Flux-qubit readout in the persistent-current basis at arbitrary bias points

Common flux-qubit readout schemes are qubit dominated, meaning they measure in the energy eigenbasis of the qubit. For various applications, measurements in a basis different from the energy eigenbasis are required. Here we present an indirect measurement protocol, which is detector dominated instead of qubit dominated, yielding projective measurements in the persistent-current basis for arbitrary bias points. We show that with our setup it is possible to perform a quantum nondemolition measurement in the persistent-current basis at all flux bias points with fidelities reaching almost $100\%$.

Figure 1

Circuit for the measurement protocol. The qubit (yellow) is coupled to the large SQUID loop of the quantum probe, here the cjj-SQUID (green). The large loop is coupled to an additional flux readout loop (blue) and an external control flux ${\mathrm{\Phi }}_c$ (red) is applied to the small SQUID loop.

Figure 2

Principle of the measurement scheme. Color code analog to Fig. 1. (1) Initialization: the qubit (yellow) and the cjj-SQUID (green) initial state are prepared. (2) Premeasurement: the coupling between the qubit and the cjj-SQUID is ramped up, such that the qubit states get entangled with corresponding pointer states. (3) Effective decoupling: the cjj-SQUID potential is turned from a single well to a double well potential, resulting in an exponential decrease of the effective coupling ${\mathrm{\Delta }}_{\mathrm{eff}}$. (4) Readout: the cjj-SQUID persistent-current state is read out with an additional flux readout device (e.g., rf-SQUID).

Figure 3

Results of the numerical simulations of the measurement protocol for $\mathrm{\Delta }/h=0.1\mathrm{\Omega }$ and linear time schedule with $g_{\max }/\mathrm{\Omega }=1$ and ${\beta }_{cjj}^{\max }=2$ and $\xi =0.1$ (corresponding, e.g., to $C_{\mathrm{\Sigma }}=15$ fF and $L=600\mathrm{pH}$). The initial qubit state is chosen to be $|0\rangle =1/\sqrt{2}(|⥀\rangle +|⥁\rangle )$. Shown is the probability distribution of projection to qubit left (red) and right (blue) persistent-current state. Numerical (solid) and analytical results (dotted).

Figure 4

Time evolution of the screening parameter ${\beta }_{\mathrm{cjj}}$, the coupling $g$, and the coupling energy ${\mathrm{\Delta }}_{\mathrm{eff}}$.

Figure 5

Measurement fidelity depending on $g_{\max }$ for different values of $m$ for the same parameters as in Fig. 3 and for different values of $\xi$. The inset shows the infidelity in log scale to illustrate how the fidelity tends to one for large couplings.

Figure 6

Decay of excited-state population during measurement for initial state $|⥀\rangle$ and the same parameters as in Fig. 3.

Figure 7

Decay of excited-state population during measurement for initial state $|⥀\rangle$ for the same parameters as in Fig. 3.

Figure 8

Comparison of the numerical (solid) and analytical (dotted) results for the same parameters as in the main text.

Figure 9

Time evolution of the parameters $\tilde{\mathrm{\Delta }}\left(t\right)$, $\mathrm{\Gamma }\left(t\right)$, and ${\phi }_p\left(t\right)$ for the same parameters as in the main text.

Figure 10

Error of the measurement result at the flux degeneracy point (left) and way from it for $\epsilon =\mathrm{\Delta }$ (right) and 100 different generations of $1/f$ noise. The yellow line shows a linear fit to visualize the average error.

Figure 11

Left: evolution of the density-matrix element ${\rho }_{⥀⥀}\left(t\right)$ for a vacuum environment (solid blue) and a $1/f$-noise environment (dotted yellow).

Figure 12

Example of a $1/f$-noise signal generated by sampling Gaussian random processes and adding a variance to account for low-frequency shifts.

Figure 13

Corresponding PSD to the $1/f$ noise signal shown in Fig. 12.