Modular tunable coupler for superconducting circuits

Recently, several demonstrations of noisy intermediate scale quantum NISQ processors involving complex systems comprising tens of coupled qubits, have cemented superconducting circuits as a leading platform for large scale quantum processing [1, 2, 3, 4, 5]. Further scaling of quantum processors and the development of distributed quantum architectures will likely benefit from reconfigurable qubit connectivity, tunable dispersive readout, bus interfacing, and transducer interconnects. For all these functionalities the use of a standalone circuit interconnect, a coupler, helps to preserve the coherence of the interconnected circuits while enabling rapid, high fidelity entangling operations between them.

Tunable couplers use an external control parameter to turn on and off an effective coupling $g_{\textrm{eff}}$. In superconducting circuits threading flux through a superconducting quantum interference device (SQuID) and applying microwave driving fields are examples of external control parameters. We highlight two broad approaches to tunable coupling: (i) couplers that use current-divider circuit elements [6, 7, 8, 9, 10, 11, 12, 13, 14, 15] (a recent well-known example is the gmon coupler [11, 12]), and (ii) couplers that interfere direct and virtual interaction pathways [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 20, 26], such as the MIT-style single transmon coupler [16, 17, 18, 19, 20, 21, 22]. For the latter case, mutual inductance or capacitance between two qubits constitutes a direct interaction pathway while interactions mediated by coupler circuitry constitute a virtual interaction pathway. Between these two coupling approaches, current divider couplers respond more linearly with flux, lending themselves to parametric coupling applications [27, 15, 13]. On the other hand, inductively connecting the qubits introduces extra flux degrees of freedom which, by extension, lead to additional noise and decoherence channels, and crosstalk between local flux bias lines. A survey of coupler modalities cited in this paper, therefore, suggests that capacitive interactions lead to cleaner and potentially easier-to-control circuitry. Moreover, for circuit-QED architectures involving transmon qubits, capacitive interactions are compatible with the fixed-frequency designs routinely employed in high-coherence circuits. Therefore, on balance, capacitive rather than inductive coupling seems to provide a significant advantage for the interference- over current divider-style coupler modality especially for multi-qubit superconducting systems.

It is worthwhile to note that in the highlighted coupler approaches, the frequencies of the qubits and the magnitude of their interaction with the coupler determine the decoupling external flux bias $\Phi_{e}^{(0)}$ that zeroes the qubit-qubit effective coupling $g_{\textrm{eff}}=0$; this necessitates circuit remodeling and optimization each time data qubits or architecture is even slightly modified. In addition, the quantum level structure of an interference coupler makes the realization of parametrically-driven exchange interactions challenging, an increasingly critical functionality explored in several demonstrations [28, 29, 30]. Motivated by these considerations several workarounds for parametric coupling, which do not involve modulating the transition frequency of the coupler itself, are gaining traction, such as modulation of the data qubit frequency itself in the presence of a fixed coupling [31, 32, 33, 34] and techniques that do not use flux driving [7, 35, 36].

In this paper, we describe and analyze a novel double transmon coupler (DTC) design, also explored in Refs. [37, 38], that provides the linearity to implement two-qubit parametric gates efficiently and possesses an internally defined zero-coupling state that is independent of the coupled data qubits or circuit resonators. Furthermore we introduce novel computationally friendly numerics and analytical expressions for the effective coupling of this circuit, as well as provide theoretical predictions for parametric and waveguide coupling use cases that are unique to this work.

The DTC uses two inductively coupled transmon qubits to combine the linearity of current divider couplers with the capacitive coupler-qubit interactions of interference-style couplers at the cost of added level structure. From this combination of attributes, we predict fast, linear, and low noise tunable coupling for the DTC that is compatible with both fixed-frequency transmon architectures and parametric coupling use cases. The design paradigm works on a simple principle: each qubit capacitively interacts with a different transmon belonging to the coupler, as illustrated in Figs. 1a-b. These two coupler transmons in turn hybridize with an interaction that can be flux-tuned from negative to positive values. The qubits thus virtually interact via simultaneous coupling to the same hybridized states. Moreover, this net virtual interaction between the qubits can be turned off by flux-tuning the coupler transmon interaction and hybridization to zero.

Crucially, the addition of a second interposing transmon in the DTC compared to the interference approach suppresses the effective capacitance coupling between data qubits. The suppression geometrically scales with each interposing coupler transmon in the dispersive limit. By controllably hybridizing the DTC’s transmons, effective coupling comparable to that of a single interposing transmon is achievable, while preserving the increased data qubit isolation. Moreover, the junction and capacitor parameters of the DTC set the functional dependence of the DTC’s flux-tuning almost exclusively, making the decoupling flux bias $\Phi_{e}^{(0)}$ depend only weakly on the transition frequencies of the qubits or qubit-DTC interaction strength (if at all). In summary, this hybridization style coupler isolates qubits in its ‘off’ configuration and its internal operations are insensitive to the frequencies and use case of the qubits to which it’s coupled: a desirable feature of modular design. We will also see that the additional transmon degree of freedom in the DTC can be leveraged to realize parametric coupling that mitigates deleterious nonlinear effects such as rectification in the presence of fast flux modulation.

The DTC circuit consists of two superconducting transmon qubits [39] sharing a common ground. The Josephson and charging energies for each transmon are denoted by $E_{j}$ and $E_{Cj}$ for $j\in\{1,2\}$. A relatively high inductance junction $E_{12}\sim 0.1\times E_{1,2}$ connects the two transmons, forming a three junction dc SQuID, see Fig. 1b. A local bias line with an externally sourced current threads a tunable magnetic field through the SQuID, which tunes the inductive interaction between the transmons,

where $E_{12}^{\prime}\sim E_{12}\cos{(\Phi_{e}/\varphi_{0})}$ is the external flux-dependent Josephson energy of the center junction [appendix A] and $\varphi_{0}=\Phi_{0}/2\pi$ is the reduced flux quantum. The plasma frequencies, $\omega_{j}=(8(E_{j}^{\prime}+E_{12}^{\prime})E_{Cj})^{1/2}$ also tune modestly with flux through flux-dependence of both $E_{12}^{\prime}$ and $E_{j}^{\prime}\sim E_{j}-E_{12}^{2}\sin^{2}{(\Phi_{e}/\varphi_{0})}/2E_{j}$. In contrast, the capacitive interaction between the coupler transmons,

remains relatively constant as a function of flux. These inductive and capacitive contributions compete to define the net exchange interaction $g_{C}-g_{L}(\Phi_{e})$, as shown in Fig. 1c [also see Eq. (13)].

To model and simulate this coupling approach, we now consider a chain of four transmon circuits. The outer transmons operate as data qubits with plasma frequencies $\{\omega_{a},\omega_{b}\}$, while the inner transmons comprise the coupler with plasma frequencies $\{\omega_{1},\omega_{2}\}$. Each qubit capacitively couples to a different coupler transmon, with strength $g_{a}$ and $g_{b}$ respectively. The effective coupling between the data qubits may then be understood as a competition between virtual interactions mediated through the $|+\rangle$ and $|-\rangle$ coupler eigenstates, as shown in Fig. 1(d). This allows tuning $g_{\textrm{eff}}$ from positive to negative values as a nearly linear function of $g_{C}-g_{L}(\Phi_{e})$, and guarantees the existence of an external flux $\Phi_{e}^{(0)}$ such that $g_{\textrm{eff}}=0$. Anharmonicity in the DTC renormalizes $g_{\textrm{eff}}$ at the $20\%$ level but otherwise the form for $g_{\textrm{eff}}$ and $\Phi_{e}^{(0)}$ is similar to that of coupled harmonic oscillators with the same connectivity. The equivalent harmonic oscillator Hamiltonian of circuit can then be represented as [see appendix B],

where $a^{\dagger}_{j}$ and $a_{j}$ are the raising and lowering operators, while the index $j$ identifies the respective qubit $j=\{a,b\}$ or DTC $j=\{1,2\}$ transmon modes.

To compute $g_{\textrm{eff}}$ we first diagonalize the coupler part of the circuit, yielding $\omega_{\pm}=\bar{\omega}\pm(\delta^{2}+(g_{C}-g_{L})^{2})^{1/2}$ where $2\bar{\omega}=\omega_{1}+\omega_{2}$ and $2\delta=\omega_{1}-\omega_{2}$ are the coupler transmon sum and difference plasma frequencies, respectively. Since $\{\omega_{1},\,\omega_{2}\}$ move together with flux, the common-mode frequency $\bar{\omega}$ also tunes with flux, whereas $\delta$ may tune only weakly, if at all. With $\delta=0$, the modulation of both $g_{L}$ and $\bar{\omega}$ are nearly equal so that one of the two coupler eigenmode plasma frequencies tunes only weakly with flux, as shown in Fig. 1c. In the other regime, when $\delta\gg|g_{C}-g_{L}|$, both coupler eigenmode plasma frequencies modulate similarly with flux, driven by flux tuning of $\bar{\omega}$.

We now discuss how the DTC can be employed to implement time-dependent parametric coupling. To this end, a parametric modulation of external flux $\Phi_{e}=(A\sin{(\omega_{d}t)}+0.25)\Phi_{0}$ can mediate Rabi-like exchange interactions with rate $g^{p}_{\textrm{eff}}$ between fixed-frequency qubits (or between other circuit elements) by tuning $\omega_{d}$ at resonance with the difference of relevant transition frequencies (Fig. 3b). The magnitude of resultant parametric coupling $g^{p}_{\textrm{eff}}$ may be understood in a rotating frame, obtained via the transformation $(a_{a}^{p},\,a_{1}^{p},\,a_{b}^{p},\,a_{2}^{p})=(a_{a},\,a_{1},\,a_{b}e^{i\omega_{d}t},\,a_{2}e^{i\omega_{d}t})$, where qubits $a$ and $b$ are rendered degenerate by the parametric driving [Fig. 4(a)]. Neglecting the fast rotating terms, the resulting stationary Hamiltonian is manifestly similar to Eq. (3) with effective coupling given by Eq. (5). Substituting $\omega_{2}^{p}=\omega_{2}-\omega_{d}$, $\omega_{b}^{p}=\omega_{b}-\omega_{d}$, and $\bar{\omega}^{p}=\bar{\omega}-\omega_{d}/2$ into $\Delta_{j}^{p}=\omega_{j}^{p}-\bar{\omega}^{p}$ and $2\delta_{j}^{p}=\omega_{1}^{p}-\omega_{2}^{p}$ with $g_{L}^{p}=(2\pi A-(2\pi A)^{3}/8)g_{L}(\Phi_{e}=0)$, we obtain the parameters given in Fig. 4(b) and the effective parametric coupling strength as,

where $(D_{j}^{p})^{2}=(\Delta_{j}^{p})^{2}-(\delta^{p})^{2}-(g_{L}^{p})^{2}/2-g_{C}^{2}$. While the qubit detuning and pump frequency need to be swept in tandem to maintain resonance $\omega_{d}=\omega_{b}-\omega_{a}$, the relative detuning between coupler transmon/qubit can be kept fixed, i.e. $|\omega_{a}-\omega_{1}|\sim|\omega_{b}-\omega_{2}|$. As a consequence, $\Delta_{j}^{p}$ and $\delta_{j}^{p}$ also remain constant in the rotating frame as plotted in Fig. 4b. Adjusting $g_{a},\,g_{b},\,\text{and}\,g_{L}^{p}$ to remain constant as other variables change [appendix A], $g^{p}_{\textrm{eff}}$ can be held constant as a function of flux. Under these circumstances numerical simulation confirm a constant $g^{p}_{\textrm{eff}}$, as shown in Fig. 4b, in qualitative agreement with Eq. (7). While numerical simulations have included terms up to sixth order in the raising and lowering operators [Fig. 6], the analytical expression in Eq. (7) is derived using a simplified harmonic oscillator model [Eq. (4)] that neglects anharmonicity. However, the expression that generated the dotted line in Fig. 4c incorporates small corrections for the anharmonicity [Eq. (55)]. When the coupling is turned on, numerical simulations show $\sqrt{2}$ stronger coupling within the second excitation manifold compared to within the first excitation manifold, qualitatively consistent with the simplified analytical model. The qualitative agreement between this model and numerical simulation implies that the anharmonicity of the coupler transmons does not play a strong role in setting $g_{\textrm{eff}}$ or $g^{p}_{\textrm{eff}}$.

Another consideration for parametric coupling is the rectification of data qubit frequency due to flux modulation. Specifically, $\omega_{\textrm{ac}}^{j}$ of the driven qubits scales inversely with $1/(\omega_{j}-\omega_{\pm})$, which can change nonlinearly with flux; if the parametric drive periodically brings $\omega_{j}$ close to $\omega_{\pm}$, it can introduce a time-averaged drive amplitude dependence to the Rabi resonance condition that manifests as distortion of the time-domain swap envelope. As shown by the simulations presented in Fig. 4c, such distortion remains negligible if the coupler-induced time-averaged frequency shifts are less than $0.1\times g^{p}_{\textrm{eff}}$. Relative to interference-based couplers, the DTC mitigates the deleterious impact of the nonlinear flux dependence shifts in two ways. First, the DTC eigenenergies tune gently with effective coupling. Second, the freedom to set the detuning and interaction rate $g_{j}$ separately for each DTC transmon/qubit pair in fabrication allows the tuning of $\omega_{\textrm{ac}}^{a}$ and $\omega_{\textrm{ac}}^{b}$ with flux to be balanced. This balancing strategy roughly maximizes $g^{p}_{\textrm{eff}}$ for a fixed magnitude of nonlinearity. The combination of these factors allows straightforward, and relatively fast, pairwise parametric coupling between non-degenerate fixed-frequency qubits, among other applications. The parameters used in Figs. 2 and 4 represent a worst case scenario for second order nonlinearity generation, where $\delta\sim 0$. Even so, parametric coupling of qubits with $80$ MHz relative detuning showed Rabi-like state evolution, such as that shown in Fig. 4d, persists as $E_{12}$ increases $10\%$ ($1\%$) for the coupler transitions placed below (above) the qubit transitions.

A parametric drive can also induce unwanted direct energy exchange between various states in the Hilbert space, i.e. $\omega_{d}\approx|\omega_{j}-\omega_{\pm}|$, that should be avoided through appropriate choice of coupler transitions in fabrication. This is an important consideration for small qubit detuning. For larger qubit detunings, some driving frequencies will excite the DTC directly $\omega_{d}\sim\omega_{\pm}$. A weak, undesirable, multi-photon exchange interactions will also occur in the second excited manifold $\mathinner{|1001\rangle}\leftrightarrow\mathinner{|0110\rangle}$ when $\bar{\omega}=(\omega_{a}+\omega_{b})/2$: choosing the level structure as depicted in Figs. 4 and 3a-b avoids this undesirable degeneracy.

The last use case that we consider is fast tunable coupling between a waveguide (or other open quantum system) and a qubit, as depicted in Fig. 3c. This use case is thematically consistent with previous demonstrations [28, 46]. For this scenario we make the left data qubit and the right DTC transmon degenerate, i.e. $\omega_{a}=\omega_{2}$, which invalidates the dispersive approximation used in Eq. (4). Instead, the effective tunable coupling is given by $g_{a}(g_{C}-g_{L})/2\delta$, which goes to zero when the flux is set to $\Phi_{e}^{(0)}$. If the degeneracy between the qubit and DTC transmon is broken, the qubit relaxation rate into the waveguide under the dispserive limit is given by $\Gamma g_{a}^{2}(g_{C}-g_{L})^{2}/D_{a}^{4}$, which is typically small. Further, for $\omega_{a}\neq\omega_{2}$, the use cases described in Figs. 3b-c can be combined to realize strong coupling to a continuum without demanding degeneracy between qubit and DTC transitions.

Our DTC layout (Fig. 1a) uses a grounded coplanar waveguide with impedance $Z_{0}=50\text{ }\Omega$ that is shorted near the three-junction SQuID loop: current in the waveguide induces a controllable flux $\Phi_{e}$ via a mutual inductance $M$ to the loop. The three junction DTC SQuID allows direct exchange coupling between the DTC states and current excitations in the flux bias line. Using the circuit model discussed in appendix D, the resultant uncorrelated relaxation rate of each DTC transmon into the bias line can be approximated as, $\Gamma_{1,j}\approx M^{2}E_{12}^{2}\cos^{2}{(\Phi_{e}/\varphi_{0})}/Z_{0}\varphi_{0}^{4}C_{j}$ for $j\in\{1,2\}$, which only doubles if $g_{C}$ is increased from zero to $g_{C}\sim g_{L}$: see the expression for $\xi_{j}$ and the derivation for $\Gamma_{1,j}$ in the appendix D. Therefore, DTCs with small mutual inductive coupling $M\leq 3\text{ pH}$ likely will not need additional low pass filters on their current bias lines so long as the thermal population of the flux bias line is low. Conveniently, $\Gamma_{1}$ also tunes to zero at $g_{L}=0$: thus, energy relaxation of the coupler into its flux bias line occurs only when $g_{\textrm{eff}}\neq 0$ in the $g_{C}\ll g_{L}$ limit.

Spontaneous emission of the DTC, whatever the underlying source of fluctuations, is a source of Purcell decay for the qubits whose rate can be approximated as, $\Gamma_{1,a}=(g_{a}/\Delta_{a})^{2}\Gamma_{1,1}$ and $\Gamma_{1,b}=(g_{b}/\Delta_{b})^{2}\Gamma_{1,2}$ where $\Gamma_{1,j}$ for $j\in\{a,1,2,b\}$ is the relaxation rate of the relevant transmon. Multi-junction transmon devices routinely have lifetimes in excess of tens of microseconds. Assuming $T_{1}>20\text{ }\mu\text{s}$ and a Purcell factor $(g_{j}/\Delta_{j})^{2}\sim 1/10$, the coupler transmons impose at worst a $200\text{ }\mu\text{s}$ energy relaxation time on each qubit. While the dispersive regime is considered to be $(g_{j}/\Delta_{j})<0.1$, violating the dispersive approximation condition requires renormalization of $(g_{j}/\Delta_{j})$ and the inclusion of fourth order terms in the Schrieffer-Wolff transformation for quantitative accuracy [47] and accounts for the deviation between Eqs. (55) and (61) and numerical simulations in Figs. 2 and 4, respectively.

The impact of DTC-induced dephasing on the qubits can be computed from the frequency response of its eigenstates in presence of fluctuations on loop flux of the SQuID. The coupler eigenenergies change by at most $4g_{L}$ over the full flux tuning range, with a maximum slope at the idling flux typically set to the deocupling bias $\Phi_{e}^{(0)}$. At the idling flux, the slope is then $|\partial\omega/\partial\Phi|=2g_{L}/\varphi_{0}$. For typical amplitudes of flux noise spectral density, $\sqrt{A_{\Phi}}\sim 2.5\mu\Phi_{0}/\sqrt{\textrm{Hz}}$, the Hahn echo dephasing rate due to $1/f$ flux noise can be estimated as $\Gamma^{E}_{\phi}=\sqrt{A_{\phi}\textrm{ln}(2)}|\partial\omega/\partial\Phi|\sim 50\times 10^{3}\text{ s}^{-1/2}$ [48]. The frequency fluctuations of the qubits due to flux- and critical-current noise of the coupler transmons are filtered by the Purcell factor $(g_{j}/\Delta_{j})^{2}$. Hence for a Purcell factor of $1/10$, $1/f$ flux noise would limit the coherence of coupled qubit modes to $\sim 200\text{ }\mu\text{s}$ at $\Phi_{e}^{(0)}$.

In summary, we have theoretically explored the use of hybridization between two transmons, a DTC, as a mechanism for mediating tunable interactions between fixed-frequency qubits. The DTC design offers low noise, along with linear and easy-to-model qubit-qubit interactions. Crucially, the decoupling condition is defined by a flux bias that nulls the hybridization, making the decoupling condition almost entirely independent of the properties of the data qubit and qubit-DTC interaction. The extra design freedom afforded by having two coupler transmons rather than one also enables implementation of fast, parametric coupling to a fixed mode or continuum. Moreover, the DTC may be attractive as a standalone qubit for its gentle eigenfrequency tuning with flux, weak exchange coupling with its current bias line, and level structure. The last benefit may also enable novel qutrit-based superconducting circuit architectures [49, 50].