While quantum computers are still a nascent technology, researchers are busy building complex machine learning algorithms to test quantum computing's capabilities. But sometimes the algorithms get stuck in mysterious dead ends: mathematical paths that you can't go forward or backward — dreaded barren plateaus.
“Understanding the problem of barren plateaus is seen as key to unlocking quantum machine learning, and our team has been working on this for five years,” said Marco Cerezo, lead scientist on the Los Alamos team. “The main problem was that while we knew these barren plateaus existed, we didn't really know what unified the source of this phenomenon. But what we've now done is to mathematically characterize why and when barren plateaus arise in variational quantum algorithms.”
Instead of the 1s and 0s that classical computers use, quantum computers use qubits, which exploit quantum phenomena such as superposition (the ability to exist in multiple states at the same time). It is believed that when combined with quantum algorithms, quantum computers could help humanity solve certain types of problems that are too difficult or time-consuming for classical computers.
Barren plateaus are a poorly understood yet common problem in quantum algorithm development: when researchers run algorithms for months, they can fail unexpectedly. Scientists have theorized about why barren plateaus exist and even employed a series of methods to avoid them. But no one knew the underlying cause of this mathematical equivalent of a dead end.
Precisely characterizing the barren plateau now gives scientists a set of guidelines to follow when creating new quantum algorithms, which will prove essential as quantum computing power expands from a maximum of 65 qubits three years ago to the more than 1,000 qubit computers now under development.
This breakthrough – a unified theory of the barren plateau – takes the guesswork out of building quantum machine learning algorithms, eliminating wasted time and resources, and addresses one of the biggest challenges facing the field of quantum computing.
Proving the theory
Barren plateaus arise most frequently in optimization algorithms. A fundamental optimization problem is that of a traveling salesman, who, given a list of addresses, must find the most efficient route to each house. To solve such optimization problems, quantum computers encode the task of finding the optimal path into a learning problem. The learning problem requires navigating an optimization landscape consisting of peaks and valleys, where peaks represent bad solutions and valleys are associated with good outcomes.
Where barren plateaus exist, mountain peaks and valleys are separated by large areas of flat, featureless plains. Optimization algorithms can get stuck on these plateaus, not knowing which direction to go. Scientists have studied this phenomenon but have been limited to specific algorithm architectures. And because there was no unified theory to explain or predict barren plateaus, researchers could only guess.
“In our previous paper, we conjectured that there is a clear relationship between certain algebraic properties of these optimization algorithms and the existence of barren plateaus,” says Martin LaRocca, who came to Los Alamos as a postdoctoral researcher to work on the barren plateau problem. “Our latest paper proves this conjecture and develops a kind of recipe that allows researchers to test their algorithms for the existence of barren plateaus.”
The team achieved this by developing an equation that can predict the barren plateaus in any quantum optimization algorithm — and, moreover, the equation also uncovers new sources of barren plateaus.
They found that general quantum optimization algorithms attempting to solve a variety of tasks encounter barren plateaus more frequently. Essentially, specialization, not generalization, is the key to avoiding barren plateaus. This breakthrough allows scientists to understand and synthesize all the known causes of barren plateaus, allowing them to avoid them when building algorithms.
This research is the first to successfully develop a unified mathematical approach to identify barren plateaus, and the results will have far-reaching implications for the rapidly developing field of quantum computing.
The research was published in the journal Nature Computational Science and was conducted in collaboration with researchers from the University of California, Davis, North Carolina State University and Pacific Northwest National Laboratory.
paper: “A barren plateau lie algebra theory for deeply parameterized quantum circuits.” Nature Communications. DOI: 10.1038/s41467-024-49909-3
Funding: Laboratory-led Research and Development Program
LA-UR-24-28066