For superconducting quantum processors consisting of low anharmonicity qubits such as transmons, we give a complete microwave description of the system in the qubit subspace. We assume that the qubits are dispersively coupled to a distributed microwave structure such that the detunings of the qubits from the internal modes of the microwave structure are stronger than their couplings. We define "qubit ports" across the terminals of the Josephson junctions and "drive ports" where transmission lines carrying drive signals reach the chip and we obtain the multiport impedance response of the linear passive part of the system between the ports. We then relate interaction parameters in between qubits and between the qubits and the environment to the entries of this multiport impedance function; in particular, we show that the exchange coupling rate J between qubits is related in a simple way to the off-diagonal entry connecting the qubit ports. Similarly, we relate couplings of the qubits to voltage drives and lossy environment to the entries connecting the qubits and the drive ports. Our treatment takes into account all the modes (possibly infinite) that might be present in the distributed electromagnetic structure and provides an efficient method for the modeling and analysis of the circuits.
For superconducting quantum processors consisting of low anharmonicity qubits such as transmons, we give a complete microwave description of the system in the qubit subspace. We assume that the qubits are dispersively coupled to a distributed microwave structure such that the detunings of the qubits from the internal modes of the microwave structure are stronger than their couplings. We define "qubit ports" across the terminals of the Josephson junctions and "drive ports" where transmission lines carrying drive signals reach the chip and we obtain the multiport impedance response of the linear passive part of the system between the ports. We then relate interaction parameters in between qubits and between the qubits and the environment to the entries of this multiport impedance function; in particular, we show that the exchange coupling rate J between qubits is related in a simple way to the off-diagonal entry connecting the qubit ports. Similarly, we relate couplings of the qubits to voltage drives and lossy environment to the entries connecting the qubits and the drive ports. Our treatment takes into account all the modes (possibly infinite) that might be present in the distributed electromagnetic structure and provides an efficient method for the modeling and analysis of the circuits.
For superconducting quantum processors consisting of low anharmonicity qubits such as transmons, we give a complete microwave description of the system in the qubit subspace. We assume that the qubits are dispersively coupled to a distributed microwave structure such that the detunings of the qubits from the internal modes of the microwave structure are stronger than their couplings. We define “qubit ports” across the terminals of the Josephson junctions and “drive ports” where transmission lines carrying drive signals reach the chip and we obtain the multiport impedance response of the linear passive part of the system between the ports. We then relate interaction parameters in between qubits and between the qubits and the environment to the entries of this multiport impedance function; in particular, we show that the exchange coupling rate J between qubits is related in a simple way to the off-diagonal entry connecting the qubit ports. Similarly, we relate couplings of the qubits to voltage drives and lossy environment to the entries connecting the qubits and the drive ports. Our treatment takes into account all the modes (possibly infinite) that might be present in the distributed electromagnetic structure and provides an efficient method for the modeling and analysis of the circuits.
Superconducting qubit circuits are one of the prime candidates for building the quantum computer. In this guide I will provide a complete microwave description of the superconducting qubit circuits consisting of the most popular superconducting qubit the "Transmon". Quantum interaction rate parameters related to simple impedance response formulas will allow microwave engineers to design and analyze superconducting qubit circuits without resorting to heavy machinery of quantum mechanical transformations. All formulas listed below are valid in the dispersive regime of low anharmonicity qubits. This work is a review of [1].
In an attempt to better leverage superconducting quantum computers, scaling efforts have become the central concern. These efforts have been further exacerbated by the increased complexity of these circuits. The added complexity can introduce parasitic couplings and resonances, which may hinder the overall performance and scalability of these devices. We explore a method of modeling and characterization based on multiport impedance functions that correspond to multi-qubit circuits. By combining vector fitting techniques with a novel method for interconnecting rational impedance functions, we are able to efficiently construct Hamiltonians for multi-qubit circuits using electromagnetic simulations. Our methods can also be applied to circuits that contain both lumped and distributed element components. The constructed Hamiltonians account for all the interactions within a circuit that are described by the impedance function. We then present characterization methods that allow us to estimate effective qubit coupling rates, state-dependent dispersive shifts of resonant modes, and qubit relaxation times.
Building on the first-order circuit quantization method [arXiv:2304.12252, arXiv:2401.09120], we provide simple formulas to construct exact Hamiltonians for Josephson-junction-based superconducting qudits capacitively, inductively, or galvanically coupled to passive linear environments. These environments may be multiport, multimode, discrete or continuous, reciprocal or nonreciprocal, and are characterized directly by their impedance or admittance matrices. In the weak-coupling regime, we further derive \emph{divergence-free} dispersive Hamiltonians for mode-resolved environments and transition-resolved weak-coupling master equations for dissipative continua. Mode structure, frequency renormalizations, environment-mediated interactions, decay rates, and directional cross couplings then follow from the same causal immittance response, while spurious Lamb-shift divergences arising from uncontrolled approximations in previous treatments are made explicit and avoided. We apply the theory to a set of illustrative circuits comprising a discrete resonator filter, finite-band metamaterial environments, nonreciprocal waveguide-QED systems, and superconducting giant atoms, for which analytical response matrices can be obtained, although the method is particularly well suited to numerical responses from electromagnetic solvers or experimental characterization. We thereby extend the black-box quantization framework to multiport, dissipative, and nonreciprocal settings, establishing a simple and scalable route toward optimized and automated electromagnetic design of large-scale superconducting quantum hardware.
We provide a systematic method for constructing effective dispersive Lindblad master equations to describe weakly anharmonic superconducting circuits coupled by a generic dissipationless nonreciprocal linear system, with effective coupling parameters and decay rates written in terms of the immittance parameters characterizing the coupler. This article extends the foundational work of Solgun et al. [IEEE Trans. Microw. Theory Techn. 67, 928 (2019)] for linear reciprocal couplers described by an impedance response. Notably, we expand the existing toolbox to incorporate nonreciprocal elements, account for direct stray coupling between immittance ports, circumvent potential singularities, and include collective dissipative effects that arise from interactions with external common environments. We illustrate the use of our results with a circuit of weakly anharmonic Josephson junctions coupled to a multiport nonreciprocal environment and a dissipative port. The results obtained here can be used for the design of complex superconducting quantum processors with nontrivial routing of quantum information, as well as analog quantum simulators of condensed matter systems.
The precise engineering of electromagnetic couplings is paramount for constructing scalable and high-fidelity superconducting quantum processors. While essential for orchestrating qubit operations, these couplings also present significant design challenges, including the mitigation of crosstalk and the management of environmental decoherence. A clear and unified theoretical framework is therefore crucial for the design, simulation, and analysis of these complex quantum circuits. This paper presents a comprehensive theoretical treatment of the fundamental electromagnetic coupling mechanisms in superconducting devices. Starting from first principles, we formulate the equations of motion and derive the input-output relations for canonical systems, including a single resonator coupled to a multi-port microwave network, interacting resonators, and coupled transmission lines. We review rigorous definitions for key parameters such as the energy decay rate (κ) and the dimensionless coupling coefficient (ζ) and connect these formalisms to practical methods of parameter extraction from electromagnetic simulations. This work provides a rigorous and pedagogical foundation for understanding and modeling linear electromagnetic interactions, serving as a vital resource for the development of advanced superconducting quantum hardware.