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arXiv:1405.1915 · Physical Review A

Tunable coupler for superconducting Xmon qubits: Perturbative nonlinear model

Michael R. Geller, Emmanuel Donate, Yu Chen, Michael T. Fang, Nelson Leung, C. Neill, P. Roushan, John M. Martinis

Published Jul 17, 2015• 83 citations
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Abstract

We study a recently demonstrated design for a high-performance tunable coupler suitable for superconducting Xmon and planar transmon qubits [Y. Chen et al., Phys. Rev. Lett. 113, 220502 (2014)]. The coupler circuit uses a single flux-biased Josephson junction and acts as a tunable current divider. We calculate the effective qubit-qubit interaction Hamiltonian by treating the nonlinearity of the qubit and coupler junctions perturbatively. We find that the qubit nonlinearity has two principal effects: The first is to suppress the magnitude of the transverse ${\ensuremath{\sigma}}^{x}\ensuremath{\bigotimes}{\ensuremath{\sigma}}^{x}$ coupling from that obtained in the harmonic approximation by about 15%, assuming typical qubit parameters. The second is to induce a small diagonal ${\ensuremath{\sigma}}^{z}\ensuremath{\bigotimes}{\ensuremath{\sigma}}^{z}$ coupling. The effects of the coupler junction nonlinearity are negligible in the parameter regime considered. The approach used here can be applied to other complex nonlinear circuits arising in the design of superconducting hardware for quantum information processing.

DOI
10.1103/physreva.92.012320

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We study a recently demonstrated design for a high-performance tunable coupler suitable for superconducting Xmon and planar transmon qubits [Y. Chen et al., Phys. Rev. Lett. 113, 220502 (2014)]. The coupler circuit uses a single flux-biased Josephson junction and acts as a tunable current divider. We calculate the effective qubit-qubit interaction Hamiltonian by treating the nonlinearity of the qubit and coupler junctions perturbatively. We find that the qubit nonlinearity has two principal effects: The first is to suppress the magnitude of the transverse \ensuremathσx\ensuremath⨂\ensuremathσx{\ensuremath{\sigma}}^{x}\ensuremath{\bigotimes}{\ensuremath{\sigma}}^{x}\ensuremathσx\ensuremath⨂\ensuremathσx coupling from that obtained in the harmonic approximation by about 15%, assuming typical qubit parameters. The second is to induce a small diagonal \ensuremathσz\ensuremath⨂\ensuremathσz{\ensuremath{\sigma}}^{z}\ensuremath{\bigotimes}{\ensuremath{\sigma}}^{z}\ensuremathσz\ensuremath⨂\ensuremathσz coupling. The effects of the coupler junction nonlinearity are negligible in the parameter regime considered. The approach used here can be applied to other complex nonlinear circuits arising in the design of superconducting hardware for quantum information processing.
Tunable Josephson harmonics open new avenues for qubit design. We demonstrate a superconducting circuit element consisting of a tunnel junction in series with a SQUID loop, yielding a Josephson potential whose harmonic content is strongly tunable by magnetic flux. Through spectroscopy of the first four qubit transitions, together with an effective single-mode model renormalized by the internal mode, we resolve a second harmonic with an amplitude up to ∼10%\sim10\%∼10% of the fundamental. We identify a flux sweet spot where the dispersive shift vanishes, achieved by balancing the dispersive couplings to the internal and qubit modes. This highly tunable element provides a route toward protected qubits and customizable nonlinear microwave devices.
The development of modular and versatile quantum interconnect hardware is a key next step in the scaling of quantum information platforms to larger size and greater functionality. For superconducting quantum systems, fast and well-controlled tunable circuit couplers will be paramount for achieving high-fidelity and resource-efficient connectivity, whether for performing two-qubit gate operations, encoding or decoding a quantum data bus, or interfacing across modalities. Here we propose a versatile and internally tunable double-transmon coupler (DTC) architecture that implements tunable coupling via flux-controlled interference in a three-junction dc superconducting quantum interference device. Crucially, the DTC possesses an internally defined zero-coupling state that is independent of the coupled data qubits or circuit resonators. This makes it particularly attractive as a modular and versatile design element for realizing fast and robust linear coupling in several applications such as high-fidelity two-qubit gate operations, qubit readout, and quantum bus interfacing.
A superconducting fluxonium circuit is an RF-superconducting quantum interference device-type flux qubit that uses a large inductance built from an array of Josephson junctions or a high kinetic inductance material. This inductance suppresses charge sensitivity exponentially and flux sensitivity quadratically. In contrast to the transmon qubit, the anharmonicity of fluxonium can be large and positive, allowing for better separation between the low energy qubit manifold of the circuit and higher-lying excited states. Here, we propose a tunable coupling scheme for implementing two-qubit gates on fixed-frequency fluxonium qubits, biased at half flux quantum. In this system, both qubits and couplers are coupled capacitively and implemented as fluxonium circuits with an additional harmonic mode. We investigate the performance of the scheme by simulating a universal two-qubit fSim gate. In the proposed approach, we rely on a planar on-chip architecture for the whole device. Our design is compatible with existing hardware for transmon-based devices with the additional advantage of lower qubit frequency facilitating high-precision gating.
Superconducting quantum circuits typically use capacitive charge-based linear coupling schemes to control interactions between elements such as qubits. While simple and effective, this coupling scheme makes it difficult to satisfy competing circuit design requirements such as maintaining large qubit anharmonicity and coherence along with a high degree of qubit connectivity and packing density. Moreover, tunable interactions using linear coupling elements produce dynamical variations in mode hybridization, which can induce non-adiabatic transitions, resulting in leakage errors and limiting gate speeds. In this work we attempt to address these challenges by proposing a junction-based coupling architecture based on SQUID (superconducting quantum interference device) couplers with relatively small Josephson energies. SQUID couplers provide intrinsic cross-Kerr interactions that can be controlled by external fluxes and that do not rely on mode hybridization. The small Josephson energies of the coupler maintain the interaction at a perturbative scale, which limits undesired higher-order mixing between coupled elements while achieving a sufficiently strong cross-Kerr interaction originating from diagonal coupling elements. Based on these properties, we show that a SQUID coupler can be used to implement a fast, adiabatic, and high-fidelity controlled-Z gate without introducing extra modes, and the operation is robust against junction asymmetry for high-frequency qubits. Although unconventional crosstalk may arise due to junction asymmetries and parasitic hybridization with spectator qubits, we show that these effects are sufficiently small for realistic circuit parameters. As an example of the utility of such junction-based coupling schemes, we present a scalable tiling strategy for a miniaturized superconducting quantum processor based on merged-element transmon qubits.
This paper presents a theoretical analysis of the recently realized tunable coupler for superconducting phase qubits [R. C. Bialczak et al., arXiv:1007.2219 (unpublished)]. The coupling can be turned off by compensating a negative mutual inductance with a tunable Josephson inductance. The main coupling in this system is of the XXXXXX type and can be zeroed exactly, while there is also a small undesired contribution of the ZZZZZZ type. We calculate both couplings as functions of the tuning parameter (bias current) and focus on the residual coupling in the OFF regime. In particular, we show that for typical experimental parameters the coupling OFF/ON ratio is few times ${10}^{\ensuremath{-}3}$, and it may be zeroed by proper choice of parameters. The remaining errors due to physical presence of the coupler are on the order of ${10}^{\ensuremath{-}6}$.
An improved tunable coupling element for building networks of coupled rf-superconducting quantum interference device (rf-SQUID) flux qubits has been experimentally demonstrated. This new form of coupler, based on the compound Josephson-junction rf-SQUID, provides a sign and magnitude tunable mutual inductance between qubits with minimal nonlinear crosstalk from the coupler tuning parameter into the qubits. Quantitative agreement is shown between an effective one-dimensional model of the coupler's potential and measurements of the coupler persistent current and susceptibility.