quantum computers can address these problems more efficiently. While promising algorithms require fault-tolerant quantum hardware, variational algorithms have emerged as viable candidates for near-term devices. The success of these algorithms hinges on multiple factors, with the design of the Ansatz being of the utmost importance. It is known that popular approaches such as the quantum approximate optimization algorithm (QAOA) and quantum annealing suffer from adiabatic bottlenecks, which lead to either larger circuit depth or evolution time. On the other hand, the evolution time of imaginary-time evolution is bounded by the inverse energy gap of the Hamiltonian, which is constant for most noncritical physical systems. In this work we propose an imaginary Hamiltonian variational Ansatz ($i\mathrm{HVA}$) inspired by quantum imaginary-time evolution to solve the MaxCut problem. We introduce a tree arrangement of the parametrized quantum gates, enabling the exact solution of arbitrary tree graphs using the one-round $i\mathrm{HVA}$. For randomly generated $D$-regular graphs, we numerically demonstrate that the $i\mathrm{HVA}$ solves the MaxCut problem with a small constant number of rounds and sublinear depth, outperforming the QAOA, which requires rounds increasing with the graph size. Furthermore, our Ansatz solves the MaxCut problem exactly for graphs with up to 24 nodes and $D\ensuremath{\le}5$, whereas only approximate solutions can be derived by the classical near-optimal Goemans-Williamson algorithm. We validate our simulated results with hardware demonstrations on a graph with 67 nodes.