Session B36: Supercondcuting Qubits: Circuit Analysis

Controlling Qubit Relaxation in 1-3 Dimensions

In the circuit QED architecture, the coupling of a qubit through its associated cavity to the cavity drive port sets the ultimate limit on coherence.  For lumped element realizations this so-called single-mode Purcell limit stringently limits qubit coherence, unless an external filter (often referred to as a Purcell filter) is used to suppress photon emission at the qubit's frequency.  However, in 2- and 3-D circuits, we can often greatly exceed the single mode Purcell limit [Houck, et.al, PRL (2008)].  In this talk, we present a method for visualizing transmon fields to find port placements which can achieve both large cavity bandwidths and long qubit lifetimes.  We find that when the qubit frequency is below the lowest cavity mode, this effect cannot be explained by interference between multiple cavity modes, but rather the variation of the qubit mode's fields in space.  We will present data from transmons in both 3-D post cavity and chip-in-tube geometries.  We have achieved coherences as high as 100 microseconds in systems with strong qubit-cavity couplings, short cavity lifetimes, and no external Purcell filtering.  This technique can also readily be extended to planar qubit/resonator designs and is thus broadly useful across many implementations of superconducting circuits.   

A Tunable Multi-Qubit Coupling Element for Superconducting Circuits

We propose a SQUID tunable microwave coplanar waveguide (CPW) resonator to

implement programmable (up to all-to-all), tunable coupling between superconducting qubits. In

order to make qualitative and quantitative predictions about the behavior of this circuit element

as a tunable coupling element we develop a theoretical description of its dynamics. We will

outline a framework for analyzing dynamics in the presence of parametric flux modulation

through its boundary SQUIDs. This nonlinearity at the resonator boundary introduces mixing

between the central resonance of the system and the sidebands at integer multiples of the

modulation frequency. These sidebands yield an effective amplitude modulation of the resonance

mode profile from which we compute the resulting time dependent coupling of the tunable

resonator to the superconducting qubits. We present analytical and numerical results for the

coupling, demonstrating quantitative agreement. This time-dependent qubit-resonator coupling

can be used as the basis for two qubit gates and interactions in digital and analog quantum

computing protocols.   

A non-reciprocal dispersive interaction in circuit QED: Part I, formulation and setup

Non-reciprocity is a valuable property for building interesting and complex quantum systems. Previous experimental studies have primarily focused on non-reciprocal excitation transfer between linear modes, which can be fully captured by a non-Hermitian effective Hamiltonian matrix. To go beyond these semi-classical descriptions, we implement a dispersive type of non-reciprocal interaction between a transmon qubit and a superconducting cavity. We achieve this by utilizing a custom designed ferrite circulator to mediate B-field tunable non-reciprocity between two cavities[1], with one of them housing the qubit. We were able to describe the input-output spectrum of this device with a 2 mode non-Hermitian linear model coupled to a qubit. By adjusting the coupling parameters we were able to put the system in a regime where the intermediary mode could be eliminated and the model could be adiabatically reduced to a master equation involving only the cavity and the qubit, utilizing a non-linear jump operator. This model can then be used to predict dispersive shifts and dephasing, showing notable asymmetry across magnetic fields due to the non-reciprocity.

[1] Y. Wang, S. van Geldern, T. Connolly, Y. Wang, A. Shilcusky, A. McDonald, A. A. Clerk and C. Wang, arXiv:2106.11283(2021).   

A non-reciprocal dispersive interaction in circuit QED: Part II, measurements and analysis

Non-reciprocity is a valuable property for building interesting and complex quantum systems.  Previous experimental studies have primarily focused on non-reciprocal excitation transfer between linear modes, which can be fully captured by a non-Hermitian effective Hamiltonian matrix. To go beyond these semi-classical descriptions, we implement a dispersive type of non-reciprocal interaction between a transmon qubit and a superconducting cavity, where the system dynamics can be captured by a cavity-qubit nonlinear jump operator in a new effective theory of non-reciprocity. This theory is a result of adiabatic elimination and its predictions must be compared to time-averaged quantities. Thus, we experimentally measured the time-averaged qubit dispersive shifts and dephasing rates from a decaying photon population, which agree excellently with model predictions, showing a notable asymmetry between positive and negative applied magnetic fields due to the non-reciprocity. We further verify the usefulness of the model by studying the dispersive shifts and dephasing rates in the steady-state regime under a continuous cavity drive, which show good agreement with theory. Furthermore, we explore potential applications of this model, such as qubit phase gate operations.   

Direct Calculation of the ZZ-interaction Rates in the Multi-Mode circuit-QED

Superconducting quantum processors might suffer from unwanted ZZ-interactions which naturally arise as a result of the finite and small anharmonicities of the Transmon qubits. Non-zero ZZ-terms in the qubit Hamiltonians can lead to the accumulation of the spurious phases during the execution of certain two-qubit gates. Recently novel qubit-qubit couplers have been proposed to suppress the ZZ-interactions while keeping a finite exchange coupling strength necessary to run the two-qubit gates. These new couplers typically involve multiple modes and their engineering requires diagonalization of the multi-mode Hamiltonians which can very quickly become computationally demanding as the number of modes of the coupler increases. Here we describe a new method for the accurate calculation of the ZZ-rates in the multi-mode circuit-QED by capturing their frequency dependence with simple relations of the entries of the multiport impedance matrix defined between the qubit ports. We observe very good agreement between the predictions of our method and the measurement data collected from the multi-qubit devices. Our method being an extension of the framework developed in (**) makes the quantum microwave engineering of the superconducting qubits more streamlined.

(*) Firat Solgun, Srikanth Srinivasan, “Direct calculation of the ZZ-interaction rates in the multi-mode circuit-QED”, arXiv:2111.05765.

(**) Firat Solgun, David P. DiVincenzo and Jay M. Gambetta, “Simple Impedance Response Formulas for the Dispersive Interaction Rates in the Effective Hamiltonians of Low Anharmonicity Superconducting Qubits”, IEEE Transactions on Microwave Theory and Techniques, Volume: 67, Issue: 3, March 2019.

*F. S. acknowledges support from Intelligence Advanced Research Projects Activity (IARPA) under contract W911NF-16-1-0114-FE.   

Majorana fermions revealing the true nature of quantum phase slip junctions

Quantum phase slips (QPS) are famously considered to be the exact dual to the Josephson effect. However, if taken at face value, this duality stipulates that charge transported across a QPS junction is continuous, as opposed to regular Josephson junctions, which perserve charge quantization. We here analyse a circuit consisting of a QPS junction shunted by a topological Josephson junction hosting Majoranas, the latter being sensitive to charge transport in units of half a Cooper pair charge. Contrary to common phenomenological approaches, we consider a microscopic derivation of the phase slip processes, using a large Josephson junction array to model the QPS junction. We find that there is a cross-over when increasing the number of junctions in the array, from a regime where the standard, dual description of phase slip junctions is valid, to a regime where charge has to be quantized in integer units of the Cooper pair charge. This cross-over isstraightforwardly visible in the eigenspectrum of the circuit as a function of an externally applied flux. Importantly, we expect that the latter (quantized) regime renders QPS junctions useless for circuit realizations of the Gottesman-Kitaev-Preskill code.   

Directional photon transport in a superconducting circuit lattice

One of the current bottlenecks in quantum computing devices based on circuit quantum electrodynamics (cQED) is in the circulators used for routing microwave photons. The circulators currently used in novel devices are bulky and require large, localized magnetic fields in order to achieve directional transport of the electromagnetic waves, making them incompatible for on-chip implementation with superconducting circuits. We address these limitations by studying a simple superconducting circuit design that achieves directional photon transport using components currently present in standard cQED devices. In a lattice of coupled microwave resonators, with an effective complex hopping term between lattice sites, we achieve directionality by engineering an effective magnetic field for photons that results in a Peierls phase. Implementations that make use of such a phase have already been proposed, using frequency mixers or magnetic fields in Josephson junction rings, but such devices are currently not considered due to their complexity. We instead investigate the possibility of using voltage-driven planar transmission lines to generate the effective Peierls phase. This will result in a simpler implementation of a superconducting circuit that can also be used for on-chip signal routing.   

Non-perturbative analytical diagonalization of Hamiltonians with application to ZZ-coupling suppression and enhancement in circuit-QED

Deriving effective Hamiltonian models plays an essential role in many quantum control and engineering problems. In this talk, we present two symbolic methods for effectively eliminating auxiliary space: the Non-perturbative Analytical Diagonalization (NPAD) and the Recursive Schrieffer-Wolff Transformation (RSWT). While NPAD makes use of the Jacobi iteration and works without the assumption of perturbation, RSWT takes advantage of the recursive structure and avoids the exponentially increasing number of terms in high-order perturbation. Both methods consist of elementary expressions and can be easily automated to obtain closed-form expressions. This opens the possibility for fast operations using adiabatic techniques such as DRAG and STA beyond the perturbative regime.

To demonstrate their application, we studied the ZZ interaction of superconducting qubits. In a near-resonant regime, the algorithm produces an analytical expression for the effective ZZ interaction strength, with an improvement of at least one order of magnitude compared to neglecting the further detuned state. In a weak-dispersive regime, we calculated perturbative correction up to the 8th order and accurately identified a regime with suppressed ZZ interaction in a simple transmon-resonator-transmon model.   

Tunable Directional Photon Scattering from a Pair of Superconducting Qubits

Nonreciprocal integrated devices provide flexibility and scalability for a wide range of on-chip applications, such as integrated photonics, quantum information processing, and nonlinear optics. We demonstrate tunable directional scattering of microwave radiation with just two transmon qubits coupled to a transmission line. When the qubits are tuned in resonance with each other such that the effective distance between them is equal to a quarter of the qubit excitation wavelength, we periodically modulate their transition frequency. The resulting interference between the two qubits enables directional forward or backward scattering of Stokes and anti-Stokes components depending on the relative phase between the local modulation tones in analogy to the optomechanical Kerker effect. Such a nonreciprocal device is compatible with modern superconducting qubit technology and can be used to route microwave radiation for the realization of chiral networks.   

Tunable Multi-qubit couplers with low residual ZZ interactions

Maintaining high gate fidelities when scaling quantum processors to a large number of qubits is key to practically useful quantum computing.  Two-qubit gates based on tunable couplers now reach fidelities above 99.5% in superconducting qubit quantum processors, in which the qubits are typically arranged on a square grid with pair-wise qubit-qubit couplers. However, involving multi-qubit couplers that provide higher connectivity may lead to more efficient algorithms. In this work, we specifically discuss the challenges imposed by residual ZZ interactions in such multi-qubit tunable-coupler architectures. We analyze possibilities to suppress these off-resonant interactions by connecting qubits to the coupler with different polarities of the coupling. Moreover, we investigate the use of additional two-qubit couplers as a way to mitigate ZZ interactions.   

Exact parameterization of qubit dynamics with coupling to a resistive element in the drive line

Understanding the coupling between a quantum circuit and its environment is crucial for qubit development. In this work we derive a complete description of the qubit dynamics, where the solution is given in terms of the experimental circuit parameters. We start from the circuit model for the qubit plus environment, the latter realized by a variable temperature attenuator [1] acting as a thermal bath with Johnson-Nyquist noise. We apply methods of circuit quantum electrodynamics for the qubit-environment combination, following Ref. 2, and obtain a Hamiltonian from which we derive a master equation in the open quantum systems formalism. This approach gives insight into how each circuit element and the drive line temperature affect the qubit dynamics. As proof of concept, we solve the master equation for the specific case of a fixed-frequency, dispersive transmon, recovering the expected result of decohering qubit without dissipation.

[1] S. Simbierowicz et al., Rev. Sci. Instrum. 92, 034708 (2021). 

[2] M. Cattaneo and G. Paraoanu, Adv. Quantum Technol. 2100054 (2021).    

Parasitic free gate

Implementation of high-performance two-qubit gates is a key factor for scalable quantum computation. However, the state-of-the-art superconducting two-qubit gates are yet far from being perfect due to the parasitic ZZ coupling.  In this paper, we propose a parasitic free (PF) gate to suppress such unwanted interaction in use of the tunable coupler. The gate is operated in two modes: in idle mode the coupler frequency is tuned such that the two qubits are effectively decoupled; in driven mode the coupler frequency is tuned so as to result in a nonzero static ZZ interaction, which later can be cancelled by the dynamical part. Our theory shows that using this method the fidelity of a CR gate is able to achieve the coherence limit.