Novel Quantum Algorithm for High-Quality Solutions to Combinatorial Optimization Problems

Researchers have proposed an innovative quantum algorithm that effectively solves combinatorial optimization problems with constraints in a short time

Conventional quantum algorithms are not feasible for solving combinatorial optimization problems (COPs) with constraints in the operation time of quantum computers. To address this issue, researchers have developed a novel algorithm called post-processing variationally scheduled quantum algorithm. The novelty of this innovative algorithm lies in the use of a post-processing technique combined with variational scheduling to achieve high-quality solutions to COPs in a short time.

Combinatorial optimization problems (COPs) have applications in many different fields such as logistics, supply chain management, machine learning, material design and drug discovery, among others, for finding the optimal solution to complex problems. These problems are usually very computationally intensive using classical computers and thus solving COPs using quantum computers has attracted significant attention from both academia and industry.

Quantum computers take advantage of the quantum property of superposition, using specialized qubits, that can exist in an infinite yet contained number of states of 0 or 1 or any combination of the two, to quickly solve large problems. However, when COPs involve constraints, conventional quantum algorithms like adiabatic quantum annealing struggle to obtain a near-optimal solution within the operation time of quantum computers. Recent advances in quantum technology have led to devices such as quantum annealers and gate-type quantum devices that provide suitable platforms for solving COPs. Unfortunately, they are susceptible to noise, which limits their applicability to quantum algorithms with low computational costs. 

To address this challenge, Assistant Professor Tatsuhiko Shirai and Professor Nozomu Togawa from the Department of Computer Science and Communications Engineering at Waseda University in Japan have recently developed a groundbreaking post-processing variationally scheduled quantum algorithm (pVSQA). “The two main methods for solving COPs with quantum devices are variational scheduling and post-processing. Our algorithm combines variational scheduling with a post-processing method that transforms infeasible solutions into feasible ones, allowing us to achieve near-optimal solutions for constrained COPs on both quantum annealers and gate-based quantum computers,” explains Dr. Shirai. Their study was published in the journal IEEE Transactions on Quantum Engineering on 13 March 2024.

The innovative pVSQA algorithm uses a quantum device to first generate a variational quantum state via quantum computation. This is then used to generate a probability distribution function which consists of all the feasible and infeasible solutions that are within the constraints of the COP. Next, the post-processing method transforms the infeasible solutions into feasible ones, leaving the probability distribution with only feasible solutions. A classical computer is then used to calculate an energy expectation value of the cost function using this new probability distribution. Repeating this calculation results in a near-optimal solution. 

The researchers analyzed the performance of this algorithm using both a simulator and real quantum devices such as a quantum annealer and a gate-type quantum device.  The experiments revealed that pVSQA achieves a near-optimal performance within a predetermined time on the simulator and outperforms conventional quantum algorithms without post-processing on real quantum devices.

Dr. Shirai highlights the potential applications of the algorithm, stating: “Drastic social transformations are urgently needed to address various social issues. Examples include the realization of a carbon-neutral society to solve climate change issues and the realization of sustainable development goals to address issues such as increased energy demand and food shortage. Efficiently solving combinatorial optimization problems is at the heart of achieving these transformations. Our new method will play a significant role in realizing these long-term social transformations.”

In conclusion, this study marks a significant step forward for using quantum computers for solving COPs, holding promise for addressing complex real-world problems across various domains.


Authors: Tatsuhiko Shirai1 and Nozomu Togawa1

Title of original paper: Post-processing variationally scheduled quantum algorithm for constrained combinatorial optimization problems

Journal: IEEE Transactions on Quantum Engineering



1Department of Computer Science and Communications Engineering, Waseda University

About Waseda University

Located in the heart of Tokyo, Waseda University is a leading private research university that has long been dedicated to academic excellence, innovative research, and civic engagement at both the local and global levels since 1882. The University has produced many changemakers in its history, including nine prime ministers and many leaders in business, science and technology, literature, sports, and film. Waseda has strong collaborations with overseas research institutions and is committed to advancing cutting-edge research and developing leaders who can contribute to the resolution of complex, global social issues. The University has set a target of achieving a zero-carbon campus by 2032, in line with the Sustainable Development Goals (SDGs) adopted by the United Nations in 2015. 

To learn more about Waseda University, visit  

About Assistant Professor Tatsuhiko Shirai

Tatsuhiko Shirai is currently an Assistant Professor at the Department of Computer and Communications Engineering at Waseda University in Japan. He obtained his master’s and Ph.D. in Physics from the University of Tokyo in 2013 and 2016 respectively. In 2022, he obtained the SLDM Research Group Excellent Paper Award. His research interests include quantum algorithms, quantum open systems, and quantum computing. He is a member of The Physical Society of Japan. 

Media Contact

Armand Aponte

Waseda University

error: Content is protected !!