Quantum control algorithm looks to explain how birds migrate

The hidden world of quantum mechanics exists at scales many orders of magnitude smaller than living organisms, and yet scientists have long theorized that quantum effects play an important role in biology. Birds’ ability to sense magnetic fields during migration is one of the most well-known mysteries in this field, with leading theories suggesting that this sensing could be achieved by exploiting quantum entanglement. By proving a mathematical principle about how to best control quantum systems, researchers at the Okinawa Institute of Science and Technology (OIST) have taken what could be the penultimate step towards finally putting this avian hypothesis to the test, while also unlocking new biological platforms for quantum computing. Their results are now published in the journal Quantum.

Professor Ugur Abdulla, head of the Analysis and Partial Differential Equations Unit at OIST, explains: “Many researchers have studied quantum effects and their role in biology, though it isn’t always easy to translate an idea or hypothesis into a laboratory experiment. We hope that by laying the mathematical foundation for controlling quantum phenomena, we can bring some of these ideas from quantum biology out of the theoretical realm and into the lab.”

The difficulty of studying how quantum entanglement can be used for navigation

Answering questions about quantum biology is no easy task. “Quantum effects are only relevant at tiny scales, much smaller than the macroscopic size of organisms. These interactions also happen incredibly fast, and are notoriously difficult to pin down even in the most advanced experiments,” says Abdulla. Despite the quantum hypothesis of avian magnetoreception being proposed nearly 50 years ago, there has yet to be conclusive experimental evidence for or against it.

The hypothesis of magnetoreception focuses on the idea that the speed of a biochemical reaction can depend on an external magnetic field. Biochemical reactions are all about rearranging electrons to form bonds, though even unbound, or “radical”, pairs of electrons can become entangled and interact with each other via their magnetic properties encoded in quantum spin. Depending on whether these magnetic fields within a radical pair point in opposite directions or the same direction leads to more or less coherent states, and affects the speed at which stable bonds are formed. External magnetic fields, even weak ones, can influence how coherent a state is, meaning birds could theoretically use the concentration of product from a biochemical reaction to perceive Earth’s magnetic field.

Understanding how a magnetic field can be used to control a quantum system, whether for a bird’s navigation or a quantum computing algorithm, falls under the study of quantum optimal control theory. “The problem we’re dealing with here is how you bring a quantum system from one state to another by using external control. This is the key problem in developing any new quantum technology, and is central to understanding quantum biology,” says Abdulla.

A recipe for optimal quantum control using magnetic fields 

Abdulla and colleagues have previously proven the Pontryagin Maximum Principle (PMP) for spins in a magnetic field, which states that in order to create a maximally coherent state, one should use a “bang-bang optimization” approach, where the applied magnetic field abruptly switches between extreme values. This principle originates in the field of complex dynamical systems, and can even be seen in everyday life: when driving a car starting from rest at point A to stop exactly at some point B, the fastest strategy is to press the accelerator pedal all the way down for a while, followed by slamming on the brakes until the car stops. "The fact that this principle remains fundamental across such diverse systems, from everyday life to the quantum realm, is remarkable," says Abdulla.

But in the same way that driving a car in this manner would be quite difficult, generating bang-bang electromagnetic fields on quantum time scales raises major technical challenges. Abdulla, along with Jose Rodrigues (OIST) and Jean-Jacques Slotine (Massachusetts Institute of Technology), were thus interested in finding out if there were other control techniques that, while not perfectly optimal, could be experimentally and biologically feasible. The mathematical proof that they now present demonstrates that PMP holds for a modified version of the quantum system that is constrained to have realistic, continuous magnetic fields. Further, this experimentally feasible technique for controlling magnetic fields can give a coherent state within 1% of the true optimum.

This mathematical framework lays the foundation upon which experimental studies may be able to create near-optimally-controlled quantum systems in the lab, whether to understand how birds migrate or to create tomorrow’s quantum computers. “We’ve extended this general mathematical principle to quantum systems, and the result is that we can now tell an experimentalist precisely how you have to use a magnetic field to test this control technique,” shares Abdulla. “It's very exciting and rewarding to reveal such a mathematical principle that will help to guide experimental work.” 

But that doesn’t mean the mathematicians’ job is done. “A mathematical model is a caricature, in that we put a spotlight on just a few aspects of the system we think are the most important. There will be some flaws in our view, but this is just the first stage; there is always room to improve on such a model, to refine our caricature, making it more and more a portrait of reality.”