Machine Learning Fine-Tunes the Quantum Math Behind Real Transmon Qubits

Quantum computers promise to solve problems that stump even the most powerful supercomputers, but the machines themselves are stubbornly imperfect. The mathematical models engineers use to describe how qubits evolve during a computation are approximations, and those approximations quietly erode the fidelity of every gate a processor performs. Now, two researchers at the Indian Institute of Science Education and Research Thiruvananthapuram have shown that a technique from the emerging field of scientific machine learning can close that gap dramatically, correcting the standard Hamiltonian of real superconducting qubits until theory and hardware agree to within a few percent.

John George Francis and Anil Shaji focused on transmon qubits, the workhorse of IBM’s quantum processors and many other superconducting platforms. Transmons are engineered from Josephson junctions and superconducting circuits to be relatively insensitive to charge noise, a design choice that has made them the most widely deployed qubit technology in the world. Yet even the best-fabricated transmons deviate from the idealized textbook description. Stray couplings, higher energy levels, and control imperfections all conspire to make the actual dynamics of the system richer and messier than the standard Hamiltonian predicts.

The team targeted one of the most important operations in the superconducting quantum computing repertoire: the cross-resonance gate. In this scheme, microwaves are applied to a target qubit at the frequency of its control qubit neighbor, creating an entangling interaction that serves as a fundamental building block for two-qubit logic. The cross-resonance interaction has a well-known effective Hamiltonian description, developed in earlier theoretical work, but that description is itself an approximation whose accuracy depends on the details of the hardware and the drive parameters. When the idealized model is used to predict what a real gate does, errors accumulate quickly.

To quantify those errors and correct them, Francis and Shaji turned to real experimental data from ibm_kyiv, one of the processors accessible on the IBM Quantum cloud platform. They collected measurement outcomes for two transmon qubits driven by microwave pulses of varying amplitudes and evolution times, spanning a range of operating conditions. Each data point captured the probabilities of the different computational basis states after the system evolved under the cross-resonance drive, providing a rich empirical fingerprint of the true dynamics that no purely theoretical model could be expected to reproduce exactly.

The heart of their method is a correction operator added to the standard Hamiltonian. Rather than guessing the correction from first principles, the researchers wrote down an ansatz, a flexible mathematical form whose matrix elements are free parameters to be optimized. The goal of the optimization is simple to state and hard to achieve: find the correction terms that make the corrected Hamiltonian’s predicted evolution match the observed hardware data as closely as possible across all the measured amplitudes and times. Because the ansatz is not unique, different parameter choices can yield equally good fits, so the team developed methods to characterize an entire equivalence class of effective corrections rather than a single answer.

Optimizing parameters inside a differential equation is exactly the kind of problem that scientific machine learning was built to solve. The researchers employed adjoint sensitivity analysis, a technique that efficiently computes how the final outcome of a dynamical simulation changes with respect to every parameter in the model, without the prohibitive cost of naive approaches. Combined with gradient descent, this allowed the correction terms to be tuned iteratively until the simulated and measured probability distributions converged. The approach draws on the framework of universal differential equations, in which physical models are embedded in differentiable programs that machine learning tools can optimize end to end.

The results are striking. With the uncorrected Hamiltonian, the worst-case average absolute error in predicting the final measurement probabilities reached 0.48, meaning the theoretical model could be wrong by nearly half a full probability unit, a level of disagreement that would render any gate calibration based on the model unreliable. After the data-driven correction was applied, that worst-case error dropped below 0.05, an improvement of nearly a factor of ten. The corrected Hamiltonian tracked the survival probabilities of all four initial two-qubit states across the full range of pulse amplitudes tested, reproducing the fine structure of the hardware’s time evolution that the standard model missed entirely.

Why does this matter beyond one pair of qubits on one machine? Gate fidelities in today’s quantum processors are limited not only by decoherence but by systematic modeling errors, and as the field marches toward error-corrected, fault-tolerant machines, every fraction of a percent of gate accuracy counts. Accurate Hamiltonian models are also the foundation of optimal control techniques, which design pulse sequences to implement gates as faithfully as possible. If the underlying model is wrong, even the most sophisticated pulse optimization will steer the quantum system to the wrong destination. A data-driven correction loop, in which the model continuously learns from the hardware it describes, offers a path to calibrations that stay faithful as devices drift and age.

The work also highlights a broader shift in how quantum engineers think about the relationship between theory and experiment. Traditional approaches either derive effective Hamiltonians perturbatively from circuit theory or fit a handful of parameters to targeted experiments. The scientific machine learning approach is more ambitious: it treats the Hamiltonian itself as a learnable object, constrained by physics but refined by data, and it embraces the fact that multiple effective descriptions may be equally valid. That equivalence-class perspective is philosophically interesting, but it is also practically useful, because it tells engineers which features of a correction are physically meaningful and which are artifacts of a particular parameterization.

Francis and Shaji’s study, published in Quantum Information Processing, was supported in part by India’s National Quantum Mission and used the high-performance computing facilities at IISER Thiruvananthapuram. The underlying data have been made openly available, allowing other groups to test and extend the method on different hardware platforms. As quantum processors scale to hundreds and eventually thousands of qubits, manual calibration of every interaction will become impossible, and techniques like this one, which let the machine teach itself the physics of its own imperfections, may become as essential to quantum computing as the qubits themselves.

Subject of Research: Data-driven correction of the cross-resonance Hamiltonian of transmon qubits using scientific machine learning

Article Title: Data-driven Hamiltonian correction of transmon qubits

Article References: Francis, J. G., & Shaji, A. (2026). Data-driven Hamiltonian correction of transmon qubits. Quantum Information Processing, 25(10), Article 328. https://doi.org/10.1007/s11128-026-05350-7

Image Credits: AI Generated

DOI: 10.1007/s11128-026-05350-7

Keywords: transmon qubits, Hamiltonian correction, cross-resonance gate, scientific machine learning, IBM Quantum, quantum computing, gradient descent, adjoint sensitivity, quantum control, superconducting qubits, gate fidelity, quantum information processing