Suppression of Dephasing by Qubit Motion in Superconducting Circuits

We suggest and demonstrate a protocol which suppresses the low-frequency dephasing by qubit motion, i.e., transfer of the logical qubit of information in a system of $n\ge 2$ physical qubits. The protocol requires only the nearest-neighbor coupling and is applicable to different qubit structures. Our analysis of its effectiveness against noises with arbitrary correlations, together with experiments using up to three superconducting qubits, shows that for the realistic uncorrelated noises, qubit motion increases the dephasing time of the logical qubit as $\sqrt{n}$. In general, the protocol provides a diagnostic tool for measurements of the noise correlations.

Figure 1

(a) Device schematic showing three phase qubits capacitively coupled to a central resonator. (b) The experimental sequence of the Ramsey fringe measurement for a single qubit. (c) Experimental sequences for relaying the logical qubit state between two physical qubits (top panel) and among three physical qubits (bottom panel). The logical qubit moves along solid lines. The relay sequence starts with an $R_{\widehat{x}}^{\pi /2}$ rotation on the first qubit to create the logical qubit state in the $x\text{−}y$ plane of the Bloch sphere, i.e., $|\mathrm{\Psi }⟩=|0⟩-i|1⟩$. Relaying the logical qubit state to the next qubit is done by two successive qubit-resonator iswap gates that takes 32 ns in total (symbolized by the double crossed arrows). Finally, an $R_{\widehat{\alpha }}^{\pi /2}$ rotation is applied to the last physical qubit in the sequence to bring the logical qubit state to the $z$ axis of the Bloch sphere for measuring the $|1⟩$-state probability of this qubit, $P_1$. Here, $\widehat{\alpha }$ refers to the effective axis that rotates in the $x\text{−}y$ plane after removing the dynamical phase, i.e., $\widehat{\alpha }=\cos ({\omega }_R\tau )\widehat{x}+\sin ({\omega }_R\tau )\widehat{y}$, where ${\omega }_R/2\pi$ is adjusted to around 25 MHz in the experiment.

Figure 2

The Ramsey fringe experimental data for sequences shown in Fig. 1. Red lines are fits according to Eq. (10). $T_2^{*}$ for a single qubit can be directly compared with ${\tau }_d$ for multiple qubits. Statistical errors, from the measured probability spread of $\sim 2\%$, are omitted for display clarity but are used to estimate the standard deviations of $T_2^{*}$ and ${\tau }_d$.

Figure 3

(a) The 2-qubit Ramsey fringe results under artificial noises, with different correlations $r_c$ as indicated. Dots are experimental data and lines are fits. Fitted ${\tau }_d^E$ values are $213\pm 4$, $251\pm 4$, $281\pm 7$, $345\pm 6$, and $485\pm 8\mathrm{ns}$ from top to bottom. (b) $(1/{\tau }_d^E{)}^2-(1/{\tau }_d^I{)}^2$ versus $r_c$ (the black squares), where ${\tau }_d^I$ is the dephasing time from the 2-qubit Ramsey fringe experiment under no artificial noise. Error bars are estimated based on uncertainties of ${\tau }_d^E$ and ${\tau }_d^I$. The numerical result (the red dots) is obtained by solving the Schrodinger-Langevin equation [32] with a time-dependent Hamiltonian, while the swap nonideality is accounted for using the Lindblad master equation. The line is a guide for the eye.