Andreev qubit readout from dynamic interference supercurrent

Abstract

Nondemolition protocols use ancilla qubits to identify the fragile quantum state of a qubit without destroying its encoded information, thus playing a crucial role in nondestructive quantum measurements particularly relevant for quantum error correction. However, the multitude of ancilla preparations, information transfers, and ancilla measurements in these protocols create an intrinsic overhead for information processing. Here we consider an Andreev qubit defined in a quantum-dot Josephson junction and show that the macroscopic time-dependent oscillatory supercurrent arising from the quantum interference of the many-body eigenstates, can be used to probe the qubit itself-arbitrarily close to the nondestructive limit-under currently available experimental capabilities. This readout of arbitrary superposition states of Andreev qubits avoids ancillae altogether and significantly reduces experimental overhead as no repetitive qubit reinitialization is needed. Our prediction of an AC-like Josephson effect without an applied external voltage, which enables the nondestructive qubit readout, is a unique macroscopic manifestation of the microscopic dynamics of the Andreev quantum state. Our findings should have an unprecedented impact on advancing research and applications involving Andreev dots, thus positioning them as promising qubit contenders for quantum processing and technologies.

Figures (3)

• Figure 1: Proposed experimental design for measuring the supercurrent flowing through a quantum-dot Josephson junction (zoom) formed in a semiconducting nanowire (purple) and embedded in a large superconducting loop (blue). The small pickup coil (copper) of the SQUID magnetometer lies above the superconducting loop and is inductively coupled to the Andreev qubit (sphere in zoom). For a precise geometry and a detailed description of our proposed setup, see Fig. 1 in the SM SM.
• Figure 2: (a) Sketch of a quantum-dot Josephson junction. A quantum dot (gray), gate-defined in an InAs nanowire (red), is tunnel coupled to proximitized InAs/Al left (L) and right (R) superconducting leads (blue) with phase bias $\phi=\phi_R-\phi_L$. The lower panel denotes the corresponding 1D tight-binding model. (b) Preparation of an arbitrary Andreev qubit state: upon resonant microwave driving [$\hbar\omega(\phi)=\mathcal{E}_{1}(\phi)-\mathcal{E}_{0}(\phi)$], the qubit undergoes Rabi oscillations between states $\vert 0\rangle$ and $\vert 1\rangle$ with Rabi frequency $\vert \Omega\vert$ [lower panel in (b)]. Arbitrary qubit states $\vert X\rangle =\cos(\theta/2)\vert 0 \rangle + \sin(\theta/2)e^{i\vartheta}\vert 1\rangle$ can be prepared on the Bloch sphere, where the azimuthal angle $\theta\in[0,\pi]$ is adjusted by the microwave pulse duration $t_R$ and the pole angle $\vartheta\in[0,2\pi)$ is controlled by the angle of the Rabi amplitude $\Omega$, i.e., $\vartheta=\text{angle}(\Omega)$. For instance, the black dot on the Rabi oscillation curve corresponds to $(\theta,\vartheta)=(\pi/6,\pi/3)$. (c) Supercurrent readout of Andreev qubits from the time evolution of the phase-biased SC $I^{}_{\vert X(t)\rangle}=I^{S}_{\vert X\rangle}+ A(\phi)\sin\theta\cos[\omega(\phi) (t-t_R)-\varphi_0(\phi)-\vartheta]$ corresponding to the arbitrary superposition $\vert X\rangle$ in (b), where $A(\phi)=2e\omega (\phi)\vert\mathcal{C}_{0,1}(\phi)\vert$, with $\mathcal{C}_{0,1}(\phi)=\langle 0 \vert \partial_{ \phi}\vert 1\rangle=\vert\mathcal{C}_{0,1}(\phi)\vert e^{i\varphi_0(\phi)}$. For a given experimental run, the phase bias $\phi$ and the gate-tunable dot level $\epsilon_D$ are kept fixed and so are $\omega(\phi)$, $\varphi_0(\phi)$, $\vert\mathcal{C}_{0,1}(\phi)\vert$, $I_{\vert 0\rangle }$ and $I_{\vert 1\rangle}$. The static supercurrent $I^{S}_{\vert X\rangle}=\cos^2(\theta/2) I_{\vert 0\rangle}+ \sin^2(\theta/2) I_{\vert 1\rangle}$, calculated from the time average of the total SC, determines $\theta$, which also controls the oscillation amplitude of the interference SC, i.e., $A(\phi)\sin\theta$. The qubit parameters, $\omega(\phi)$ and $\vartheta$, can be read out from the oscillation period $2\pi/\omega(\phi)$ and the interference SC $A(\phi)\sin\theta\cos \vartheta$ at $\omega (\phi) (t-t_R)=\varphi_0(\phi)+2n\pi$, respectively. Other parameters are the same as Fig. \ref{['WSC']}.
• Figure 3: (a) Ideal supercurrent $I_{\vert X (t)\rangle}$ readout for different initial qubit states $\vert X\rangle$, as parametrized by the Bloch-sphere angles $(\theta,\vartheta)$ [Fig. \ref{['FIGSTORY']}(b)]. Both angles can be used to control the supercurrent, e.g., as $\vartheta$ changes from $0$ (red) to $\pi$ (cyan) (for $\theta=0.5\pi$), the supercurrent undergoes a $\pi$ shift, while by varying $\theta$ from $0.5\pi$ (red) to $0.2\pi$ (for ($\vartheta=0$) reduces its amplitude and modifies its static part [Eq. \ref{['averagedsupercurrent']}] from $I^S_{\vert X\rangle}/\mathcal{I}_0\simeq 0.028$ to $I^S_{\vert X\rangle}/\mathcal{I}_0\simeq-0.068$. (b) Real‑time supercurrent $I_{\vert X'_{\imath}(t_{\imath})\rangle}$[i.e., \ref{['vfdavfd']}] under repeated quantum measurements for several values of the number of measurements $M$, where $\sigma_I=0.02\mathcal{I}_0$ and $(\theta,\vartheta)=(\pi/3,\pi/2)$. Here we consider realistic parameters $\phi=1.18\pi$ and $\epsilon_D= 4\Delta$, resulting in $\hbar \omega(\phi)\simeq 0.89\Delta$, $\varphi_0(\phi)\simeq 0.82\pi$, $\vert \mathcal{C}_{0,1}(\phi) \vert\simeq 0.31$, $I_{\vert 0\rangle}/\mathcal{I}_0=-0.091$, and $I_{\vert 1\rangle}/\mathcal{I}_0=0.147$. Other parameters: $\Delta=1$, $t=t_R$, and $N=903$.