Hybrid algorithm finds steady solutions

In the approach proposed by the team, the quantum computer acts once to calculate the properties of the nonequilibrium steady states. Based on the outputs, the classical computer runs the optimisation program to find solutions to the problem. Image credit: Shutterstock.com/29october

Simulating chemical and biological systems is one area where researchers expect quantum computers to be useful. CQT researchers have finessed a new algorithm that could help current quantum computers solve these problems more efficiently.

In their paper published in Physical Review Letters on 12 June 2023, the group of CQT Principal Investigator Kwek Leong Chuan tackles what is known as ‘nonequilibrium steady states’ (NESS). NESS refers to states of a system that do not settle into equilibrium but have a steady behaviour over time. They are important for understanding the behaviour of complex systems and can model some chemical reactions and biological processes such as photosynthesis.

“You could also think of a swing with friction that is constantly being pushed,” says CQT PhD student Jonathan Lau Wei Zhong. “The external environment constantly acts on the system so it is not in equilibrium, but there is a pattern and a steady state.”

Jonathan and fellow PhD student Lim Kian Hwee are joint first authors of the paper. They worked together with Kwek, CQT alumnus Kishor Bharti and CQT Visiting Senior Research Fellow Sai Vinjanampathy.

Finding NESS

Researchers have already proposed some ways to solve NESS problems. However, some of their algorithms require millions of qubits acting in a ‘fault-tolerant’ way, far beyond what current noisy intermediate-scale quantum (NISQ) devices offer. Today’s NISQ quantum computers typically have only tens of qubits labouring under uncorrected noise.

On the other hand, algorithms designed for such NISQ devices also have limitations. Variational quantum algorithms are one popular method. These are hybrid algorithms that make use of both classical and quantum computers. Typically, the classical computer optimises parameters for the quantum computer – but this is resource intensive, making use of a classical-quantum feedback loop. Additionally, the type of optimisation deployed in these algorithms is known to be hard for classical computers and is not well studied.

The researchers reframe the NESS problem as a feasibility semidefinite program for the classical computer. This type of optimisation program is mathematically well-understood. It gives systematic ways for the researchers to enforce constraints during optimisation and solve problems with multiple NESS solutions.

In this approach, the quantum computer just acts once to calculate the states’ properties, doing away with the quantum-classical feedback loop. Based on the outputs, the classical computer then runs the optimisation program to find solutions.

Quantum assist

The team demonstrated their algorithm with some toy examples, considering a two-qubit and five-qubit transverse field Ising model. Extending their algorithm to solve for multiple NESS solutions, they tested it with simulations of system sizes of up to eight qubits. Their algorithm gave results in agreement with the exact solutions.

Kian Hwee says, “If we keep thinking in the framework of variational methods, we are not going to get anywhere. Instead, we reframe the problem to give the classical computer what it is good at doing and the quantum computer what it is good at doing.”

Besides a greater understanding of NESS problems, the researchers also expect their work to extend studies of NISQ algorithms to simulating open quantum systems.