Scientists frequently turn to mathematics to explain and model everyday events. These often include processes that unfold over time, such as the fluctuations of financial markets or the spread of a disease. For complex phenomena where past events and long-term effects are major factors, fractional differential equations (FDEs) offer a way to capture every stage of the process in numbers.
“Solving an FDE is challenging because it requires keeping track of everything that has happened before—like remembering every step you’ve taken on a long hike. The longer the hike, the more memory you need,” said Fong Yew Leong, a Principal Scientist at the A*STAR Institute of High Performance Computing (A*STAR IHPC).
These memory costs can be a nightmare for even the most powerful conventional computers we have today. As such, Leong and colleagues at A*STAR IHPC and the Singapore University of Technology and Design have turned to quantum computers, which have the potential to solve mathematical problems in new and faster ways over classical computers.
However, today’s quantum computers are still limited in size and easily affected by hardware noise, which can muddy their solutions for FDEs. The team believes the answer lies in variational quantum algorithms (VQAs), a class of mathematical approaches which leverage the best of both quantum and conventional computing.
“VQAs work by setting up a guess-and-check system: a quantum computer guesses a possible solution, while a classical computer checks how good that guess is and adjusts the settings until the solution is found,” Leong explained.
The team developed a VQA based on a divide-and-conquer strategy to solving FDEs, tapping into both the processing power of quantum computing and the precision of its classical counterpart. In their VQA, a quantum computer evaluates quantum state overlaps, then classical computers sum up the resulting values for numerical integration. Where conventional methods would rely on large solution vectors to store information, the team’s approach uses shorter parametric vectors to reduce memory costs when tracking long-term history in FDEs.
To test their classical-quantum hybrid approach under real-world conditions, Leong and colleagues implemented their algorithm on IBM quantum hardware. As they expected, the noise issues faced by current quantum hardware proved a formidable challenge and limited the size of the FDE that their VQA could solve.
Still, the researchers remain hopeful that as error-resilient quantum devices emerge, the performance of VQAs and other near-term quantum algorithms will also improve significantly. According to Leong, such hardware improvements are already on the horizon, as the next generation of fault-tolerant quantum computers could offer greater resistance to noise and errors.
The team now looks to further optimise their algorithms, extending their work to these fault-tolerant devices.
“This will allow us to move beyond the limitations of near-term quantum hardware and design algorithms that are more robust, scalable and practical for real-world applications,” Leong said.
The A*STAR-affiliated researchers contributing to this research are from the A*STAR Institute of High Performance Computing (A*STAR IHPC).
