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IEEE MTT-S International Conference on Numerical Electromagnetic and Multiphysics Modeling and Optimization

Finite Element Time Domain Discretization of a Semiclassical Maxwell-Schrödinger Model of a Transmon Qubit

Thomas E. Roth

Published Jun 28, 2023• 5 citations
Publisher

Abstract

In superconducting circuit quantum computers, classical microwave pulses are used to control and measure the state of the qubits. The design of microwave pulses for these purposes typically use simple theoretical or numerical models that do not account for the self-consistent interactions of how the presence of the transmon modifies the applied microwave field. In this work, we present the formulation and finite element time domain discretization of a semiclassical Maxwell-Schrödinger model that can self-consistently characterize the interactions between a transmon and classical microwave pulses. Our numerical results demonstrate the validity of this approach for describing the central effects related to the control and measurement of the state of a transmon. In the future, our numerical method can be used to efficiently investigate a wider range of operating regimes than existing approaches are valid for to search for more effective control and measurement protocols.

DOI
10.1109/nemo56117.2023.10202378

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In superconducting circuit quantum computers, classical microwave pulses are used to control and measure the state of the qubits. The design of microwave pulses for these purposes typically use simple theoretical or numerical models that do not account for the self-consistent interactions of how the presence of the transmon modifies the applied microwave field. In this work, we present the formulation and finite element time domain discretization of a semiclassical Maxwell-Schrödinger model that can self-consistently characterize the interactions between a transmon and classical microwave pulses. Our numerical results demonstrate the validity of this approach for describing the central effects related to the control and measurement of the state of a transmon. In the future, our numerical method can be used to efficiently investigate a wider range of operating regimes than existing approaches are valid for to search for more effective control and measurement protocols.
In superconducting circuit quantum information technologies, classical microwave pulses are applied to control and measure the qubit states. Currently, the design of these microwave pulses use simple theoretical or numerical models that do not account for the self-consistent interactions of how the qubit state modifies the applied microwave pulse. In this work, we present the formulation and finite element time domain discretization of a semiclassical Maxwell-Schrödinger method for describing these self-consistent dynamics for the case of a superconducting qubit capacitively coupled to a general transmission line network. We validate the proposed method by characterizing key effects related to common control and measurement approaches for transmon and fluxonium qubits in systems that are amenable to theoretical analysis. Our numerical results also highlight scenarios where including the self-consistent interactions are essential. By treating the microwaves classically, our method is substantially more efficient than fully-quantum methods for the many situations where the quantum statistics of the microwaves are not needed. Further, our approach does not require any reformulations when the transmission line system is modified. In the future, our method can be used to rapidly explore broader design spaces to search for more effective control and measurement protocols for superconducting qubits.
In superconducting circuit quantum information technologies, classical microwave pulses are applied to control and measure the qubit states. Currently, the design of these microwave pulses uses simple theoretical or numerical models that do not account for the self-consistent interactions of how the qubit state modifies the applied microwave pulse. In this work, we present the formulation and finite element time domain discretization of a semiclassical Maxwell-Schrödinger method for describing these self-consistent dynamics for the case of a superconducting qubit capacitively coupled to a general transmission line network. We validate the proposed method by characterizing key effects related to common control and measurement approaches for transmon and fluxonium qubits in systems that are amenable to theoretical analysis. Our numerical results also highlight scenarios where including the self-consistent interactions is essential. By treating the microwaves classically, our method is substantially more efficient than fully-quantum methods for the many situations where the quantum statistics of the microwaves are not needed. Further, our approach does not require any reformulations when the transmission line system is modified. In the future, our method can be used to rapidly explore broader design spaces to search for more effective control and measurement protocols for superconducting qubits.
The control and readout of a transmon qubit through applied microwave pulses is a key capability in operating many quantum technologies. Currently, the design of microwave pulses for these purposes is done through simple theoretical or numerical models that do not account for how the transmon modifies the applied microwave field. In this work, we present the formulation and discretization of a self-consistent semiclassical model that solves the Schrödinger equation describing qubit dynamics in tandem with a one-dimensional wave equation describing transmission line dynamics. Numerical results are presented using this Maxwell-Schrödinger method to characterize the control and readout of a transmon qubit. We show that our method matches simple theoretical predictions in relevant operating regimes and also look at situations where simplified models break down. In the future, our method can be used to explore these broader operating regimes to search for more effective control and readout protocols.
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.
In superconducting circuit quantum systems, transmon qubits are typically controlled through applying classical microwave pulses on a transmission line capacitively coupled to the qubit. In order to design a pulse that achieves the desired result, modeling the coupled dynamics of the transmission line and transmon is necessary. Existing models consider the dynamics of both portions of the system, but often fail to account for effects such as the back-action of each subsystem on the other. This work begins to study this effect using a 1-D finite element time-domain solution to the equations governing the transmission line flux and transmon phase in a fully classical treatment, which will be extended to a full-wave semiclassical model in a future work.
Transmon qubits integrated into microwave circuitry are a leading quantum computing architecture. However, significant engineering improvements are still needed. General-purpose numerical methods are becoming of interest to tackle these challenges, but existing fully-quantum methods are highly inefficient. In this work, we formulate a semiclassical Maxwell-Schrödinger model as an alternative approach to efficiently analyze the interactions of transmons capacitively coupled to transmission line networks. Additionally, we present a suitable discretization and time marching strategy. We validate our approach by comparing it to theoretical models relevant to the dispersive operating regime used in most hardware implementations. Our results show that our method can accurately predict effects related to transmon state measurements that are typically derived using fully-quantum models. In the future, our approach can be used to rapidly explore broader design spaces to improve the speed and fidelity of qubit state measurements, which is a key obstacle toward achieving fault-tolerant quantum computation.