Resource-Efficient Noise Spectroscopy for Generic Quantum Dephasing Environments

Abstract

We present a resource-efficient method based on repetitive weak measurements to directly measure the noise spectrum of a generic quantum environment that causes qubit phase decoherence. The weak measurement is induced by a Ramsey interferometry measurement (RIM) on the qubit and periodically applied during the free evolution of the environment. We prove that the measurement correlation of such repetitive RIMs approximately corresponds to a direct sampling of the noise correlation function, thus enabling direct noise spectroscopy of the environment. Compared to dynamical-decoupling-based noise spectroscopy, this method can efficiently measure the full noise spectrum with the detected frequency range not limited by qubit coherence time. This method is also more resource-efficient than the correlation spectroscopy, as for the same detection accuracy with $N$ sampling times, it takes total detection time $O(N)$ while the latter one takes time $O(N^2)$. We numerically demonstrate this method for both bosonic and spin baths.

Figures (3)

• Figure 1: Schematic of direct noise spectroscopy for a generic quantum dephasing environment via repetitive weak measurements. Each cycle contains a RIM on a probe qubit and a free evolution of the environment, denoted as "R" and "Free", respectively. When the RIM induces a weak measurement on the environment, the measurement correlations between the first and the other RIMs approximately constitute a direct sampling of the environmental noise correlation function.
• Figure 2: Direct qubit noise spectroscopy for (a,b) a bosonic bath and (c,d) a spin cluster bath. The blue solid lines represent the exact noise correlation functions (multiplied by a small global factor $4\tau_1^2$) in (a,c) and the corresponding noise spectra in (b,d), while the red circles represent the direct noise spectroscopy results obtained from Monte Carlo simulations of repetitive RIMs (with $4\times 10^6$ samples). For the Ohmic bosonic bath, we simulate only a finite set of discretized modes, and the residual tail in $|\tilde{S}(\omega)|$ arises from the finite-time discrete Fourier transform. The unitless parameters for the bosonic bath are $N_\omega=48$, $\alpha=0.4$, $\omega_{\rm max}=2$, $\tau_1=0.2$, $\tau=0.9$ , $\beta=1$, and the spin cluster contains five nuclear spins in the maximally mixed state with $\tau_1=1.8\,\,\mu$s, $\tau=300\,\,\mu$s, $B=0.1$ T and the leading theoretical frequencies $\omega_{ij}$ are indicated by dashed lines.
• Figure 3: Schematic illustration of the total detection time for (a) correlation spectroscopy and (b) noise spectroscopy via repetitive weak measurements. (c) Total detection time as a function of estimation error for different noise rates of the spin bath in Fig. \ref{['fig:boson']}(d): $\Gamma\tau=0$ (solid line), $\Gamma\tau= 10^{-3}$ (dashed line), and $\Gamma \tau =5\times10^{-3}$ (dash–dotted line), where we set $\Gamma_1=\Gamma_\phi=\Gamma$. The estimation error is defined as $||S-S^*||_2/||S||_2$, where $S$ is the ideal noise spectrum obtained from a long but finite-time evolution, $S^*$ is the noise spectrum obtained by different strategies, and $||\cdot||_2$ represents the Euclidean norm.