A new protocol for estimating Hamiltonian parameters in superconducting quantum processors could improve accuracy
Overview of experiments and identification algorithms. (a) The time evolution under the target Hamiltonian h0 is implemented on part of a Google Sycamore chip (gray) using the pulse sequence shown in the center. (b) The expected values of the canonical coordinates xm and pm of each qubit m over time are estimated from measurements with different ψn as input states. (c) The data at each time t0 shown in (b) can be interpreted as a (complex-valued) matrix with entries indexed by the measurement qubit and the initial excited qubits m and n. The identification algorithm proceeds in two steps. 1. From the matrix time series, the Hamiltonian eigenfrequencies are extracted using the newly introduced algorithm tensorESPRIT, introduced in SM, or an adapted version of the ESPRIT algorithm. The blue line shows the denoised high-resolution signal as “recognized” by the algorithm. 2. After removing the initial ramp using some fixed-time data, the Hamiltonian eigenspace is reconstructed using a non-convex optimization algorithm for orthogonal groups. Obtain the diagonal orthogonal estimate of the final ramp. From the extracted frequencies and the reconstructed eigenspace, we can compute the identified Hamiltonian h^, which describes the measured time evolution and the tomographic estimate of the initial ramp. Credit: Hangleiter et al.
Researchers from Freie Universität Berlin, University of Maryland, NIST, Google AI, and Abu Dhabi set out to reliably estimate the free Hamiltonian parameters of boson excitations in superconducting quantum simulators. The protocol they developed, outlined in a paper previously published on arXiv, could help achieve highly accurate quantum simulations that go beyond the limits of classical computers.
“I was at a conference in Brazil when I got a call from a friend of mine from the Google AI team,” lead author Jens Eisert told Phys.org.
“They were trying to calibrate the Sycamore superconducting quantum chip with the method of Hamiltonian learning and encountered serious obstacles and were looking for help. A lot of research has been done on both the idea of analog quantum simulation and the system identification methodology. So I was really intrigued.”
When Isert first began considering the problem presented by his friends, he thought it should be easy to solve. However, he quickly realizes that it is more difficult than expected, as the frequencies of the Hamiltonian operators in the team’s system have not been recovered accurately enough to identify the unknown Hamiltonian from the available data. Ta.
“I invited two very smart PhD students, Ingo Roth and Dominik Hungreiter, and together we quickly found a solution using the idea of super-resolution. “In principle, that is, until the data came in,” Isert said.
“Then it took us several more years to figure out how to make the idea of Hamiltonian learning robust enough to apply to real-world large-scale experiments.
“Meanwhile, another PhD student, Jonas Fukusa, joined us, and the other two had long since graduated. That helped Pedram Roshan, the lead experimenter on the Google AI effort, persevere. They kept trying and provided us with good data. Eventually, after a few years, we found a solution to the question posed on a Zoom call a few years ago.”
To learn Hamiltonian mechanics in a superconducting quantum simulator, Isert and his colleagues employed a variety of techniques. First, the researchers used super-resolution, a method of increasing the resolution of eigenvalue estimation, to obtain the correct Hamiltonian frequency.
They then recovered the eigenspace of the Hamiltonian operator using a technique known as manifold optimization, which actually recovered the Hamiltonian. Manifold optimization requires the use of specialized optimization algorithms to tackle complex problems where the variables lie on a manifold (a smooth, curved space) rather than a standard Euclidean space.
“To achieve a robust estimate, we combined several ideas,” Isert explained.
“It was important to even understand the processes of switching on and off. These processes are neither perfect nor instantaneous (and not even unitary), so trying to apply a Hamiltonian evolution that is not partially Hamiltonian… The result is as follows: Finally, a new signal processing method called TensorEsprit allows for robust recovery up to large system sizes.
FU Berlin. Credit: Jens Eisert
In their paper, the researchers introduce a new technique for implementing super-resolution, which they call TensorEsprit. By combining this method with manifold optimization techniques, we were able to reliably determine the Hamiltonian parameters for up to 14 coupled superconducting qubits distributed across two Sycamore processors.
“In the early stages, it was important to fully understand the importance of the Hamilton learning method,” Eissert said.
“A meaningful eigenspace can only be recovered if the eigenvalues are known very precisely. We are in pain as to why so few publications present data for Hamiltonian learning at this late stage of the project. Now that I understand it, please use it as practical data.”
Initial tests performed by the researchers suggest that their proposed technique is scalable and could be robustly applied to large-scale quantum processors. Their work could inspire the development of similar approaches for characterizing Hamiltonian parameters in quantum processors.
As part of their next research, Eisert and his colleagues plan to apply their method to interacting quantum systems. They are also working on applying similar ideas from tensor networks, first introduced by physicist Immanuel Block, to quantum systems made of cold atoms.
“I think this field will be important in the future,” Isert added. “An old, but often underappreciated question is what the Hamiltonian of a system actually is. This question has already been asked in the basic lectures on quantum mechanics. Even when we characterize something, it is often assumed to be known, an assumption that is often not.
“At the end of the day, experiments only produce data. Therefore, quantum mechanics only has predictive power if you know the Hamiltonian exactly. Therefore, the question of how can we learn the Hamiltonian from data? A problem arises.”
In addition to contributing to the conceptual understanding of Hamiltonian operators, the researchers’ future work could help develop quantum technologies. In fact, supporting the characterization of analog quantum simulators could open new avenues for achieving high-precision quantum simulations.
“Analog quantum simulation allows us to study complex quantum systems and materials in new ways by recreating them under very precise conditions in the laboratory,” Isert explained.
“But this idea only makes sense and is associated with accurate predictions if we know the Hamiltonian that accurately characterizes the system.”
Details: Dominik Hangleiter et al, Reliably learning Hamiltonian mechanics for superconducting quantum processors, arXiv (2024). DOI: 10.48550/arxiv.2108.08319
Magazine information: arXiv
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