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arXiv:2111.05765 · Physical Review Applied

Direct Calculation of Z Z Interaction Rates in Multimode Circuit Quantum Electrodynamics

Firat Solgun, Srikanth J. Srinivasan

Published Oct 11, 2022• 13 citations
arXivHTMLPDF

Abstract

Hamiltonians of the superconducting qubits of transmon type involve nonzero $ZZ$ interaction terms due to their finite and small anharmonicities. These terms might lead to the unwanted accumulation of spurious phases during the execution of the two-qubit gates. Exact calculation of the $ZZ$ interaction rates requires the full diagonalization of the circuit Hamiltonians, which very quickly becomes computationally demanding as the number of the modes in the coupler circuit increases. Here we propose a direct analytical method for the accurate estimation of the $ZZ$ interaction rates between low-anharmonicity qubits in the dispersive limit of the multimode circuit quantum electrodynamics (circuit QED). We observe very good agreement between the predictions of our method and the measurement data collected from the multiqubit devices.

DOI
10.1103/physrevapplied.18.044025

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Hamiltonians of the superconducting qubits of transmon type involve nonzero ZZZZZZ interaction terms due to their finite and small anharmonicities. These terms might lead to the unwanted accumulation of spurious phases during the execution of the two-qubit gates. Exact calculation of the ZZZZZZ interaction rates requires the full diagonalization of the circuit Hamiltonians, which very quickly becomes computationally demanding as the number of the modes in the coupler circuit increases. Here we propose a direct analytical method for the accurate estimation of the ZZZZZZ interaction rates between low-anharmonicity qubits in the dispersive limit of the multimode circuit quantum electrodynamics (circuit QED). We observe very good agreement between the predictions of our method and the measurement data collected from the multiqubit devices.
Superconducting qubits are one of the most mature platforms for achieving practical quantum computers, but significant performance improvements are still needed. To improve the engineering of these systems, three-dimensional (3D) full-wave computational electromagnetics analyses are increasingly being turned to. Unfortunately, existing analysis approaches often rely on full-wave simulations using eigenmode solvers that are typically cumbersome, not robust, and computationally prohibitive if devices with more than a few qubits are to be analyzed. To improve the characterization of superconducting circuits while circumventing these drawbacks, this work begins the development of an alternative modeling framework that we illustrate in the context of evaluating the qubit-qubit exchange coupling rate between transmon qubits. This quantity is a key design parameter that determines the entanglement rate for fast multiqubit gate performance and also affects decoherence sources like qubit crosstalk. Our modeling framework uses a field-based formalism in the context of macroscopic quantum electrodynamics, which we use to show that the qubit-qubit exchange coupling rate can be related to the electromagnetic dyadic Green's function linking the qubits together. We further show how the needed quantity involving the dyadic Green's function can be related to the impedance response of the system that can be easily and efficiently computed with classical computational electromagnetics tools. We demonstrate the validity and efficacy of this approach by simulating four practical multiqubit superconducting circuits and evaluating their qubit-qubit exchange coupling rates. We validate our results against a 3D numerical diagonalization method and against experimental data where available. We also demonstrate the impact of the qubit-qubit exchange coupling rate on qubit crosstalk by simulating a multicoupler device and identifying operating points where the qubit crosstalk becomes zero.
Superconducting circuits consisting of a few low-anharmonic transmons coupled to readout and bus resonators can perform basic quantum computations. Since the number of qubits in such circuits is limited to not more than a few tens, the qubits can be designed to operate within the dispersive regime where frequency detunings are much stronger than coupling strengths. However, scaling up the number of qubits will bring the circuit out of the regime, and this invalidates current theories. We develop a formalism that allows to consistently diagonalize the superconducting circuit Hamiltonian beyond the dispersive regime. This will allow to study qubit-qubit interaction unperturbatively, therefore, our formalism remains valid and accurate at small or even negligible frequency detuning; thus, our formalism serves as a theoretical ground for designing qubit characteristics for scaling up the number of qubits in superconducting circuits. We study the most important circuits with single- and two-qubit gates, i.e., a single transmon coupled to a resonator and two transmons sharing a bus resonator. Surprisingly, our formalism allows to determine the circuit characteristics, such as dressed frequencies and Kerr couplings, in closed-form formulas that not only reproduce perturbative results, but also extrapolate beyond the dispersive regime and can ultimately reproduce (and even modify) the Jaynes-Cumming results at resonant frequencies.
Superconducting qubits are one of the leading platforms for creating practical quantum computers. However, these architectures face significant engineering challenges as they scale up and move towards devices capable of error correction. To aid these efforts, three-dimensional (3D) full-wave computational electromagnetics tools are increasingly being utilized to characterize these devices; however, standard approaches relying on eigenmode solvers are often computationally prohibitive and lack the scalability required to analyse devices beyond a few qubits. To overcome these limitations, this thesis introduces an efficient, alternative modeling framework for superconducting circuits, applied specifically to the evaluation of the qubit-qubit exchange coupling rate in transmon architectures. This coupling rate is a key design parameter that governs the entanglement speed for multi-qubit gates and influences error channels such as qubit crosstalk. Leveraging a field-based formalism within the broader context of macroscopic quantum electrodynamics, we establish that the exchange coupling rate is directly related to the electromagnetic dyadic Green’s function. We then map the imaginary part of the Green’s function to the system’s classical impedance response. Since the impedance response can be evaluated quickly and accurately in standard full-wave solvers, this new formulation provides a scalable way to characterize large-scale superconducting devices. We validate this framework by simulating four practical multi-qubit circuits, benchmarking the extracted coupling rates against both 3D numerical diagonalization methods and experimental data. We also demonstrate how the exchange coupling rate impacts qubit crosstalk, and characterize a multi-coupler device to find operational sweet spots, where the crosstalk is minimized while maintaining strong enough qubit-qubit coupling to enable fast multi-qubit gates. We then conclude this thesis and outline some plans for future work.
In this thesis we propose new methods in multi-qubit multi-mode circuit quantum electrodynamics (circuit-QED) architectures. In Chapter 2 we describe a direct parity measurement method for three qubits, which can be realized in 2D circuit-QED with a possible extension to four qubits in a 3D circuit-QED setup for the implementation of the surface code. In Chapter 3 we show how to derive Hamiltonians and compute relaxation rates of the multi-mode superconducting microwave circuits consisting of single Josephson junctions using an exact impedance synthesis technique (the Brune synthesis) and applying previous formalisms for lumped element circuit quantization. In the rest of the thesis we extend our method in Chapter 3 to multi-junction (multi-qubit) multi-mode circuits through the use of state-space descriptions which allows us to quantize any multiport microwave superconducting circuit with a reciprocal lossy impedance response.
Devices built using superconducting circuits are a popular approach being pursued to develop quantum information processing hardware. These systems involve the quantized interactions of electromagnetic fields with large coherent collections of charges in superconductors, and so are typically referred to as macroscopic circuit quantum electrodynamics (QED) devices or simply circuit QED devices. Although significant progress has been made recently, substantial engineering efforts are required to improve performance and scale these devices so they can address problems of practical interest. Performing these engineering efforts is difficult due to the absence of rigorous numerical modeling approaches. This work begins to address this by providing a new mathematical description of a commonly used circuit QED system, a transmon qubit coupled to microwave transmission lines. Expressed in terms of three-dimensional vector fields, our model is better suited to developing rigorous numerical solvers than approximate lumped element circuit descriptions used in the literature. We present details on the quantization of our model, and derive equations of motion for the coupled field-transmon system. These results can be used in developing full-wave solvers in the future. To make this work more accessible, we assume a limited amount of training in quantum physics and provide many background details.
Circuit quantum electrodynamics (QED) studies the interaction of artificial atoms, open transmission lines, and electromagnetic resonators fabricated from superconducting electronics. While the theory of an artificial atom coupled to one mode of a resonator is well studied, considering multiple modes leads to divergences which are not well understood. Here, we introduce a first-principles model of a multimode resonator coupled to a Josephson junction atom. Studying the model in the absence of any cutoff, in which the coupling rate to mode number nnn scales as n\sqrt{n}n​ for nnn up to $\ensuremath{\infty}$, we find that quantities such as the Lamb shift do not diverge due to a natural rescaling of the bare atomic parameters that arises directly from the circuit analysis. Introducing a cutoff in the coupling from a nonzero capacitance of the Josephson junction, we provide a physical interpretation of the decoupling of higher modes in the context of circuit analysis. In addition to explaining the convergence of the quantum Rabi model with no cutoff, our work also provides a useful framework for analyzing the ultrastrong coupling regime of a multimode circuit QED.