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arXiv:2504.10651 · IEEE International Conference on Quantum Computing and Engineering (QCE)

An Adaptive Weighted Qite-Vqe Algorithm for Combinatorial Optimization Problems

Ningyi Xie, Xinwei Lee, Tiejin Chen, Yoshiyuki Saito, Nobuyoshi Asai, Dongsheng Cail

Published Aug 30, 2025• 1 citation
arXivHTMLPDF

Abstract

The variational quantum eigensolver (VQE) is an algorithm for finding the ground states of a given Hamiltonian. Its application to binary-formulated combinatorial optimization (CO) has been widely studied in recent years. However, typical VQE approaches for CO problems often suffer from local minima or barren plateaus, limiting their ability to achieve optimal solutions. The quantum imaginary time evolution (QITE) offers an alternative approach for effective ground-state preparation but requires large circuits to approximate non-unitary operations. Although compressed QITE (cQITE) reduces circuit depth, accumulated errors eventually cause energy increases. To address these challenges, we propose an Adaptive Weighted QITE-VQE (AWQV) algorithm that integrates the VQE gradients with the cQITE updates through an adaptive weighting scheme during optimization. In numerical simulations for MaxCut on unweighted regular graphs, AWQV achieves near-optimal approximation ratios, while for weighted Erdős-Rényi instances, it outperforms the classical Goemans-Williamson algorithm.

DOI
10.1109/qce65121.2025.00032

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The variational quantum eigensolver (VQE) is an algorithm for finding the ground states of a given Hamiltonian. Its application to binary-formulated combinatorial optimization (CO) has been widely studied in recent years. However, typical VQE approaches for CO problems often suffer from local minima or barren plateaus, limiting their ability to achieve optimal solutions. The quantum imaginary time evolution (QITE) offers an alternative approach for effective ground-state preparation but requires large circuits to approximate non-unitary operations. Although compressed QITE (cQITE) reduces circuit depth, accumulated errors eventually cause energy increases. To address these challenges, we propose an Adaptive Weighted QITE-VQE (AWQV) algorithm that integrates the VQE gradients with the cQITE updates through an adaptive weighting scheme during optimization. In numerical simulations for MaxCut on unweighted regular graphs, AWQV achieves near-optimal approximation ratios, while for weighted Erdős-Rényi instances, it outperforms the classical Goemans-Williamson algorithm.
Solving combinatorial optimization problems using variational quantum algorithms (VQAs) has emerged as a promising research direction. Since the introduction of the Quantum Approximate Optimization Algorithm (QAOA), numerous variants have been proposed to enhance its performance. QAOA was later extended to the Quantum Alternating Operator Ansatz (QAOA+), which generalizes the initial state, phase-separation operator, and mixer to address constrained problems without relying on the standard Quadratic Unconstrained Binary Optimization (QUBO) formulation. However, QAOA+ often requires additional ancilla qubits and a large number of multi-controlled Toffoli gates to prepare the superposition of feasible states, resulting in deep circuits that are challenging for near-term quantum devices. Furthermore, VQAs are generally hindered by issues such as barren plateaus and suboptimal local minima. Recently, Quantum Imaginary Time Evolution (QITE), a groundstate preparation algorithm, has been explored as an alternative to QAOA and its variants. QITE has demonstrated improved performance in quantum chemistry problems and has been applied to unconstrained combinatorial problems such as MaxCut. In this work, we apply the variational form of QITE (VarQITE) to solve the Multiple Knapsack Problem (MKP), a constrained problem, using a Max-Cut-tailored ansatz. To the best of our knowledge, this is the first attempt to address constrained optimization using VarQITE. We show that VarQITE achieves significantly lower mean optimality gaps compared to QAOA and other conventional methods. Moreover, we demonstrate that scaling the Hamiltonian coefficients can further reduce optimization costs and accelerate convergence.
We use quantum imaginary-time evolution (QITE) to solve polynomial unconstrained binary optimization (PUBO) problems. We show that a linear ansatz yields good results for a wide range of PUBO problems, often outperforming standard classical methods, such as the Goemans-Williamson (GW) algorithm. We obtain numerical results for the low-autocorrelation binary sequences (LABS) and weighted MaxCut combinatorial optimization problems, thus extending an earlier demonstration of successful application of QITE on MaxCut for unweighted graphs. We find the performance of QITE on the LABS problem with a separable ansatz comparable to QAOA at level 10 for up to 18 vertices and do not see a significant advantage with an entangling ansatz. On weighted MaxCut, QITE with a separable ansatz often outperforms the GW algorithm on graphs with up to 150 vertices.
Abstract The variational quantum eigensolver (VQE) is one of the most prominent algorithms using near-term quantum devices, designed to find the ground state of a Hamiltonian. In VQE, a classical optimizer iteratively updates the parameters in the quantum circuit. Among various optimization methods, the quantum natural gradient descent (QNG) stands out as a promising optimization approach for VQE. However, standard QNG only leverages the quantum Fisher information of the entire system and treats each subsystem equally in the optimization process, without accounting for the different weights and contributions of each subsystem corresponding to each local term in the Hamiltonian. To address this limitation, we propose a Weighted Approximate QNG (WA-QNG) method tailored for k -local Hamiltonians. In this paper, we theoretically analyze the potential advantages of WA-QNG compared to QNG from three distinct perspectives and reveal its connection with the Gauss–Newton method. We also show it outperforms the standard QNG descent in the numerical simulations for seeking the ground state of the Hamiltonian.
Variational Quantum Eigensolver (VQE) is widely used in near-term hardware. However, their performances remain limited by the poor trainability and are dependent on random parameter initialization. In this work, we propose a warm start method inspired by imaginary time evolution, allowing for determining initial parameters that prioritize lower energy states in a resource-efficient way. Using classical simulations, we demonstrate that this warm start method significantly improves the success rate and reduces the number of iterations required for the convergence of VQE. The numerical results also indicate that the warm start approach effectively mitigates statistical errors arising from a finite number of measurements, and to a certain extent alleviates the effect of barren plateaus.
The variational quantum eigensolver (VQE) is a hybrid quantum-classical algorithm designed for current and near-term quantum devices. Despite its initial success, there is a lack of understanding involving several of its key aspects. There are problems with VQE that forbid a favourable scaling towards quantum advantage. In order to alleviate the problems, we propose and extensively test a quantum annealing inspired heuristic that supplements VQE. The improved VQE enables an efficient initial state preparation mechanism, in a recursive manner, for a quasi-dynamical unitary evolution. We conduct an in-depth scaling analysis of finding the ground state energies with increasing lattice sizes of the Heisenberg model, employing simulations of up to 404040 qubits that manipulate the complete state vector. For the current devices, we further propose a benchmarking toolkit using a mean-field model and test it on IBM Q devices. The improved VQE avoids barren plateaus, exits local minima, and works with low-depth circuits. Realistic gate execution times estimate a longer computational time to complete the same computation on a fully functional error-free quantum computer than on a quantum computer emulator implemented on a classical computer. However, our proposal can be expected to help accurate estimations of the ground state energies beyond 505050 qubits when the complete state vector can no longer be stored on a classical computer, thus enabling quantum advantage.
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 (iHVAi\mathrm{HVA}iHVA) 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 iHVAi\mathrm{HVA}iHVA. For randomly generated DDD-regular graphs, we numerically demonstrate that the iHVAi\mathrm{HVA}iHVA 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.