Partial-Measurement Backaction and Nonclassical Weak Values in a Superconducting Circuit

We realize indirect partial measurement of a transmon qubit in circuit quantum electrodynamics by interaction with an ancilla qubit and projective ancilla measurement with a dedicated readout resonator. Accurate control of the interaction and ancilla measurement basis allows tailoring the measurement strength and operator. The tradeoff between measurement strength and qubit backaction is characterized through the distortion of a qubit Rabi oscillation imposed by ancilla measurement in different bases. Combining partial and projective qubit measurements, we provide the solid-state demonstration of the correspondence between a nonclassical weak value and the violation of a Leggett-Garg inequality.

Figure 1

(a) Two-transmon, three-resonator cQED processor. Resonators $H_1$ and $H_2$ allow individual readout of qubits $Q_1$ and $Q_2$ via a common feedline. A resonator bus $B$ (single-photon quality factor 210 000) couples to both qubits. Local flux-bias lines (ports 3 and 4) allow independent tuning of qubit transition frequencies with $\sim 1\mathrm{GHz}$ bandwidth [37]. (b) Scheme for two-step indirect measurement of one qubit ($Q$) through partial entanglement with an ancilla qubit ($A$) followed by projective measurement of $A$. (c) Bloch-sphere illustration of the evolution of $A$ during the interaction step, for $Q$ in $|0_Q⟩$ and $|1_Q⟩$.

Figure 2

(a) Pulse sequence realizing and testing the indirect measurement [Fig. 1b] of qubit $Q=Q_1$ with ancilla $A=Q_2$. $Q$ is first prepared in state $|\theta ⟩=\cos (\theta /2)|0_Q⟩+\sin (\theta /2)|1_Q⟩$. [(b)–(d)] Ensemble-averaged ancilla measurement in the (b) $X_A$, (c) $Y_A$, and (d) $Z_A$ bases, achieved using premeasurement rotation $U_i=R_{y_A}(-\pi /2)$, $R_{x_A}(\pi /2)$ and identity, respectively. In the experiment, $R_{x_A}(-\pi /2)$ and $U_i$ are compiled into one rotation to reduce the effects of qubit decoherence. (e) Partial-measurement backaction reduces the contrast in the final projective measurement of $Q$. Ideally, $⟨X_Q^{\prime }⟩=⟨Z_Q⟩\cos (ϵ/2)$, independent of $M_A$ basis ($X_A$ here, see the Supplemental Material [29] for the three bases). Inset: Parametric plot of oscillation amplitudes $\Delta X_Q^{\prime }$ and $\Delta X_A^{\prime }$. $\Delta X_Q^{\prime }$ $(\Delta X_A^{\prime })$ decreases (increases) as $ϵ$ increases in the range [0°,180°]. Solid (dashed) curves correspond to the model with (without) decoherence during the gate.

Figure 3

Partial-measurement kickback. The kickback on qubit $Q$ induced by partial measurement depends on the interaction strength $ϵ$, the ancilla measurement basis, and measurement result $M_A$. Left and right panels correspond to $ϵ=45°$ and $90°$, respectively. [(a),(b)] Conditioning on the result $M_A$ of ancilla measurement in the $X_A$ basis reveals distorted Rabi oscillations of $Q$. A positive (negative) result retards (advances) the oscillation for $\theta \in [0°,180°]$ and advances (retards) it for $\theta \in [180°,360°]$. [(c),(d)] Distortions for measurement in the $Y_A$ basis. In this case, the kickback is a $z$ rotation by $\pm ϵ/2$, causing an identical reduction of contrast in the conditioned Rabi oscillations. [(e),(f)] Distortions for measurement in the $Z_A$ basis. Ideally, $M_A=+1$ has no kickback on $Q$, whereas $M_A=-1$ causes a $z$ rotation of $\pi$. To extract $⟨X_Q^{\prime }⟩$, we correct for readout errors of $Q$ using standard calibration procedures [29, 30]. The difference in contrast for $⟨X_Q^{\prime }⟩$ for $M_A\pm 1$ is due to asymmetric readout errors of $A$ (Supplemental Material [29]).

Figure 4

Observation of nonclassical weak values and Leggett-Garg inequality violations (measurement strength $ϵ=45°$). (a) Pulse sequence. Note that qubit roles are swapped ($Q=Q_2$ and $A=Q_1$) compared to previous figures in order to minimize errors when conditioning on $M_Q=\pm 1$. (b) Dependence of modified weak value $W_{\mathrm{m}}$ (see text for definition and normalization procedure) on $Q$-rotation angle $\theta$. The MAR bound $|W_{\mathrm{m}}|\le 1$ is amply exceeded. From the quantum perspective, the extrema in $W_{\mathrm{m}}$ at I and II can be understood using the Bloch spheres in (c) and (d), respectively. Ideally, at I (II), the $M_A=-1(+1)$ kickback aligns $Q$ with the $+x_Q$ axis, perfectly correlating $M_Q=-1$ with $M_A=+1(-1)$. (e) Measured averaged Leggett-Garg operators $B_{1,2}$ defined in text. One of the inequalities $B_{1,2}\le 1$ is violated whenever nonclassical $W_{\mathrm{m}}$ is observed.