Researchers must be able to define what they’re studying; this holds true whether you’re describing birds, chemical elements, or the building blocks of our universe.
In the field of quantum matter, a new frontier of research is emerging with the onset of ‘artificial’ quantum systems in the form of quantum computers. The development of these new systems means that there are brand new phases of matter to define. For Perimeter researchers, pinning down these new phases, in particular something called mixed state phases, is laying the foundation for progress in quantum error correction.
Artificial quantum matter
Quantum matter is all about understanding phenomena that arise from many quantum degrees of freedom, says Timothy Hsieh, Director of Perimeter Institute’s Clay Riddell Centre for Quantum Matter. In these scenarios, there aren’t just one or two particles interacting, but potentially millions. Superconductors are one example, where many active electrons interact with each other via electromagnetism to produce new and interesting effects.
But quantum computers are a type of artificial quantum system, and they're changing the game. The individual qubits used to store information on quantum computers can interact with one another and display new properties.
To add to the complexity, researchers have to contend with noise introduced by the surrounding environment into their quantum computers.
“The interesting thing is we really have to deal with the fact that these are open quantum systems,” says Hsieh. “You not only have all these degrees of freedom inside the quantum computer that we’re operating, but they are coupled to some environment like the laboratory it’s situated in.”
“This interaction between the quantum system that we want to keep coherent, and this larger environment is really at the heart of being able to build a practical quantum computer that’s robust to noise.”
Trouble with translation
A 2024 paper co-authored by Hsieh describes how in all quantum simulators and quantum computers, environmental interactions inevitably drive the system into what’s called a mixed state – a somewhat messy situation where the environment has interfered with the quantum system.
There isn’t a commonly agreed upon definition for what constitutes a mixed state phase, explains the paper’s lead author, Shengqi Sang. Sang completed his PhD at Perimeter Institute studying under Hsieh and is now a postdoctoral fellow at Stanford University.
“A mixed state means an ensemble of pure states,” says Sang. “And a pure state is usually what people mean when they use the term quantum state. They’re the mathematical descriptions of a quantum system.”
Finding a definition for mixed state phases is a challenge, says Sang, because the mathematical tools used to define pure states do not translate over to mixed states.
“In pure states, the definition relies on something called the unitary circuit,” he says. “That is generally reversible, and that’s not true in general for its counterpart in mixed states called a channel circuit. It is not reversible.”
“It requires more tools in the theory of quantum information to properly define and study properties of mixed states,” he adds.
Learning how to define and diagnose mixed states
“My group has been working on coming up with a formalism for not only defining mixed state phases but also trying to see when the phases are stable and how to diagnose that stability in these new kinds of phases in open quantum systems,” says Hsieh.
In their 2024 paper, Sang, Hsieh, and their co-author proposed a new way of showing that two mixed states are in the same phase. They adopted an existing definition of mixed state phases: that two mixed states are in the same phase if there’s a local channel that takes you from one to the other, and another one that takes you back, explains Hsieh.
“Basically, if they’re connected by two-way local operations, that’s a working definition of mixed phases.”
The real test was to show this new method would work in action.
“We applied it on some simpler examples to show that this definition can work, and it gave us new understandings of these examples. It’s a first step,” says Sang.
But their new method might just have applications in quantum computing. “Once you propose a definition, you need to show it is useful and convenient,” says Sang.
A second work co-authored by Sang and Hsieh addresses how to determine when a mixed state is stable. Imagine you have a pure ‘code state’ of a quantum computer that’s been subjected to noise and evolved into a mixed state, says Hsieh. The researchers were interested in diagnosing the conditions in which that noisy state is guaranteed to be in the same phase as the clean state. The pair succeeded in coming up with a diagnosis based on a correlation measure called conditional mutual information.
“It’s nice to have a guarantee of stability, because then it means that any quantum information in the original state is preserved, even in the presence of noise,” says Hsieh.
The Wild West of quantum matter
As we’ve learned, the transition to a mixed state is inevitable in an open quantum system. Environmental factors like heat, light, and air molecules are all sources of noise.
“If you have a quantum computer, you have to put it somewhere. So you put it on some substrate, and then it just inevitably interacts with the substrate,” says Sang. “And that will turn the pure state of your quantum processor into a mixed state.”
But these mixed states are proving to be a source of inspiration. Hsieh says the topic is generating conversations between quantum matter and quantum information researchers.
“What I find pretty exciting is that the insights we get from these two works suggest a new diagnostic for quantum error correcting codes.”
Sang agrees that mixed state research should provide interesting outcomes for quantum information. “One of the things I really hope will happen is that the research of mixed state quantum phases can help us build more efficient protocols for fault-tolerant quantum computation.”
Finally, there’s a sense of discovery and exploration surrounding mixed state research.
“It’s really a Wild West frontier, where we’re dealing with basic questions and laying down the foundations,” says Hsieh.
About PI
Perimeter Institute is the world’s largest research hub devoted to theoretical physics. The independent Institute was founded in 1999 to foster breakthroughs in the fundamental understanding of our universe, from the smallest particles to the entire cosmos. Research at Perimeter is motivated by the understanding that fundamental science advances human knowledge and catalyzes innovation, and that today’s theoretical physics is tomorrow’s technology. Located in the Region of Waterloo, the not-for-profit Institute is a unique public-private endeavour, including the Governments of Ontario and Canada, that enables cutting-edge research, trains the next generation of scientific pioneers, and shares the power of physics through award-winning educational outreach and public engagement.
