Prediction and Retrodiction for a Continuously Monitored Superconducting Qubit

The quantum state of a superconducting transmon qubit inside a three-dimensional cavity is monitored by transmission of a microwave field through the cavity. The information inferred from the measurement record is incorporated in a density matrix ${\rho }_t$, which is conditioned on probe results until $t$, and in an auxiliary matrix $E_t$, which is conditioned on probe results obtained after $t$. Here, we obtain these matrices from experimental data and we illustrate their application to predict and retrodict the outcome of weak and strong qubit measurements.

Figure 1

Time evolution in a monitored system. (a) Simplified experimental setup consisting of a transmon circuit coupled to a waveguide cavity. (b) We prepare the qubit in an initial state [$\mathrm{Tr}({\rho }_i{\sigma }_x)\simeq +1$] and propagate $\rho$ forward in time, which makes accurate predictions about a final projective measurement (in the ${\sigma }_z$ basis) labeled $M$. The dashed line is the prediction based on a single quantum trajectory, and the solid line is the result from projective measurements on an ensemble of experiments that have similar values of ${\rho }_t$.

Figure 2

Retrodiction in a monitored system. (a) To test retrodictions made by $E$ we prepare different states ${\rho }_i$ and conduct a subsequent projective measurement $M$. We propagate $E$ backwards from the final state $E_T=\widehat{I}/2$ to $E_0$ for variable periods of time ($T$). This yields a retrodiction (shown as dashed lines for two different experiments) for the outcome of $M$. The solid line, which is based on an ensemble of experiments that yielded similar values of $E_0$ confirms the retrodictions based on the single measurement record. (b) We prepare two different initial states [$\mathrm{Tr}({\rho }_i{\sigma }_x)\simeq +1$, red, $\mathrm{Tr}({\rho }_i{\sigma }_y)\simeq +1$, blue], and compare the retrodictions, $P_p(+z)$, based on $5\mu \mathrm{s}$ of probing, to the outcomes of measurements $M$ that yielded similar values of $E_0$. In the lower panel, we display histograms of $P_p(+z)$ for different propagation times. As more of the record is included, the retrodicted probabilities assume a wider range of values.

Figure 3

Conventional and past quantum state predictions for the measurement ${\mathrm{\Omega }}_V$ conducted at time $t$. (a) The experiment sequence initializes the qubit along $+x$ and probes the cavity while the qubit transition is driven with a constant Rabi frequency. Each experiment yields a value $V$ resulting from the ${\mathrm{\Omega }}_V$ measurement and predicted mean value $⟨V⟩$ (which is based on $\rho$), and $⟨V{⟩}_p$ (which is based on $\rho$ and $E$). We plot $V$ versus $⟨V⟩$ (b) and $⟨V{⟩}_p$ (c) and find that the conditional average of $V$ (open circles) is in agreement with the expected mean value given by the dashed line. Note that $⟨V{⟩}_p$ makes predictions for the mean value that fall outside of the spectral range of the qubit observable (in the pink region).

Figure 4

Bloch vector representation of the matrix elements of $\rho$ and $E$. For each iteration of the experiment, a line joins the coordinates $\{\mathrm{Tr}(\rho {\sigma }_x),\mathrm{Tr}(\rho {\sigma }_z)\}=\{⟨{\sigma }_x⟩,⟨{\sigma }_z⟩\}$ (closed circles) and $[1/\mathrm{Tr}(E)]\{\mathrm{Tr}(E{\sigma }_x),\mathrm{Tr}(E{\sigma }_z)\}$ (open circles). The closed circles represent the state of the system at time $t$ based on ${\rho }_t$, and the open circles represent the corresponding quantity based on $E_{t+\mathrm{\Delta }t}$. The color indicates the value of $⟨V{⟩}_p$ for each pair of states. Panel (a) displays some of the the matrix elements that yield normal predictions ($|⟨V{⟩}_p|\le 1$), and panel (b) displays a sample of matrix elements that yield anomalous ($|⟨V{⟩}_p|>1$) predictions.