A research team led by Yale physics researchers Victor Albert ’17 and Liang Jiang recently published findings that could aid in the creation of quantum computers.

The team used linear algebra to derive the results, originally focusing on a stabilized double-well open quantum system — which can be compared to two depressions in a plane — to predict the final states of objects which are initially in a quantum state anywhere in the plane. The location and velocity of objects in a quantum state cannot simultaneously be determined. The study, which was published in the journal Physical Review X, proved that the researchers’ methods for determining these positions could be applied to all similar open quantum systems with multiple-well steady states. These methods show, given an object’s initial position, whether or not it will remain in a quantum state.

According to the Institute of Physics, superposition refers to the property of quantum particles that allows for simultaneous existence across multiple states of energy, position and speed. Albert further explained this phenomenon by describing the behavior of a ball dropped above two wells.

“Imagine you’re a quantum ball dropped above the two wells. You can roll to both wells sort of simultaneously,” Albert said. “There’s some quantum probability that you roll to one well and some quantum probability that you roll to the other well, and there’s always some nonzero probability that you start in one well and end up in the other.”

Albert explained that one of the team’s objectives was to find the initial scenarios which would allow such an object to maintain its initial quantum superposition between the two wells as it rolls into the wells — the steady states of the system. They found that the answer was dependent upon the coefficients of vector subspaces in the investigative model.

Ultimately, the team concluded that if an object initially exists in a quantum state and is either entirely outside or entirely within both of the wells in a two-well system, it will retain its property of superposition and remain in a quantum state. If an object initially lies simultaneously within and outside of either well, it will be unable to maintain its quantum state as the outside portion of the object moves into the well.

According to the Jiang Lab’s study, it was previously thought that the flow of information into such wells would lead to the inevitable destruction of an object’s quantum properties, but the team’s research shows that it is possible to preserve quantum information in some circumstances. Albert added that this finding may help engineers decipher what is and is not possible in transmitting information via quantum computers.

To explain operations necessary for quantum computing, Albert used an analogy in which an individual is holding a pendulum swinging in a constant direction relative to the person holding it. If the individual is initially standing at the north pole, moves southward to the equator, turns 90 degrees, continues moving in that direction, then turns 90 degrees again to return to the north pole, the pendulum will end up in its initial position but with a new orientation, he said.

By moving in this closed loop, one would have performed an operation on the pendulum and illustrated the concept of holonomy, a consequence of a space’s curvature that measures the extent that closed loop paths disturb data being transferred along them, Albert said. This concept can be applied to quantum spaces because like the sphere in Albert’s analogy, quantum spaces have curvature.

“Sure, you could have just turned yourself 90 degrees at the north pole, but in a quantum computer you can’t just do that — you need to perform a computation by going around a closed loop on your parameter space,” Albert said. “It could be that all we have available are closed loops and we have to get back to where we started because we need to go back to the original subspace, but if we move slowly enough around a closed loop, we can change our orientation and maintain the quantum state because of the curvature of our manifold.”

The concept of holonomy is often applied in other disciplines, Albert said, such as robotics. However, the team’s research was unique in that it proved that holonomy could be applied for all cases of stabilized subspaces.

While there are currently many technological platforms for quantum computing in research, Albert’s team worked specifically in the class of reservoir engineering, which involves altering the steady states of a system in order to move information within them and into them while preserving the quantum state of the information.