Incorporating multi-qubit exchange coupling effects between transmon qubits in Maxwell-Schrödinger numerical methods

TL;DR

The paper addresses accurate numerical modeling of multi-qubit entanglement in superconducting transmon devices by embedding multi-qubit exchange coupling into Maxwell-Schrödinger simulations. Using a first-principles derivation and Schrieffer-Wolff transformation, it derives an effective qubit-qubit exchange term $J_{ij}$ that can be computed via impedance parameters $Z_{lm}$, avoiding heavy modal sums. It shows that back-action of qubits on the Maxwell field leads to a classical crosstalk mechanism that can dominate multi-qubit dynamics in cross-resonance gates, and Maxwell-Schrödinger can capture this non-Markovian open-system behavior. Numerical results on a two-qubit cross-resonance circuit validate the approach against fully quantum simulations and demonstrate parameter dependencies (e.g., $C_R$, $\Delta$, $\delta$) on the crosstalk. The work suggests paths toward more realistic device optimization and integration with quantum control, including potential 3D full-wave solvers and tensor-network techniques for scalability, to enable reliable multi-qubit operation in superconducting processors.

Abstract

Superconducting qubits have emerged as a leading platform for realizing quantum computers. Accurate modeling of these devices is essential for predicting performance, improving design, and optimizing control. Many modeling approaches currently rely on lumped circuit approximations or other simplified treatments that can be limited in resolving the interplay between the qubit dynamics and the electromagnetic circuitry, leading to significant experimental deviations from numerical predictions at times. To address many of these limitations, methods that self-consistently solve the Schrödinger equation for qubit dynamics with the classical Maxwell's equations have been developed and shown to accurately predict a wide range of effects related to superconducting qubit control and readout. Despite these successes, these methods have not been able to consider multi-qubit effects that give rise to qubit-qubit entanglement. Here, we address this by rigorously deriving how multi-qubit coupling effects between transmon qubits can be embedded into Maxwell-Schrödinger methods. To support this, we build on earlier first-principles derivations of Maxwell-Schrödinger methods for the specific case of two transmon qubits coupled together through a common electromagnetic system in the dispersive regime. To aid in validating aspects of the Maxwell-Schrödinger framework, we also provide a new interpretation of Maxwell-Schrödinger methods as an efficient simulation strategy to capture the class of non-Markovian open quantum system dynamics. Our results demonstrate that these effects can give rise to strong classical crosstalk that can significantly alter multi-qubit dynamics, which we demonstrate for the cross-resonance gate. These classical crosstalk effects have been noted in cross-resonance experiments, but previous quantum theory and device analysis could not explain their origin.

Figures (12)

• Figure 1: Schematic for deriving the Maxwell-Schrödinger method for a transmon and transmission line system.
• Figure 2: Device schematic with two transmons coupled capacitively to a common transmission line resonator of length $d_\mathrm{R}$. Each transmon is also connected to an independent drive source through additional transmission lines. This circuit enables a cross-resonance gate between the transmons, where for our simulations, Transmon 1 serves as the "control transmon" and Transmon 2 is the "target transmon".
• Figure 3: Simulation results of the basic cross-resonance effect. On each plot, results from two different simulations are shown, corresponding to initial states of $\ket{00}$ or $\ket{10}$. Depending on the state of the control transmon, the Rabi oscillation frequency for the target transmon changes. (a) Results are plotted for our Maxwell-Schrödinger framework with back-action artificially turned off (MS: No BA) and for the standard time evolution using QuTiP of the total Hamiltonian $\hat{H}_J + \hat{H}_{d1}$. (b) The same simulations are repeated but with the Maxwell-Schrödinger framework with back-action turned on (MS: BA) and for the standard time evolution using QuTiP of the total Hamiltonian $\hat{H}_J + \hat{H}_{d2}$ that includes a classical crosstalk term. As noted in previous experiments sheldon2016proceduremagesan2020effective, the presence of a classical crosstalk term can significantly modify the cross-resonance effect.
• Figure 4: The envelope of the voltage $V_\mathrm{C}(t)$ that is responsible for the classical crosstalk in Fig. \ref{['sec3a:Image:Uni_Rabi']}. This voltage is naturally evaluated in the Maxwell-Schrödinger framework as the back-action of the control and target transmons into the transmission line resonator. We monitor this voltage for simulations when only the back-action of the control transmon is considered (Control BA) and when both the control and target transmons back-action are considered (Total BA). These are compared against the envelope of the external classical drive that was required to make the results of the drive Hamiltonian $\hat{H}_{d2}$ in our quantum simulation in Fig. \ref{['sec3a:Image:Uni_Rabi']}(b) match the Maxwell-Schrödinger results. These envelopes are shown for initial states of (a) $\ket{00}$ and (b) $\ket{10}$.
• Figure 5: Maxwell-Schrödinger simulations with back-action (MS: BA) and without (MS: No BA) for the circuit of Fig. \ref{['fig:two-qubit-schematic']} while varying the coupling capacitances to the resonator. Each simulation starts in the $\ket{00}$ state and we apply a $340\;\mathrm{ns}$ duration pulse. The device is symmetric so that $C_\mathrm{R}^{(1)}=C_\mathrm{R}^{(2)}=C_\mathrm{R}$ and $C_\mathrm{D}^{(1)}=C_\mathrm{D}^{(2)}=0.1\;\mathrm{fF}$. Qubit frequencies are fixed at $\bar{q}_{0,1}^{(1)}/2\pi=4.91\;\mathrm{GHz}$ and $\bar{q}^{(2)}_{0,1}/2\pi=5.11\;\mathrm{GHz}$. The resonator coupling capacitances are (a) $C_\mathrm{R}=3\;\mathrm{fF}$, (b) $C_\mathrm{R}=4\;\mathrm{fF}$, and (c) $C_\mathrm{R}=5\;\mathrm{fF}$.