Stabilization of Qubit Relaxation Rates by Frequency Modulation

Temporal, spectral, and sample-to-sample fluctuations in coherence properties of qubits form an outstanding challenge for the development of upscaled fault-tolerant quantum computers. A ubiquitous source for these fluctuations in superconducting qubits is a set of atomic scale defects with a two-level structure. Here we propose a way to mitigate these fluctuations and stabilize the qubit performance. We show that frequency modulation of a qubit or, alternatively, of the two-level defects, leads to averaging of the qubit relaxation rate over a wide interval of frequencies.

Figure 1

Qubit relaxation rate as a function of its central energy splitting $E_0$, obtained from a numerical integration of the Schrödinger equation, Eq. (4), for a bath of 40 weakly coupled TLSs with a constant level spacing $\mathrm{\Delta }/\mathrm{\Omega }=5/3$, and for various values of $A/\mathrm{\Omega }$, as indicated in the legend. The relaxation rate ${\mathrm{\Gamma }}_{0.99}$ (in units of $\mathrm{\Omega }$) is calculated as $1/T$, where $T$ is the time for which $|a(T)|=0.99$. The couplings $g_n/\mathrm{\Omega }$ and relaxation rates ${\gamma }_n/\mathrm{\Omega }$ are randomly chosen from homogeneous distributions in the ranges $[2/3,10/3]\times {10}^{-2}$ and $[2/3,10/3]\times {10}^{-1}$, respectively. The qubit central energy splitting is counted relative to some typical energy splitting of the qubit, $E^{\ast }$.

Figure 2

Qubit relaxation rate as a function of its central energy splitting $E_0$, obtained from a numerical integration of the Schrödinger equation, Eq. (4), for various values of $A/\mathrm{\Omega }$, as indicated in the legend. The relaxation rate ${\mathrm{\Gamma }}_{0.99}$ (in units of $\mathrm{\Omega }$) is calculated as $1/T$, where $T$ is the time for which $|a(T)|=0.99$. The bath consists of a single strongly coupled TLS with (a) $g/\mathrm{\Omega }=2/3\times {10}^{-1}$ and $\gamma /\mathrm{\Omega }=0.02$ and (b) $g/\mathrm{\Omega }=0.2$ and $\gamma /\mathrm{\Omega }=2/3\times {10}^{-1}$. The inset in (a) is an enlargement of the undriven case ($A=0$), showing a flat peak with height $\propto g$ for $|E_0-\epsilon |<g$ and Lorentzian tails for $|E_0-\epsilon |>g$ (note that $\epsilon$ is taken as the origin of the horizontal axis).

Figure 3

Qubit relaxation rate as a function of its central energy splitting $E_0$ (counted relative to some typical energy splitting of the qubit, $E^{\ast }$), obtained from a numerical integration of the Schrödinger equation, Eq. (4), for various values of $A/\mathrm{\Omega }$, as indicated in the legend. The relaxation rate ${\mathrm{\Gamma }}_{0.99}$ (in units of $\mathrm{\Omega }$) is calculated as $1/T$, where $T$ is the time for which $|a(T)|=0.99$. The calculation is performed for a bath of 41 TLSs with a constant level spacing $\mathrm{\Delta }/\mathrm{\Omega }=5/3$, 40 weakly coupled TLSs with couplings $g_n/\mathrm{\Omega }$ and relaxation rates ${\gamma }_n/\mathrm{\Omega }$ randomly chosen from homogeneous distributions in the ranges $[2/3,10/3]\times {10}^{-2}$ and $[2/3,10/3]\times {10}^{-1}$, respectively [similar realization as in Fig. 1], and a single strongly coupled TLS at $(E_0-E^{\ast })/\mathrm{\Omega }=-25$ with $g/\mathrm{\Omega }=2/3\times {10}^{-1}$ and $\gamma /\mathrm{\Omega }=0.02$ [as in Fig. 2].