Reversing quantum information loss using lessons from machine learning

Quantum systems are fragile. When a qubit interacts with a noisy environment, the information it carries can easily be lost, scrambled in such a way that retrieving the information is virtually impossible.

But not in all scenarios.

PhD student Einar Gabbassov has been exploring cases where quantum information loss can be reversed, taking inspiration from image generation models like DALL-E or SORA. These are known as diffusion models. They are trained by allowing an image to gradually be turned into noise, and then they attempt to work backwards from the noise to recreate a clear picture.

“The classical forward and reverse processes used by image generation models are described by a stochastic differential equation, and this equation was derived in the 1970s. It's pretty old mathematics, later adapted to machine learning,” says Gabbassov. “In quantum mechanics, we have equations that describe the forward processes of information loss, but there are no known analogs of the quantum reverse process. I wondered whether it was possible.”

In a new paper published this week, Gabbassov derived the stochastic Schrödinger equations that describe the quantum reverse process.

The key to making it possible, he explains, is to carefully monitor the environment affecting a quantum system. Imagine, for example, that we have two qubits. One qubit is the quantum system whose information we’d like to retrieve, and the second qubit represents the environment. As time goes on, they may become weakly entangled.

We can take a measurement of the environment qubit, causing its wavefunction to collapse to a state of zero or one, instead of remaining in a quantum superposition. Due to entanglement, the system qubit will also be perturbed, but because we measured what happened to the environment, we know exactly how the system was perturbed.

“You can repeat this over and over again, and as long as you retain this measurement record of the environment qubit, then we essentially understand the stochastic trajectory of the system,” he says.

If you know the trajectory, you can reverse it.

The equations that describe this reverse process ensure that the path back to the original state isn’t necessarily the exact path it took when it evolved forward. It wanders back stochastically (think: probabilistically), but with a drift that nudges it eventually back towards the original quantum state, even though it is dealing with the same noisy conditions that caused the information loss in the first place.
 

So what makes these stochastic Schrödinger equations different from the usual way of looking at open quantum systems (systems that interact with their environment)?

“The well-established formalism of quantum channels and master equations that describes systems evolving under noise only describes average dynamics, and discards information about the environment. Deriving a reverse process is not possible with this formalism, unless you make some other assumptions. But that’s what a stochastic Schrödinger equation can give you. If you work at the level of stochastic trajectories, then you can build a reverse process,” Gabbassov explains.

These new equations offer new perspectives on information loss and recovery.

First, fundamentally, we now know that reversing the loss of quantum information is physically possible in some scenarios, as described by these new stochastic Schrödinger equations. And second, now that the actual stochastic Schrödinger equations have been derived based on first principles, they can be used in quantum generative machine learning.

“it’s useful to have an explicit equation for generative machine learning, because before it was all ad hoc or variational. People were coming up with clever circuits to simulate the reverse process, but now we have the actual equation that describes the reverse process,” says Gabbassov.

Einar Gabbassov is a PhD student at the University of Waterloo’s Faculty of Mathematics with affiliations at the Institute for Quantum Computing and Perimeter Institute for Theoretical Physics. The paper was published in Physical Review Research.