Imaginary Hamiltonian variational ansatz for combinatorial optimization problems

TL;DR

In this work, an imaginary Hamiltonian variational ansatz inspired by quantum imaginary time evolution to solve the MaxCut problem is proposed and numerically demonstrated that the iHVA solves the MaxCut problem with a small constant number of rounds and sublinear depth, outperforming QAOA, which requires rounds increasing with the graph size.

Abstract

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 having the utmost importance. It is known that popular approaches such as quantum approximate optimization algorithm (QAOA) and quantum annealing suffer from adiabatic bottlenecks, that 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 non-critical physical systems. In this work, we propose imaginary Hamiltonian variational ansatz ($i$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 $i$HVA. For randomly generated $D$-regular graphs, we numerically demonstrate that the $i$HVA solves the MaxCut problem with a small constant number of rounds and sublinear depth, outperforming QAOA, which requires rounds increasing with the graph size. Furthermore, our ansatz solves MaxCut exactly for graphs with up to 24 nodes and $D \leq 5$, 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.

Figures (17)

• Figure 1: Comparison of $i$HVA and QAOA ansatz finding the ground state of (a) $H_1=-Z$, and (b) $H_2=ZZ$. (a) Gradient descent trajectories of $i$HVA(red) and QAOA(blue) on the Bloch sphere starting at $\theta=0$. (b) The energy landscape of QAOA(left) and $i$HVA(right). The arrows indicate trajectories of the gradient descent.
• Figure 2: (a) Examples of a tree graph (upper row) and a 3-regular graph (lower row). (b) Oriented spanning trees of the tree graph and the 3-regular graph. Each oriented edge connects a child node at its tip and a parent node at its tail. A root node is the topmost node in an oriented tree that has no parent node, and the leaf node does not have child nodes. The height of an oriented tree is the length of the longest downward path from the unique root node to one of the leaf nodes. (c) The MaxCut solution of the tree graph and the 3-regular graph.
• Figure 3: (a) The oriented tree for the tree graph in Fig. \ref{['fig:graph']}(a) and the corresponding tree arrangement of the $ZY$ gates. Each colored rectangle is a $ZY$ exponential $e^{-i\theta_{l,ij}Z_iY_j/2}$ with the value of $\theta_{l,ij}$ shown at the center of the rectangle. This oriented tree has the node $3$ as the root, which is the lowest among all orientations of the tree. (b) The highest-oriented tree of the tree graph has node $0$ as the root node. Its corresponding tree arrangement of $ZY$ gates is illustrated.
• Figure 4: (a) Construction of the $i$HVA-tree solving MaxCut of arbitrary graphs. These three steps give one round $U^{(l)}_{ZY}$ of the $i$HVA-tree. Each two-qubit gate represents one $ZY$ exponential $e^{-i\theta_{l,ij}Z_iY_j/2}$. (b) $i$HVA-tree with two rounds. The first round has two parts that arrange $ZY$ gates by the procedure in (a). The second round is constructed by reversing the orientation of $ZY$ gates to $YZ$.
• Figure 5: Simulated results for the approximation ratio $\alpha$ of 3-regular graphs as a function of $p$ circuit rounds. The ansätze considered here include $i$HVA-tree, $i$HVA-stagger, and ma-QAOA, with results marked by red lower-triangle, green cross, and blue upper-triangle, respectively. Each subplot corresponds to a fixed number of nodes $N$, with $50$ randomly generated 3-regular graphs. The box plot is used to reflect the statistical properties of the $50$ ratios. For $N\leq 14$, approximation ratios achieved by $i$HVA-tree are all close to $1$ as $p\geq 2$.

Theorems & Definitions (3)

• Theorem 1
• Theorem 2
• Lemma 1