Modulated longitudinal gates on encoded spin qubits via curvature couplings to a superconducting cavity

We propose entangling operations based on the energy curvature couplings of encoded spin qubits to a superconducting cavity, exploring the nonlinear qubit response to a gate voltage variation. For a two-qubit ($n$-qubit) entangling gate we explore acquired geometric phases via a time-modulated longitudinal ${\sigma }_z$ coupling, offering gate times of tens of nanoseconds even when the qubits and the cavity are far detuned. No dipole moment is necessary: the qubit transverse ${\sigma }_x$ coupling to the resonator is zero at the full sweet spot of the encoded spin qubit of interest (a triple quantum dot three-electron exchange-only qubit or a double quantum dot singlet-triplet qubit). This approach allows always-on, exchange-only qubits, for example, to stay on their “sweet spots” during gate operations, minimizing the charge noise and eliminating an always-on static longitudinal qubit-qubit coupling. We calculate the main gate errors due to the (1) diffusion (Johnson) noise and (2) damping of the resonator, the (3) $1/f$-charge noise qubit gate dephasing and $1/f$ noise on the longitudinal coupling, (4) qubit dephasing and AC-Stark frequency shifts via photon fluctuations in the resonator, and (5) spin-dependent resonator frequency shifts (via a “dispersivelike” static curvature coupling), most of them associated with the nonzero qubit energy curvature (quantum capacitance). Using spin-echo-like error suppression at optimal regimes, gate infidelities of ${10}^{-2}–{10}^{-3}$ can be achieved with experimentally existing parameters. The proposed schemes seem suitable for remote spin-to-spin entanglement of two spin qubits or a cluster of spin qubits: an important resource of quantum computing.

Figure 1

(a) A TQD exchange only qubit (solid, blue box) or a DQD S-T qubit (dashed, red box) capacitively coupled to a SC resonator (see experimental Refs. [39, 40] for a similar layout). The curvature couplings to a SC resonator arise via the resonator quantized voltage drop ${\widehat{V}}_r=V_{\mathrm{vac}}(\widehat{a}+{\widehat{a}}^{†})$ on the coupling (middle) dot, essentially as quantum capacitance couplings: these are (i) the dynamical longitudinal coupling, Eq. (4), ${\tilde{g}}_{\parallel }{\sigma }_z(\widehat{a}+{\widehat{a}}^{†})\sim {\widehat{V}}_r{V}_m\left(t\right)$, that arises via additional qubit gate modulation and (ii) the always-on dispersivelike (quantum capacitance) static coupling $\delta \omega {\widehat{a}}^{†}\widehat{a}{\sigma }_z\sim {\widehat{V}}_r^2$, Eq. (8). Both couplings can be expressed via the qubit energy curvature $C_q\equiv e^2\frac{{\partial }^2{E}_q}{\partial {\varepsilon }_m^2}\propto \frac{t_c^2}{U_{\mathrm{charge}}^3}$ that can be significant, $\gtrsim 30\mathrm{aF}$ (see Ref. [62] and Table 1), for typical dot charging energy $U_{\mathrm{charge}}\gtrsim 0.4\mathrm{meV}$ and interdot tunnelings $t_c\gtrsim 40\mu \mathrm{eV}$, reachable experimentally. For a TQD qubit the relevant voltage detuning is ${\varepsilon }_m\equiv e[(V_3+V_1)/2-V_2]$ and for a DQD qubit it is ${\varepsilon }_v\equiv e(V_1-V_2)$. (b) Full sweet spot regime of a TQD exchange only qubit (solid, red circle). The energy splitting $E_q({\varepsilon }_v,{\varepsilon }_m)$ as a function of the qubit gate voltage detunings ${\varepsilon }_v\equiv e(V_3-V_1)/2$ and ${\varepsilon }_m$ (in units of the dots' charging energy $U_{\mathrm{ch}}$). At the full sweet spot [26] where $\frac{\partial E_q}{\partial {\varepsilon }_v},\frac{\partial E_q}{\partial {\varepsilon }_m}=0$, while its transverse dipole moment is also zero [62], the TQD qubit is insensitive (in first order) to gate voltage fluctuations, and at the same time qubit dephasing through phonon relaxation and coupling to two-level fluctuators is also minimized as the transition dipole moment is zero [62] as well. A similar sweet spot of the DQD qubit (not shown) with respect to the detuning ${\varepsilon }_v\equiv e(V_1-V_2)$ will be referred to as a symmetric operating point (SOP) [28, 29].

Figure 2

Two TQD spin qubits [or DQD S-T spin qubits, Fig. 1] can be entangled by modulating their respective dot gate voltages $V_2^{\left(j\right)}$ by a modulation $V_m^{\left(j\right)}\left(t\right)={\tilde{V}}_m^{\left(j\right)}cos({\omega }_{m}t+{\phi }_m)$, for a finite gate time $t_g=2\pi /\delta \equiv 2\pi /({\omega }_r-{\omega }_m)$, where $\delta \equiv {\omega }_r-{\omega }_m$ is the frequency detuning. For a specially chosen detuning $\delta ={\delta }_{\pi }\equiv \pi \sqrt{4N{\tilde{g}}_{\parallel }^{\left(1\right)}{\tilde{g}}_{\parallel }^{\left(2\right)}}$, the accumulated geometric phases for each two-qubit spin state amount to a controlled $\pi$-phase gate, see Eq. (23) and Appendix pp1. The defining gate voltages for each qubit as well as the coupling to the (high $Q$-factor) resonator can be different in general (see text). The qubit frequencies can be different, and they can be strongly detuned from the resonator (beyond the usual dispersive limit).

Figure 3

Gate time $t_{\pi }$ of a two-qubit controlled $\pi$-phase gate for TQD exchange only qubits [Eq. (23)] is plotted vs interdot tunneling amplitude $t_l=t_r$ and dots' charging energy $U_{\mathrm{ch}}$, for fixed qubits' modulation ${\tilde{V}}_m=0.1\mathrm{mV}$, resonator frequency ${\omega }_r\simeq 6.3\mathrm{GHz}$, resonator inductance $L_r=50\mathrm{nH}$ (impedance $Z_r\simeq {\omega }_r{L}_r\simeq 1.85\mathrm{k}\mathrm{\Omega }$; compare, e.g., with Ref. [81]), and qubits'-resonator lever arm ${\alpha }_c\simeq 0.14$ that amounts to a reachable coupling ratio of $\eta /\hslash ={\alpha }_c\sqrt{\frac{Z_r}{\hslash /e^2}}\simeq 0.1$. The gate time $t_{\pi }$ ranges from 15 to 150 ns for $t_l\simeq [35,110]\mu \mathrm{eV}$ and $U_{\mathrm{ch}}\in [0.4,0.5]\mathrm{meV}$, featuring relatively high tunneling amplitudes and relatively low dots' charging energies. The scaling of the gate time $t_{\pi }\sim 1/\sqrt{L_r}$ leads to a moderate increase for smaller resonator inductances $L_r$. By reaching higher experimental values for ${\alpha }_c, {\omega }_r$, and ${\tilde{V}}_m$ one can bring the dots' tunneling and charging energy to experimentally more favorable ranges [9]. Since for a DQD S-T qubit the scaling of the longitudinal coupling [Eq. (4)] with parameters is exactly the same [62], one expects similar ranges for these systems.

Figure 4

The nonideal oscillator trajectories in the phase space $\{\mathrm{Re}\left[{\alpha }_s\left(t\right)\right],\mathrm{Im}\left[{\alpha }_s\left(t\right)\right]\}$ for two subsequent cycles, each of duration $t_g=2\pi /\delta$. The ideal evolution ${\alpha }_s^{\mathrm{id}}\left(t\right)$ obtains deviations $\delta {\alpha }_s\left(t\right)$ (red solid or dashed decaying circles) due to (i) resonator damping $\kappa$ [Eqs. (27) and (35)], and (ii) qubit-induced spin-dependent resonator frequency shifts $\delta {\omega }_s$ [Eqs. (1), (27), and (35)] via the always-on dispersivelike curvature coupling [Eq. (7)]. By changing the modulation phase for every odd cycle: ${\phi }_m^{\prime }={\phi }_m+\pi$, the spin-dependent resonator driving amplitude ${\mathrm{\Omega }}_{\parallel ,s}$ flips sign (red, dashed decaying circle; sf. also Fig. 10) that allows partial cancellation of the deviations, for small $\kappa , \delta {\omega }_s$. The (iii) resonator thermal (Johnson) noise is shown schematically (black, solid line) as a noisy deviation from the ideal trajectory ${\alpha }_s^{\mathrm{id}}\left(t\right)$. The noisy trajectory is generated by a random force Hamiltonian ${\mathcal{H}}_f=-{\xi }_f\left(t\right)\widehat{x}$ [see Eq. (36) and Appendix pp4], which is an unraveling of the diffusion term in the ensemble-averaged evolution [Eq. (27)].

Figure 5

The infidelity $\delta {\epsilon }_{\kappa ,\delta \omega }^{2\mathrm{Qb}}\left(\frac{\delta \omega }{{\tilde{g}}_{\parallel }},0\right)$ [Eqs. (39) and (44)] is a growing function of the ratio $\frac{\delta \omega }{{\tilde{g}}_{\parallel }}=\frac{\eta }{\hslash }\frac{\hslash {\omega }_r}{e{\tilde{V}}_m}$. The values of $\frac{\delta \omega }{{\tilde{g}}_{\parallel }}=0.035,0.011$ at which $\delta {\epsilon }_{\kappa ,\delta \omega }=0.01,0.001$, respectively, are calculated for resonator driving and phase [Eq. (6)] at which the qubits' spin-independent curvature couplings ${\tilde{g}}_0^{\left(j\right)}$ are canceled, see Eq. (43).

Figure 6

The infidelity $\delta {\epsilon }_{\kappa ,\delta \omega }^{2\mathrm{Qb}}({\omega }_r,{\tilde{V}}_m)$ for the range of resonator frequency ${\omega }_r\in [2,10]\mathrm{GHz}$ and qubits gate modulation voltage (amplitude) ${\tilde{V}}_m\in [0.05,0.25]\mathrm{mV}$. Contours where the infidelity reaches the values of 0.1, 0.01, 0.001 are also shown. To keep $\delta {\epsilon }_{\kappa ,\delta \omega }$ constant requires keeping the ratio $\frac{{\omega }_r^{3/2}}{{\tilde{V}}_m}$ constant (see text).

Figure 7

Density plot of the resonator (Johnson) noise infidelity $\delta {\epsilon }_{\xi }(t_{\pi },logQ)$ for a resonator frequency ${\omega }_r/2\pi \simeq 6.3\mathrm{GHz}$, range of the two-qubit gate time $t_{\pi }\in [15,150]\mathrm{ns}$, and resonator $Q$ factor of $Q\in [3.4\times {10}^3,3.4\times {10}^6]$. Contours where the infidelity reaches the values of 0.1, 0.01, 0.001 are also shown.

Figure 8

Schematic of the two-qubit energy levels and the associated charge noise dephasing rates. The individual qubit dephasings are ${\tilde{\mathrm{\Gamma }}}_{\varphi }^{\left(1\right)}={\tilde{\mathrm{\Gamma }}}_{34}$ and ${\tilde{\mathrm{\Gamma }}}_{\varphi }^{\left(2\right)}={\tilde{\mathrm{\Gamma }}}_{24}$. For uncorrelated white noise ${\tilde{\mathrm{\Gamma }}}_{14}={\tilde{\mathrm{\Gamma }}}_{\varphi }^{\left(1\right)}+{\tilde{\mathrm{\Gamma }}}_{\varphi }^{\left(2\right)}$, see Eq. (70) and Appendix pp5-s2. For correlated white noise and for uncorrelated $1/f$ noise ${\tilde{\mathrm{\Gamma }}}_{14}\ne {\tilde{\mathrm{\Gamma }}}_{\varphi }^{\left(1\right)}+{\tilde{\mathrm{\Gamma }}}_{\varphi }^{\left(2\right)}$, see Eqs. (57) and (76), respectively.

Figure 9

Density plot of the two-qubit controlled $\pi$-phase gate $1/f$ charge noise infidelity $\delta {\varepsilon }_{\varphi ,1/f}^{2\mathrm{Qb}}(t_{\pi },T_2^{*})$, Eq. (78), for a resonator frequency ${\omega }_r/2\pi \simeq 6.3\mathrm{GHz}$, range of the two-qubit ($\pi$-phase) gate time $t_{\pi }\in [15,150]\mathrm{ns}$, and qubit's dephasing times $T_2^{*}\in [200,1500]\mathrm{ns}$, where $T_2^{*}\equiv 1/{\tilde{\mathrm{\Gamma }}}_{\varphi }$, Eq. (79). Contours where the infidelity reaches the values of 0.1, 0.01, 0.001 are shown, which corresponds to the relations $t_{\pi }=0.354T_2^{*},0.112T_2^{*},0.0354T_2^{*}$, respectively.

Figure 10

The ideal circular oscillator trajectories for two different initial phases: ${\phi }_m$ and ${\phi }_m^{\prime }={\phi }_m+\pi$ (red, solid circles). The radius of the circles is $\frac{\mathrm{\Omega }}{\delta }$ and their centers lie on a circle with the same radius put at the origin (black, dashed circle). Interchanging the phase ${\phi }_m$ at each odd circle allows cancellation of small imperfections of the ideal evolution (see Fig. 4 and Appendix pp2).