Self-consistent noise characterization of quantum devices

Characterizing and understanding the environment affecting quantum systems is critical to elucidate its physical properties and engineer better quantum devices. We develop an approach to reduce the quantum environment causing single-qubit dephasing to a simple yet predictive noise model. Our approach, inspired by quantum noise spectroscopy, is to define a “self-consistent” classical noise spectrum, that is, compatible with all observed decoherence under various qubit dynamics. We demonstrate the power and limits of our approach by characterizing, with nanoscale spatial resolution, the noise experienced by two electronic spins in diamond that, despite their proximity, surprisingly reveal the presence of a complex quantum spin environment, both classically reducible and not. Our results overcome the limitations of existing noise spectroscopy methods and highlight the importance of finding predictive models to accurately characterize the underlying environment. Extending our work to multiqubit systems would enable spatially resolved quantum sensing of complex environments and quantum device characterization, notably to identify correlated noise between qubits, which is crucial for practical realization of quantum error correction.

Figure 1

Reducing a quantum environment to a self-consistent classical noise model. To model a quantum environment, we attempt to develop a classical noise model $S\left(\omega \right)$ that is consistent with the set of all observed decoherence under various controlled dynamics. When such a “self-consistent” noise model is possible, as demonstrated in this paper experimentally for an NV electronic spin in diamond but not a nearby interacting electronic spin X several nanometers away, we further verify that the self-consistent model has predictive power even under new dynamics, confirming that it accurately models the underlying quantum bath.

Figure 2

Self-consistent noise model of an NV electronic spin in diamond. The (minimal) self-consistent noise model $S_{\left(\text{min}\right)}^{\text{NV}}\left(\omega \right)$ is presented, along with noise model candidates $S_{1,2}$ (consistent with R-E but not CPMG dynamics) and $S_{\text{CP}}$ (vice versa). While as shown both R-E- and CPMG-based methods fail to yield the correct noise model due to their limitations, by reconciling them our method succeeds. (a) The noise models are shown against the measured decoherence (markers) under multiple CPMG dynamics, $S\left({\omega }_m\right)$. Note that $S_{1,2}$ fail to be predictive under higher-frequency noise. (b) and (c) Decay under Ramsey (b) and spin echo (c) dynamics is measured (blue circles) and fitted (red curve) to perform R-E-noise spectroscopy. Note that $S_{\text{CP}}$ fails to be predictive under Ramsey or echo dynamics (zero or low-frequency noise). The controlled qubit dynamics (pulse sequence) is shown below; green boxes indicate the minimal experimental measurements used to inform $S_{\text{min}}^{\text{NV}}\left(\omega \right)$, which is further predictive of new dynamics (see Fig. 3). The green box in (a) contains $S_{\text{CP}}\left({\omega }_m\right)=\{17.5,12,10.5\}{\text{ms}}^{-1}$ at ${\omega }_m=\left(2\pi \right)\{0.05,0.08\overline{3},0.10\}\text{MHz}$, respectively.

Figure 3

Predictive power of the self-consistent noise model. As a crucial check that the developed self-consistent noise model $S_{\left(\text{min}\right)}^{\text{NV}}\left(\omega \right)$ is an accurate model of the underlying bath, we demonstrate that it is predictive even under new qubit dynamics, namely, in (a) higher-frequency CPMG dynamics at ${\omega }_m=0.183\text{MHz}$ (not used to inform $S_{\text{min}}^{\text{NV}}$) as well as (b) Walsh dynamics of sequency 5, while R-E-noise spectroscopy fails ($S_{1,2}$). (c) Walsh filter function. The Walsh filter $|{\tilde{f}}_{W(k,\lambda )}{\left(\omega \right)|}^2$ at time point $T=N\lambda =120\mu \text{s}$ (gold) for sequency $k=5$ (cycle length $\lambda =8\tau$) is plotted against the noise spectrum directly measured from the data $S_{\text{CP}}^{\mathrm{NV}}\left({\omega }_m\right)$ and its fit (blue). The filter function is scaled to more easily visualize which parts of the noise spectrum it is sampling. Here the filter was numerically generated by taking its finite-time Fourier transform of the Walsh time-domain function [37].

Figure 4

Quantum bath of dark electronic spin X. By investigating the spin echo (a) and CPMG (b) dynamics with varying driving (Rabi) strengths $\mathrm{\Omega }$, we verify the quantum nature of the local environment of the X spin, realized by the presence of near-resonant and interacting (NRI) spins. (a) X echo dynamics reveals the presence of NRI spins: At sufficiently high Rabi strength ${\mathrm{\Omega }}_h$, small-amplitude oscillations are observed, akin to SEDOR and DEER experiments (the curve is a guide to the eye). However, at sufficiently low Rabi strength ${\mathrm{\Omega }}_l$, the monotonic decay expected under single-qubit echo is recovered (the curve shows the fit to decays under $S_{0,1}^{\text{X}}$ which overlap). (b) X CPMG dynamics further reveals the effect of NRI spins, where typically one expects to observe higher coherence time $T_2$ with increasing Rabi power $\mathrm{\Omega }$ due to higher control $\pi$-pulse fidelity $F_{\pi }\left(\mathrm{\Omega }\right)$ (Appendix pp1). However, here we observe the opposite behavior, where the higher $F_{\pi }$ leads to lower $T_2$, because the larger ${\mathrm{\Omega }}_h$ recouples the X spin to a larger electronic-spin environment. (c) Finally, while the quantum bath precludes a classical noise model, by suppressing the X spin interaction with the quantum NRI spins—which is possible by suppressing ${\mathrm{\Omega }}_{}$—we successfully recover a classical model for the X qubit over a restricted frequency range, following the same protocol as discussed with the NV.

Figure 5

Characterization of $\pi$-pulse fidelities $F_{\pi }$. The measured NV (red squares) and X (circles) $\pi$-pulse fidelities $F_{\pi }$ are shown. The NV is controlled by a single-tone $\pi$ pulse resonant with the $m_s=\{0,-1\}$ transition and strength ${\mathrm{\Omega }}_0=6.76\text{MHz}$. By fitting the signal to $\langle {\sigma }_z\left(N\right)\rangle ={\beta }_0{(-n_z)}^N+c_0$, we extract $F_{\pi }=\left|\text{Tr}\left[U_{\pi }^{†}R\right]\right|/2=\sqrt{(1-n_z)/2}=0.987\left(0\right)$ for the NV. On the other hand, the X spin is modulated by a two-tone driving on resonance with the nuclear hyperfine splitting, to effectively remove the nuclear spin degree of freedom [41]. As expected, the control fidelity of the X spin is higher at higher Rabi frequency ${\mathrm{\Omega }}_h=10{\mathrm{\Omega }}_l=2.5\text{MHz}$: $F_{\pi }\left({\mathrm{\Omega }}_h\right)=0.992\left(2\right)$ (purple) $>F_{\pi }\left({\mathrm{\Omega }}_l\right)=0.955\left(7\right)$ (blue).