Calculating Multi-Qubit Exchange Coupling Rates for Transmon Qubits Using a Field-Based Formalism
Ghazi Khan, Thomas E. Roth · 2024
Superconducting qubits are one of the most mature approaches to realize quantum computers. However, to meet the stringent performance requirements, these systems are becoming increasingly complex. To help solve this engineering problem, high-fidelity numerical methods are becoming a key resource. Unfortunately, existing approaches use full-wave computational electromagnetics (CEM) methods in inefficient ways that become impractical if even a few qubits are considered. Here, we derive a simple formula that can be efficiently evaluated using typical full-wave CEM tools to numerically calculate the exchange coupling rate between multiple transmons; which is a key system parameter involved in the design of multi-qubit entangling gates in modern quantum processors. Our approach is applicable to the common case of transmons operating in the dispersive regime where the operating frequency of the electromagnetic (EM) resonator is far detuned from the transition frequencies of the qubits coupled to it. We start from our field-theory description of these systems (T. E. Roth and W. C. Chew, “Macroscopic circuit quantum electrodynamics: A new look toward developing full-wave numerical models,” IEEE Journal on Multiscale and Multiphysics Computational Techniques, Vol. 6, 109-124,2021) with multiple qubits and perform a Schrieffer-Wolf transform to rewrite the system Hamiltonian into the dispersive regime. This re-expresses the indirect qubit-field coupling of the original Hamiltonian as direct qubit-qubit interactions with an exchange coupling rate that depends on properties of the EM system the qubits are embedded in. From here, typical field-theory techniques can be used to relate this exchange coupling rate to the EM Green's function, which can further be related to the impedance matrix between the qubits. This quantity can be efficiently computed with standard full-wave CEM tools, which is in contrast to prevailing approaches that rely on performing EM eigenmode decompositions. We qualitatively verified our formula using a geometry (Fig. 1, left) based on (Filipp, Stefan, et al. “Multimode mediated qubit-qubit coupling and dark-state symmetries in circuit quantum electrodynamics.” Physical Review A 83.6 (2011): 063827), and have also quantitatively tested it against other calculation methodologies (Fig. 2, right) (Solgun, Firat, David P. DiVincenzo, and Jay M. Gambetta. “Simple impedance response formulas for the dispersive interaction rates in the effective Hamiltonians of low anharmonicity superconducting qubits.” IEEE Transactions on Microwave Theory and Techniques 67.3 (2019): 928–948).