Enhancing qubit readout through dissipative sub-Poissonian dynamics

Single-shot qubit readout typically combines high readout contrast with long-lived readout signals, leading to large signal-to-noise ratios and high readout fidelities. In recent years, it has been demonstrated that both readout contrast and readout signal lifetime, and thus the signal-to-noise ratio, can be enhanced by forcing the qubit state to transition through intermediate states. In this work, we demonstrate that the sub-Poissonian relaxation statistics introduced by intermediate states can reduce the single-shot readout error rate by orders of magnitude even when there is no increase in signal-to-noise ratio. These results hold for moderate values of the signal-to-noise ratio ($\mathcal{S}\lesssim 100$) and a small number of intermediate states ($N\lesssim 10$). The ideas presented here could have important implications for readout schemes relying on the detection of transient charge states, such as spin-to-charge conversion schemes for semiconductor spin qubits and parity-to-charge conversion schemes for topologically protected Majorana qubits.

Figure 1

Minimal model of qubit readout. (a) Level diagram of the minimal model. The detector registers a signal $I\left(t\right)=I_{\pm }+\delta I\left(t\right)$ when the qubit state is $\left|\pm \right.〉$ at time $t$. Here $\delta I\left(t\right)$ is Gaussian white noise. The excited state $\left|+\right.〉$ relaxes to the ground state with probability per unit time $\mathrm{\Gamma }$. (b) Instances of the readout signal $I\left(t\right)$ for both qubit states. The average readout signal for $\left|-\right.〉$ (dotted red line) is a constant $I_{-}$ for all readout times $t>0$. The average readout signal for $\left|+\right.〉$ (dashed blue line) is initially $I_{+}$ but jumps to $I_{-}$ at a random time $\tau >0$ due to relaxation. The probability distribution $P_0\left(\tau \right)$ of the jump time $\tau$ is exponential (solid black line).

Figure 2

Model of qubit relaxation through $N$ intermediate states. (a) The system transitions from $\left|+\right.〉$ to $\left|-\right.〉$ by cascading through the intermediate states. All transitions in the sequence occur with probability per unit time ${\mathrm{\Gamma }}_N=(N+1)\mathrm{\Gamma }$. It is assumed that all states except $\left|-\right.〉$ yield the same average readout signal $I_{+}$. (b) Instances of the readout signal for both qubit states. As before, the average readout signal for $\left|-\right.〉$ (dotted red line) is a constant $I_{-}$. The average readout signal for $\left|+\right.〉$ (dashed blue line) is initially $I_{+}$ but jumps to $I_{-}$ at a time $\tau >0$ when the system transitions from the last intermediate state to $\left|-\right.〉$. Contrary to Fig. 1, however, the jump time $\tau$ now follows the Gamma distribution $P_N\left(\tau \right)$ (solid black line).

Figure 3

Average single-shot readout error rate Eq. (1) on a logarithmic scale as a function of the SNR for different numbers $N$ of intermediate states. The reported values of $\epsilon$ are calculated as described in the main text and as detailed in Appendix pp1.

Figure 4

Sub-Poissonian charge dynamics in a chain of quantum dots [35]. Metallic gates are used to maintain strong capacitive coupling between distant dots while the charge hops from one quantum dot to the next (circles) with probability per unit time ${\mathrm{\Gamma }}_N$. The resulting statistics for the total transition time $\tau$ are given by Eq. (7).

Figure 5

Average error rate $\epsilon$ as a function of the number of intermediate states $N$ using the time-averaged method Eq. (5) (blue dots) and using the optimal method Eq. (B3) (red squares). The SNR is $\mathcal{S}=20$. In the optimal case, each error rate is obtained by simulating $5\times {10}^4$ signals $I\left(t\right)$ for each state $\left|\pm \right.〉$. The error bars for the optimal method are given by the standard deviation of a binomial process with frequency $\epsilon$.

Figure 6

Average error rate $\epsilon$ as a function of the asymmetry ratio ${\mathcal{S}}^{\left(0\right)}/{\mathcal{S}}^{\left(1\right)}$ in the case of $N=1$ intermediate state. The asymmetry in SNR arises from an asymmetry in contrast $\mathrm{\Delta }{I}^{\left(i\right)}$ (blue dots) or from an asymmetry in transition rates ${\mathrm{\Gamma }}_N^{\left(i\right)}$ (red squares). The total SNR is $\mathcal{S}={\mathcal{S}}^{\left(0\right)}+{\mathcal{S}}^{\left(1\right)}=20$. Each error rate is obtained by simulating ${10}^4$ signals $I\left(t\right)$ for each state $\left|\pm \right.〉$ and then applying the optimal processing method, Eqs. (B2) and (B3). The error bars are given by the standard deviation of a binomial process with frequency $\epsilon$. The horizontal lines are the error rates obtained from simulations of the symmetric case (solid) and completely asymmetric cases (dashed and dotted) with ${10}^5$ samples for each state. The completely asymmetric cases correspond to the readout dynamics discussed in Refs. [38, 39].