Quantum-limited measurement of spin qubits via curvature couplings to a cavity

Existing schemes for semiconductor spin qubit readout involve either spin-to-charge conversion or electric dipole coupling to a superconducting resonator. The former requires destructive readout while the latter suffers from enhanced qubit dephasing, limiting the potential performance of quantum measurement: an open challenge in semiconductor quantum computing. Here we propose a coupling to a superconducting resonator based on qubit energy curvature versus gate voltage, which enables quantum nondemolition (QND) readout while the qubit resides in its full sweet spot and with zero dipole coupling (offering resilience to charge noise). Readout strengths of tens to hundred megahertz (MHz) can be reached, at least ten times larger than in the previously proposed schemes. We show how this curvature interaction generates two strictly QND readout approaches: (i) via a “dispersive-like” (quantum capacitance) spin-dependent resonator energy shift, similar in implementation to circuit-QED dispersive readout, but avoiding the Purcell effect; and (ii) with the addition of qubit gate voltage modulation, a longitudinal readout which can be selectively switched on, for a chosen n-qubit subsystem. Neither scheme requires the qubit and resonator to be close in frequency.

Figure 1

(a) Coupling a microwave SC resonator to a triple quantum dot (TQD) always-on, exchange-only (AEON) qubit at its full sweet spot, where the qubit dipole moment (both transverse and longitudinal) is zeroed, minimizing the qubit relaxation and dephasing. The coupling is via the middle dot gate voltage [27] $V_2$ and is proportional to the qubit's finite energy curvature vs QD voltage detuning, $C_q\equiv {\partial }^2{E}_q/\partial V_m^2$ (a quantum capacitance), where $V_m\equiv (V_3+V_1)/2-V_2$. Readout is achieved either (1) via a “dispersive-like” spin-dependent curvature coupling, $\hslash \delta \omega {\widehat{a}}^{†}\widehat{a}{\sigma }_z$ while driving the resonator (with strength ${\epsilon }_d)$ similar to circuit-QED; or (2) via modulating $V_2$ near the resonator frequency, creating a dynamical longitudinal coupling, $\hslash \left[g_{\parallel }\left(t\right){\sigma }_z+g_0\left(t\right)\right](\widehat{a}+{\widehat{a}}^{†})$ (also proportional to $C_q)$. Photons put into the resonator are scattered off the (far off-resonant) spin qubit and measured via a homodyne signal, $I_m\left(t\right)$, with $\beta$ and $\phi$ being the strength and phase of the local oscillator. (b) A TQD AEON spin qubit [28] with levels $E_{\pm }({\varepsilon }_v,{\varepsilon }_m)$ (normalized by the interdot tunneling) stays during readout at a full sweet spot (in yellow), at gate energy detunings ${\varepsilon }_v^0=0$ and ${\varepsilon }_m^0=e{V}_m^0$ (Appendix pp1), with finite energy curvature. The doubly occupation states contributing to the qubit dipole moment out of the sweet spot (e.g., in the RX regime) are separated by a large energy gap, $U_{\mathrm{charge}}\gtrsim 100\mathrm{GHz}$, and exactly cancel at the sweet spot.

Figure 2

(a) Coupling a microwave SC resonator to a double quantum dot (DQD) singlet-triplet qubit operating at its symmetric operating point (SOP). Similar to the TQD (Fig. 1) case, the qubit full dipole moment is zeroed at the sweet spot, minimizing the charge noise (via relaxation and dephasing). The coupling is through the right dot voltage [59] $V_2$ and is proportional to the qubit finite energy curvature vs dots' detuning $C_q^{\mathrm{DQD}}\equiv e^2{\partial }^2{E}_q/\partial {\varepsilon }_v^2$, where ${\varepsilon }_v\equiv e(V_1-V_2)$. Readout is achieved in complete analogy to the TQD case (Fig. 1), using the two effective spin-dependent interactions (each proportional to $C_q^{\mathrm{DQD}})$: (1) a quantum capacitance coupling, $\hslash \delta \omega {\widehat{a}}^{†}\widehat{a}{\sigma }_z$, or (2) dynamical longitudinal coupling, $\hslash \left[g_{\parallel }\left(t\right){\sigma }_z+g_0\left(t\right)\right](\widehat{a}+{\widehat{a}}^{†})$, see Fig. 1 and Eqs. (2) and (4). (b) DQD S-T qubit with levels $E_{\pm }\left({\varepsilon }_v\right)$ (normalized by the interdot tunneling $t_c)$ stays during readout at the SOP (a full sweet spot, in yellow), at gate energy detuning ${\varepsilon }_v^0$ (Appendix pp3), with finite energy curvature. The two doubly occupation states, $S(0,2)$ and $S(2,0)$ that contribute to the qubit dipole moment (e.g., in the charge degeneracy point) are separated by a large energy gap, $U_{\mathrm{charge}}\gtrsim 100\mathrm{GHz}$, and cancel at the sweet spot.

Figure 3

The longitudinal measurement rate ${\mathrm{\Gamma }}_{\parallel }\left({\omega }_r\right)$, Eq. (E6), in units of $B/\sqrt{A}$, see Eq. (E4), as a function of the resonator frequency ${\omega }_r$ for different optimal frequencies ${\omega }_{r0}=22.2,31.4,44.4,62.9\mathrm{GHz}\left(Q=100\right)$. For a fixed resonator frequency, ${\omega }_r/2\pi =10\mathrm{GHz}$ (vertical dashed line) a maximal rate is achieved for ${\omega }_{r0}$ approaching ${\omega }_r$ from above.