Bayesian Network Analysis and Inference via Quantum-supported Optimization (BAIQO)

Medical research is an essential contribution of science to the well-being of humanity. Every day, research is conducted into new methods of treating diseases. Testing drugs and treatment approaches is not without risk and is therefore carried out in medical studies. The study design is important for the success and accuracy of the studies. There are many factors that influence the outcome of a study. Examples of factors include the number of doctors, participating hospitals, and the prevalence of the disease, but also factors that are difficult to measure, such as awareness of the disease, for example through media coverage or information campaigns. In order to better control the design of studies, project partner Merck KGaA has created a decision tree from a large data set of old studies that models the influence of various factors on the target parameters. Mathematically, this structure is a Bayesian network.

This structure allows the target parameters to be simulated given the input parameters. This means that, given this number of doctors, hospitals, and prevalence, how high is the probability of a successful study? However, it is not possible to reverse the process and optimize individual input factors in the sense of: How many doctors and hospitals are needed to achieve a certain probability of success?

Alternative computational approaches must be used to invert the underlying Bayesian network. Quantum computing is one such approach. Quantum computers work with reversible operations, meaning that an operation can be inverted directly. The desired optimization can then be performed by partially inverting the Bayesian network.

The project was carried out in collaboration with Merck KGaA. The university partner in the Chair of Mobile and Distributed Systems at Ludwig Maximilian University in Munich focused primarily on quantum circuits, while Merck KGaA concentrated on the optimization and application of Bayesian networks.

Quantum circuits are necessary for calculating problems on quantum computers. These describe the operations to be performed on the qubits. The quantum circuits must be created individually for each problem, as there is still no direct translation from problem formulation to circuit elements. For the project implementation, we mainly use optimizable circuits inspired by machine learning methods. Individual circuit elements contain variables that are changeable. Using classical optimization methods, it is possible to adjust the values until the circuit corresponds to the model and can map the desired function or Bayesian network.

A decomposition of a matrix quantum circuit. Taken from the publication: M. B. Mansky et al, “Near-optimal Quantum Circuit Construction via Cartan Decomposition, Phys. Rev. A 108 (5) 052607, doi: 10.1103/PhysRevA.108.052607

In the course of the project, it became apparent that there are many different ways of representing the same network. This results in a high degree of symmetry in the solution space, meaning that many different solutions produce the same solution quality. This diversity has a negative effect on network modeling, and the learning approach no longer converges. To address this problem, symmetry-invariant circuits are considered, i.e., those that map all possible solution variants to a single one. This drastically reduces the search space. This approach of symmetry reduction is difficult or impossible to achieve with classical approaches. The construction of quantum circuits from individual elements allows the symmetries of the problem to be directly incorporated into the design of the approach, thereby achieving a better solution quality. The work on symmetry-invariant approaches is also reflected in the results published in the project.

Many of the results from the project represent a further development of quantum circuits in general. The inclusion of symmetries in circuit design leads to improved convergence on problems with high symmetry, regardless of the background of Bayesian networks. In total, the project has produced eleven scientific papers, two of which have been published in journals, five in scientific conference proceedings, and four are still awaiting publication outside of preprint servers. In addition, the quantum circuits researched are being implemented on real hardware. The results show that the hardware currently available is not yet sufficient for use on complex problems.

The published results range from the creation of quantum circuits from a target matrix to the efficient construction of symmetry-invariant quantum circuits, to their properties and behavior in quantum machine learning. The results represent important advances in the application of quantum circuits to complex problems and impressively demonstrate that the structure of the quantum circuit has a significant influence on the quality of the results. Among the secondary results of the project, the work on sampling problems on quantum computers stands out. This review paper presents the entire range of possible applications of sampling from complex distributions.