New protocols for estimating Hamiltonian parameters of a superconducting quantum processor could improve accuracy

Researchers from Freie Universität Berlin, University of Maryland and NIST, Google AI and Abu Dhabi set out to robustly estimate the free Hamiltonian parameters of bosonic excitations in a superconducting quantum simulator. The protocols they developed, described in a pre-published article on arXivcould contribute to the realization of high-precision quantum simulations beyond the limits of classical computers.

“I was at a conference in Brazil when I received a call from friends on the Google AI team,” Jens Eisert, first author of the paper, told Phys.org.

“They were trying to calibrate their Sycamore superconducting quantum chip with Hamiltonian learning methods and encountered serious obstacles and were calling for help. Having worked extensively on both analog quantum simulation ideas and identification methodology systems, I was really intrigued.”

When Eisert started thinking about the problem his friends were giving him, he thought it should be easy to solve. Yet he soon realized that this would be more difficult than expected, because the frequencies of the Hamiltonian operator in the team’s system were not recovered accurately enough to identify the unknown Hamiltonian from the data available.

“I invited two extremely smart doctoral students, Ingo Roth and Dominik Hangleiter, and together we quickly found a solution using superresolution ideas, in principle until the data arrived,” Eisert said.

“It then took a few more years before we understood how to make the Hamiltonian learning ideas robust enough that they could be applied to real large-scale experiments.

“Meanwhile, another PhD student joined us, Jonas Fuksa, and the other two had long since graduated. This helped Pedram Roushan, the experimental lead for Google’s AI efforts, stay persisting and providing excellent data. In the end, years later, we found a solution to the question raised on the Zoom call years earlier.

To learn the Hamiltonian dynamics of a superconducting quantum simulator, Eisert and his colleagues used a variety of techniques. First, the researchers used superresolution, a method for improving the resolution of eigenvalue estimation, to achieve the correct Hamiltonian frequencies.

They then used a technique known as multiple optimization to recover the eigenspaces of the Hamiltonian operator, thereby recovering the Hamiltonian. Manifold optimization involves the use of specialized optimization algorithms to solve complex problems for which the variables lie on a manifold (smooth and curved space), rather than in standard Euclidean space.

“To get robust estimates, we combined a number of ideas,” Eisert explained.

“Even understanding the processes of switching on and off was important, because these processes are not perfect and instantaneous (and not even unitary), so if one tries to adapt to a Hamiltonian evolution which, in part, is not at all Hamiltonian, we obtain a Ultimately, new signal processing methods that we called TensorEsprit enabled robust recovery up to large systems.

In their article, the researchers present a new technique for implementing super-resolution, which they have called TensorEsprit. By combining this technique with a multiple optimization method, they were able to robustly identify Hamiltonian parameters for up to 14 coupled superconducting qubits distributed across two Sycamore processors.

“Early on, it was important to understand the importance of Hamiltonian learning methods,” Eisert said.

“We can then only recover the eigenspaces in a meaningful way if the eigenvalues ​​are known with extreme precision. In the later phases of the project, we learned the hard way why there are so few publications presenting data from Hamiltonian learning: it’s just very difficult to make it work for practical data.

Initial tests carried out by the researchers suggest that the proposed techniques could be scalable and robustly applicable to large quantum processors. Their work could inspire the development of similar approaches to characterize the Hamiltonian parameters of quantum processors.

In their next studies, Eisert and his colleagues plan to apply their methods to interacting quantum systems. They are also working to apply similar ideas derived from tensor networks to quantum systems made of cold atoms, which were first introduced by physicist Immanuel Bloch.

“I think this area will become important in the future,” Eisert added. “An old and yet often underestimated question is that of knowing what a Hamiltonian of a system really is. This question is already asked in basic courses on quantum mechanics. Because even if it characterizes the system, we often assumes that it is known, an assumption which is often not the case.

“Ultimately, experiments only produce data, and so in quantum mechanics you only have predictive power if you know the Hamiltonian exactly. So the question arises how to learn it from data.”

In addition to contributing to the conceptual understanding of Hamiltonian operators, the researchers’ future studies could inform the development of quantum technologies. By facilitating the characterization of analog quantum simulators, they could indeed open new avenues for carrying out high-precision quantum simulations.

“Quantum analog simulation allows complex quantum systems and materials to be studied in a new way by recreating them under extremely precise conditions in the laboratory,” explained Eisert.

“Yet this idea only makes sense – and is only associated with accurate predictions – if you know the Hamiltonian that exactly characterizes the system.”

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