Classical optimization with imaginary time block encoding on quantum computers: The MaxCut problem

TL;DR

The paper addresses solving MaxCut-type combinatorial optimization with quantum computers by introducing ITE-BE, a non-variational, block-encoded imaginary-time evolution method to prepare the ground state of the diagonal cost Hamiltonian $H_C$. Two protocols are studied: purely ITE-BE and QAOA+ITE-BE, evaluated on unweighted 3-regular graphs, showing that imaginary-time evolution increases overlap with the ground state and that shallow QAOA circuits boosted by ITE-BE can outperform deeper QAOA. The authors derive an exact first-order Trotter decomposition for $e^{- au H_C}$ within a block-encoding framework and present a deterministic variant for input $|+ angle|+ angle$, enabling non-post-selected operation. Results indicate that with circuit depth $O(|E|)$ the ITE-BE methods converge to the ground state as $ au$ grows, highlighting a potential pathway to quantum advantage for MaxCut while acknowledging post-selection as a practical bottleneck. The work suggests extending ITE-BE to other combinatorial problems and combining non-unitary evolution with classical optimization strategies to improve performance and scalability.

Abstract

Optimization problems in finance, physics and computer science are typically very hard to tackle in classical computing and quantum computing could help speed up computations and provide efficient methods for tackling large problems. Typically, to treat the problem with a quantum computer, the optimal solution is cast as the ground state of a diagonal Hamiltonian. We develop a new method, called ITE-BE, based on a recent imaginary time algorithm, which requires no variational parameter optimization as all parameters can be derived analytically from the target Hamiltonian. We also demonstrate that our method can be successfully combined with other quantum algorithms such as quantum approximate optimization algorithm (QAOA). For illustration, here we study the MaxCut problem. We find that the QAOA ansatz increases the post-selection success of ITE-BE, and shallow QAOA circuits, when boosted with ITE-BE, achieve better performance than deeper QAOA circuits. For the special case of the transverse initial state, we adapt our block encoding scheme to allow for a deterministic application of the first layer of the circuit.

Figures (7)

• Figure 1: Block-encoding-based circuit implementations for the imaginary-time propagator $e^{-\tau w_{jk}Z_jZ_k/2}$ with (a) a generic input state $|\psi\rangle$ and (b) the input state $|+\rangle_j \otimes |+\rangle_k$. Circuit parameters are given in Eq. \ref{['eq:rbm_para']} and Eq. \ref{['eq:rbm_corr_para']}. These block encodings can be applied for a generic Pauli string propagator $e^{- K \prod_i \sigma_i}$.
• Figure 2: The protocol of QAOA+ITE-BE method with a $p-$level QAOA circuit state preparation for ITE-BE rather than using the equal superposition state $|+\rangle^{\otimes n}$.
• Figure 3: The averaged approximation ratio $r$, probability of obtaining optimal solution $p_{\rm opt}$, and the post-selection success rate for randomly generated unweighed $3$-regular graphs within three standard deviations of the mean for ITE-BE method. Here, the approximation ratio starts from $0.5$ at $\tau=0$, which is expected according to Section 6.2.1 of Ref. prob_and_computing. This method guarantees convergence to the optimal solution, while the post-selection success rate decays exponentially. The convergence becomes slower and the success rate decreases when the problem size $N$ increases.
• Figure 4: The averaged approximation ratio $r$, probability of obtaining optimal solution $p_{\rm opt}$, and the post-selection success rate within three standard deviations of the mean for the QAOA+ITE-BE method. Here we use $p-$level QAOA circuit for state preparation, where $p = 4,6,8,10,12$. The QAOA input state leads to faster convergence of ITE-BE and a post-selection rate that is one order of magnitude higher than using an equal superposition state to solve same graph for $p=12$. As comparison, we show the results from purely ITE-BE method labeled as $p=0$ in above figures, which indicates that the ITE-BE has faster convergence to the optimal solution and nearly $10$ times higher post-selection rate when using $p$-level QAOA input state.
• Figure 5: The averaged performance of pure ITE-BE method on 40 randomly generated bipartite $3-$regular graphs for $N = 6, 8, 10$ and $12$. The ITE-BE method converges faster than non-bipartite graphs and the success rate reaches a plateau after imaginary time $\tau \sim 0.6$.