Frame-Based Filter-Function Formalism for Quantum Characterization and Control

We introduce a theoretical framework for resource-efficient characterization and control of non-Markovian open quantum systems, which naturally allows for the integration of given, experimentally motivated, control capabilities and constraints. This is achieved by developing a transfer filter-function formalism based on the general notion of a frame and by appropriately tying the choice of frame to the available control. While recovering the standard frequency-based filter-function formalism as a special instance, this control-adapted generalization affords intrinsic flexibility and, crucially, it permits an efficient representation of the relevant control matrix elements and dynamical integrals if an appropriate finite-size frame condition is obeyed. Our frame-based formulation overcomes important limitations of existing approaches. In particular, we show how to implement quantum noise spectroscopy in the presence of nonstationary noise sources, and how to effectively achieve control-driven model reduction for noise-optimized prediction and quantum gate design.

Popular Summary

Materializing the promises of quantum technologies largely relies on our ability to execute operations with extremely high-accuracy fidelities. The main obstacle in achieving this is the unavoidable interaction of the quantum system of interest with its environment. While many strategies to achieve high-accuracy operations exist, a particularly promising strategy is one in which the system of interest is used to characterize its own environment—a procedure known as quantum noise spectroscopy (QNS)—such that the acquired information can be leveraged to perform optimal control.

Despite recent advances, QNS protocols remain far from complete since characterizing the effect of the environment on increasingly more complex systems and under the effect of realistic control constraints becomes rapidly intractable. Here we overcome many of these limitations by formalizing a simple intuition. Given a set of available control capabilities, only certain components of the noise contribute to the dynamics of the quantum system of interest. Such information is, simultaneously, what can possibly be inferred by studying the system’s dynamics, and what suffices to perform optimal control on the system, given the aforementioned capabilities. We achieve this by integrating the notion of “frames”—a staple approach in signal processing theory—with the theory of open quantum systems, which results in a powerful control-driven model-reduced representation of open quantum systems.

Our framework paves the way for the development of efficient approaches to characterize and optimally control realistic open quantum systems of growing complexity, as required by noisy intermediate-scale quantum era devices.

Figure 1

(a) Theoretical versus (b) digitally reconstructed version of the nonstationary correlation function $⟨\beta (t_1)\beta (t_2)⟩$ in units of ${\mathrm{rad}}^2{\mathrm{s}}^{-2}$ for a time resolution of $T/M$, $T=40\mathrm{ms}$ and $M=16$. The solid magenta curve in (a) highlights where $⟨\beta (t_1)\beta (t_2)⟩$ is not differentiable.

Figure 2

(a) Actual antisymmetric (quantum) and (b) symmetric (classical) components of the correlation function, and their corresponding digital reconstructions [(c) and (d), respectively] in units of ${10}^{18}{\mathrm{Hz}}^2$ for $T=1536\mathrm{ps}$ and $M=16$. In our numerical simulations we choose: $w_1=125\pi /96\mathrm{GHz}$, $w_2=7.5w_1$, ${\mathrm{\Omega }}_2=60{\mathrm{\Omega }}_1$, ${\mathrm{\Omega }}_1=125\pi /128\mathrm{GHz}$, couplings ${\overline{g}}_1/\hslash =976.56\mathrm{MHz}$, ${\overline{g}}_2/\hslash =345.27\mathrm{MHz}$, and inverse temperatures $\hslash {\beta }_1=61.44\mathrm{ps}$, $\hslash {\beta }_2=2.05\mathrm{ps}$ (corresponding to $T_1=0.12\mathrm{K}$ and $T_2=3.73\mathrm{K}$). The reconstruction with the chosen resolution is capable of capturing the slow oscillations, but not the fast oscillations.

Figure 3

(a) Actual antisymmetric (quantum) component of the correlation function and (b) its corresponding digital reconstruction in units of $({10}^9\mathrm{GHz}{)}^2$ for $T=16\mathrm{ps}$ and $M=16.$ While the resolution is $\tau =1\mathrm{ps}=16\mathrm{ps}/16=1536\mathrm{ps}/1536$, equivalent to performing a digital reconstruction on a total time of $1536\mathrm{ps}$ with a $1536\times 1536$ grid, note that in the reconstruction with $T=16\mathrm{ps}$, the slow oscillations cannot be appreciated.

Figure 4

The model assumes a correlation function (upper left) $C_{+}^{(a)}(t_1-t_2)=⟨[B^{(a)}(t_1),B^{(a)}(t_2){]}_{+}⟩$ unknown to the experimenter. Given $\mathcal{C}$ (hence $\mathcal{F}$), one can calculate the ideal frame projection ${\overline{S}}_a^{+}(n_1,n_2)$ (bottom middle) and the corresponding time-domain representation $[B^{(a)}(t_1),B^{(a)}(t_2){]}_{+}{|}_{\mathcal{F}}$ (top middle). A CA-QNS protocol provides an accurate estimate ${\hat{\overline{S}}}_a^{+}(n_1,n_2)/{\hslash }^2$ (bottom right) of ${\overline{S}}_a^{+}(n_1,n_2)/{\hslash }^2$, with a maximum absolute error of $5.38\times {10}^{-4}$. For multiplicative noise, our estimates are ${\hat{\overline{S}}}_m^{+}(1,1)/{\hslash }^2=1.125\times {10}^{-3}{\mathrm{Hz}}^2$ and ${\hat{\overline{S}}}_m^{+}(2,1)/{\hslash }^2=2.251\times {10}^{-3}{\mathrm{Hz}}^2$, with relative errors less than $0.001\mathrm{\%}$. The magnitude of these errors depends on a nonzero $\epsilon$. The distance $\mathcal{E}(P;T)$ between the actual dynamics—calculated using $[B^{(\alpha )}(t_1),B^{(\alpha )}(t_2){]}_{+}$—and the predicted dynamics—using ${\hat{\overline{S}}}_{\alpha }^{+}(n_1,n_2)$—at the time $T$ is small. We showcase the prediction error by randomly selecting pulse directions [here we use ${\overrightarrow{n}}^{(1)}=(0.658,0.751,-0.052)$ and ${\overrightarrow{n}}^{(2)}=(-0.411,0.000,0.912)$] and sweeping over all pulse angles allowed by $\mathcal{C}$ (bottom left). All figures are given in units of ${\mathrm{kHz}}^2$.

Figure 5

Illustrative example of the possible issues from incompatibility between a control sequence of interest and QNS-inferred information. Red dots represent sampled points from comb-based SP QNS for a representative Gaussian power spectrum, $S(\omega )=\exp [-{\omega }^2/(3\times {10}^{11})]/7042.$ Shown are also three possible interpolations $S_{\mathrm{inter}}^{(i)}(\omega )$ consistent with them. Namely, (i) $S_{\mathrm{inter}}^{(1)}(\omega )=S(\omega )$ (blue line), (ii) $S_{\mathrm{inter}}^{(2)}(\omega )=S(\omega )(1+{\sin }^4[\frac{1}{2}\pi \omega \times {10}^{-5}])$ (brown dashed line), and (iii) $S_{\mathrm{inter}}^{(3)}(\omega )=S(\omega )(1-{\sin }^4[\frac{1}{2}\pi \omega \times {10}^{-5}])$ (green dash-dot line). Given in the legend are the predicted expectation values of ${\sigma }_x$, obtained by using the initial state ${\rho }_S=({\sigma }_0+{\sigma }_x)/2$ and the expression $E[{\sigma }_x(T){]}_{{\rho }_S}=\exp [-\int d\omega |F^{(1)}(\omega ,T){|}^2{S}_{\mathrm{inter}}^{(i)}(\omega )].$ The predicted value can vary significantly depending on the interpolation being considered if the control sequence being studied is not “of the form” of those used in the QNS protocol. In this case, this means ensuring that $|F^{(1)}(\omega ,T){|}^2$ (in gray) has peaks out of phase with the interpolation points. An example of such a filter [12] can be generated by $M\gg 1$ repetitions of a $\pi$-pulse sequence with cycle time $T_c=(20\pi )/3\mu \mathrm{s}.$ Shown here is the filter for $M=10$ repetitions of a Hahn-echo sequence given by $[T_c/2-{\sigma }_x-T_c/2-{\sigma }_x]$.

Figure 6

Left: the first eight Walsh functions in sequency ordering. The gray solid lines represent zero for each function. The names of the traditional $\pi$-pulse sequences [46, 47] are written next to the plot. Right: each Walsh function is represented with its values in intervals.

Figure 7

(a) The five functions in the dual frame ${\tilde{\mathcal{F}}}^{(a)}$, obtained via the Moore-Penrose pseudoinverse. (b) Seven examples (orange thick) of $\cos ({\theta }_1{\psi }_t^{(1)})$ and $\sin ({\theta }_1{\psi }_t^{(1)})$ with randomly generated ${\theta }_1$ and their approximated frame representation (black dashed) with ${\tilde{N}}_{\mathrm{\#}}=2$.

Figure 8

Frame elements in ${\mathcal{F}}_{\mathrm{\#}}^{(a)}$ for $M=2$ and ${\tilde{N}}_{\mathrm{\#}}=2$. Horizontal axes are time, $t(\mu s)$.