Why This Quantum Pioneer Thinks We Need More People Working on Quantum Algorithms

By Olivia Lanes and Robert Davis

There are a lot of classical computer algorithms — probably too many to count. However, researchers have so far devised perhaps only dozens of algorithms for quantum computers. One might assume this is because the first classical algorithm meant to be run on a machine was published over 150 years ago, while the first quantum algorithms didn’t appear until the past few decades. But maturity isn’t the only reason for the discrepancy, according to quantum computing pioneer Dorit Aharonov.

“Today’s quantum ecosystem is extremely vibrant, and there is a lot of energy and focus on particular questions,” Aharonov said. “I think there are advantages and disadvantages to that. I would like to see more stamina from the community when it comes to looking into the more difficult questions that have less immediate reward — like, for example, deeply understanding quantum algorithms.”

Aharonov has long argued that, despite all its progress, the quantum computing research community still understands relatively little about devising quantum algorithms that offer meaningful advantage over classical methods. “Quantum algorithmic advantage manifests when two conditions are met,” Aharonov said. “One is that quantum computation can somehow access some structure in the problem, and two is that classical computation cannot — or it’s not easy for it to access that structure.” The challenge is that today’s researchers only know of a handful of problem structures that meet those criteria.

When asked why there aren’t more researchers working to develop new approaches to quantum algorithms, Aharonov offers a simple explanation: It’s very, very difficult.

To be sure, there are many people working today on certain types of quantum algorithms, such as VQE algorithms and quantum approximation optimization algorithms (QAOA). However, Aharonov says that the search for new approaches to quantum algorithm design have not received the same attention in recent years. She suggested that researchers may avoid these areas because they require deep knowledge of many different topics in advanced mathematics, including combinatorics, random walks, and group representation theory, to name just a few.

“The thing is that people have tried a lot with few successes, but I don’t think that too many people have tried enough,” Aharonov said. “I think there are more algorithms down that road — very interesting ones — but it will probably be difficult to come up with them.”

For Aharonov, the depth and breadth of the mathematical theory that fits into quantum computing research is one of the key qualities that makes quantum computation so distinct from other areas of computer science.

“One very, very interesting thing about quantum computation is that it touches so many different fields in mathematics,” she said. “It’s not like that in classical computation. It’s really something that is special for quantum computation because it’s somehow ‘complete’ — quantum computation is some kind of completion, mathematically, of classical computation. I think of this as maybe similar to the fact that the complex numbers are an algebraic closure of the real numbers. And quite similarly again, the problems in the quantum model often have richer connections between them; they somehow ring ‘right,’ mathematically.” This, ultimately, is why Aharonov feels quantum computation is so beautiful.

“There are so many different universal models of quantum computers, each one based on a completely different algorithmic approach — like adiabatic computation, quantum walks, measurement-based quantum computation, topological quantum computation, and more — and they are all essentially equivalent,” she said. “You don’t really know which direction the next quantum algorithm will come from.”

Stoquastic Hamiltonians are common in nature, and they are easier to work with than other kinds of Hamiltonians. However, it’s an open question as to whether universal AQC is possible with a stoquastic Hamiltonian, or whether stoquastic adiabatic optimization is capable of achieving an exponential speedup over classical optimization techniques. In their paper, Gilyén et al. were able to demonstrate a sub-exponential speedup for a stoquastic AQC algorithm that searches an undirected graph with vertices labeled by n-bit strings to find a specific “Exit” vertex given the “Entrance” vertex.

That’s a significant achievement, but Aharonov notes that it comes with a few important caveats. For one thing, it’s important to bear in mind that the researchers’ results concern a contrived “toy” problem, rather than a problem whose solution might provide some real-world value. Still, Aharonov believes the paper could lead to important insights.

“Previous papers, and in particular Hastings’ first papers on topological obstructions, manage to show that stoquastic adiabatic computation is capable of understanding something about the global structure of a problem. The Gilyén, Hastings & Vazirani paper actually manages to show that this could lead to a provable, almost-exponential advantage! I think this is very interesting,” Aharonov said, adding that further study could improve our understanding of how AQC with stoquastic Hamiltonians is able to understand the topological or global structures of the Hamiltonian, or of the graph upon which the Hamiltonian is defined.

Aharonov suggested that the Gilyén et al. paper raises new hopes about how strong adiabatic stoquastic Hamiltonian computation really is — though providing stoquastic adiabatic quantum algorithms for more practical problems may prove challenging. “The Gilyén, Hastings and Vazirani algorithm might be very impractical in the sense of just implementing it,” she said, “even if just for demonstrations.”

It will take much more research to answer questions such as those raised by the Gilyén et al. paper. However, Aharonov suggests that the beauty and elegance of quantum algorithms makes the pursuit of these answers well worth the effort. Asked to name an example of something from the field of quantum algorithms that she finds especially striking, Aharonov turned to what she says is a common favorite among quantum computing researchers, Kitaev’s Phase Estimation algorithm.

“The phase estimation algorithm demonstrates one particularly amazing feature or ability in quantum algorithms, which is the ability to use a single quantum bit, entangled to no matter how large a quantum system, to estimate the inner product between two possible states of that system. So if you have two states, which could each be as complex as a description of the entire universe, and you entangle them to a single qubit, the state of that qubit somehow manages to know something about the inner product of these two extremely complicated states.”

For Aharonov, it’s observations like these that make the search for new quantum algorithms so rewarding, and she is hopeful that more researchers will join the hunt. However, she also hopes those researchers will resist the urge to follow the crowd, focusing all their energy on the topics and questions that are receiving the most attention. “I’m sure that there are many, many questions that someone original and creative could think about, and that no one else is interested in,” Aharonov said. “So I would say invent your own questions. Think about the questions, not just about the answers. When it comes to quantum computing, I believe that some of the most interesting questions have not yet been asked.”

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