Dispersive Readout of Majorana Qubits

We analyze a readout scheme for Majorana qubits based on dispersive coupling to a resonator. We consider two variants of Majorana qubits: the Majorana transmon and the Majorana box qubit. In both cases, the qubit-resonator interaction can produce sizeable dispersive shifts in the megahertz range for reasonable system parameters, allowing for submicrosecond readout with high fidelity. For Majorana transmons, the light-matter interaction used for readout manifestly conserves Majorana parity, which leads to a notion of quantum nondemolition (QND) readout that is stronger than for conventional charge qubits. In contrast, Majorana box qubits only recover an approximately QND readout mechanism in the dispersive limit where the resonator detuning is large. We also compare dispersive readout to longitudinal readout for the Majorana box qubit. We show that the latter gives faster and higher fidelity readout for reasonable parameters, while having the additional advantage of being manifestly QND, and so may prove to be a better readout mechanism for these systems.

Popular Summary

Majorana zero modes are quasiparticles that may emerge in certain condensed matter systems, with properties that could be very favorable for quantum computing. Quantum information can be stored nonlocally between a set of Majorana zero modes, in a way that makes the information inaccessible from the environment, thus protecting it from noise and external disturbances. On the other hand, this nonlocal protection also makes the manipulation of quantum information more challenging. Of particular importance is the ability to measure properties of the quantum system, without introducing unwanted disturbances. By measuring joint properties of the Majorana zero modes, logical operations can be executed on the encoded quantum information in a protected manner. We show, using numerical simulations, how to perform such measurements that are both accurate and fast, within a practical and scalable architecture.

In our paper, we introduce measurement schemes borrowed from circuit quantum electrodynamics, using superconducting circuits and microwave resonators to perform measurements. This approach has been extremely successful for other quantum computing platforms, including superconducting and semiconducting qubits. We show theoretically that the same approach can be used to measure the joint parity of Majorana zero modes with high accuracy and on a timescale that is expected to be short compared to the coherence time of Majorana qubits.

An important implication of our work is that it shows that tried and tested measurement techniques, which have already been used in quantum computing chips with dozens of superconducting qubits, can also be used to measure the more exotic Majorana zero modes.

Figure 1

Illustration of readout for two Majorana qubit variants. Topological superconductors (gray rectangles) host MZMs ${\hat{\gamma }}_i$, and qubit readout corresponds to a measurement of $i{\hat{\gamma }}_2{\hat{\gamma }}_3$. Majorana wavefunctions can be made to overlap either by (a) direct tunneling or (b) tunneling to a proximal quantum dot (gray circle). The resonator (illustrated by an $LC$ oscillator) is capacitively coupled to the island charge. Alternative resonator-island coupling geometries are possible, including coupling to the quantum dot in (b). (a) Majorana transmon qubit: the topological superconductors form two distinct superconducting islands, shunted by a Josephson junction with energy $E_J$ (one of the islands may be grounded). The level diagram illustrates low-energy eigenstates labeled $|g,\pm ⟩$ and $|e,\pm ⟩$. The blue and red arrows indicate allowed transitions induced by the qubit-resonator interaction. These allowed transitions conserve Majorana parity. (b) Majorana box qubit: the topological superconductors are shunted by a trivial superconductor, and the device forms a single superconducting island. The level diagram labels dressed eigenstates of the coupled superconducting island-dot system. In this case, the resonator induces transitions between dressed states of different dot occupancy and Majorana parity.

Figure 2

The low-level spectrum of a Majorana transmon qubit as a function of the offset charge parameter $n_g$, where $E_C/h=250$ MHz, $E_J/E_C=50$ (${\omega }_t/2\pi \simeq 5$ GHz), and (a) $E_M=0$ and (b) $E_M/\hslash {\omega }_t=0.05$ (${\omega }_{mt}/{\omega }_t\simeq 0.1$). When $E_M=0$, each level in the transmon spectrum is two-fold degenerate. This degeneracy is lifted when $E_M>0$. The resulting level structure can be thought of as two parallel transmon ladders, for $i{\hat{\gamma }}_2{\hat{\gamma }}_3=\pm 1$.

Figure 3

Dispersive frequency shift of a readout resonator coupled to a Majorana transmon qubit ${\chi }_{mt}$ (purple) in comparison to those of a conventional transmon qubit ${\chi }_t$ (gold), where $E_C/h=250$ MHz, $E_J/E_C=50$ (${\omega }_t/2\pi \simeq 5$ GHz), $n_g=0$, and $\lambda /2\pi =100$ MHz ($g_t/2\pi \simeq 200$ MHz). In (a),(b),(c) ${\phi }_x=0$ and in (b),(c) $\mathrm{\Delta }/g_t=-10$. Circle, triangle, and square markers indicate equivalent points between plots. The top plots show ${\chi }_{mt}$ and ${\chi }_t$ as a function of (a) $\mathrm{\Delta }/g_t$ and (b) ${\mathrm{\Delta }}_t/g_t$ for three values of $E_M$ where ${\phi }_x=0$. In (a) $\mathrm{\Delta }$ is the detuning from the relevant transition frequency (${\omega }_t$ for the transmon and ${\omega }_{+}$ for the Majorana transmon). In (b) ${\mathrm{\Delta }}_t={\omega }_t-{\omega }_r$, plotted such that the saddle regimes are visible. The magnitude of ${\chi }_{mt}$ in comparison to ${\chi }_t$ is plotted as a function of (c) $2E_M/\hslash {\omega }_t$ ($\simeq {\omega }_{mt}/{\omega }_t$) and (d) ${\phi }_x/\pi$.

Figure 4

The low-level spectrum of a Majorana box qubit as a function of the offset charge parameter $n_g$, where $\delta /h=(E_{\mathrm{tot}}+\epsilon )/h=5$ GHz, ${\phi }_x=\pi /2$, $t_{L,R}=t$, and (a) $t=0$ and (b) $t/\delta \simeq 0.2$ (${\epsilon }_m/h\simeq 0.5$ GHz). When MZMs interact via the quantum dot $(t>0)$, the charge state $|N,n_d=0,\pm ⟩$ hybridizes with $|N-1,n_d=1,\mp ⟩$.

Figure 5

Dispersive frequency shifts of a readout resonator coupled to a Majorana box qubit ${\chi }_{mb}$ (green) for $\delta /h=(E_{\mathrm{tot}}+\epsilon )/h=5$ GHz, $n_g=0$, $t_{L,R}=t$, and $\lambda /2\pi =100$ MHz. In (a),(b) ${\phi }_x=0$ and in (b),(c) $\mathrm{\Delta }/g_{+}=-10$. Circle, triangle, and square markers indicate equivalent points between plots. We show ${\chi }_{mb}$ as a function of (a) detuning $\mathrm{\Delta }=f_{+}/\hslash -{\omega }_r$ for three values of $t/\delta$, (b) $t/\delta$ for $\mathrm{\Delta }/|g_{+}|=-10$, (c) ${\phi }_x=0$. Analytical results (dotted) obtained from Eq. (28) are also shown. A very small discrepancy arises due to the rotating-wave approximation used in Eq. (28).

Figure 6

Timescales and fidelities for resonator-based readout of Majorana qubits. (a) Dispersive shift of a readout resonator coupled to a Majorana transmon qubit ${\chi }_{mt}$ (purple), a Majorana box qubit ${\chi }_{mb}$ (green), and a conventional transmon ${\chi }_t$ (gold) as a function of the qubit frequency ${\omega }_q/2\pi$ for $n_g=0$, ${\phi }_x=0$, $\lambda /2\pi =100$ MHz, and fixed detuning $\mathrm{\Delta }/g=-10$. For the Majorana and conventional transmon we have set $E_C/h=250$ MHz and $E_J/E_C=50$. For the Majorana box qubit, we have set $\delta /h=(E_{\mathrm{tot}}+\epsilon )/h=5$ GHz. (b) Infidelity $1-F$ of each readout scheme as a function of readout time $\tau$ for ${\omega }_q/2\pi =1$ GHz, $\overline{n}/n_{\mathrm{crit}}=1/5$, and $\kappa =2\chi$. Longitudinal case calculated for ${\tilde{g}}_z/2\pi =10$ MHz, as detailed in Appendix pp4. (c) Readout time required to reach a measurement fidelity of 99.99% for each dispersive scheme in (a) as a function of ${\omega }_q/2\pi$.

Figure 7

Low-energy spectrum and dispersive shifts of a Majorana transmon qubit with an “indirect” interaction model. System parameters are identical to those in Fig. 3. Here $\epsilon /h=20$ GHz and $t_{L,R}=t$ is tuned such that the energy splitting between the two lowest levels, $\hslash {\omega }_{mt}$, is equal to the corresponding case from Fig. 3.

Figure 8

Longitudinal coupling strength for the Majorana box qubit $g_z=g_m/2$ as a function of $t/\delta$ and ${\phi }_x$. For (a), we have set ${\phi }_x=0$ and, for (b), we have set $t/\delta \simeq 0.5$. Dashed lines indicate a modulation of ${\tilde{g}}_z=10$ MHz, which was used in Fig. 6. This corresponds to a modulation of roughly (a) $\tilde{t}/\delta \simeq 0.1$ or (b) ${\tilde{\phi }}_x=\pi /10$.