Shadow Tomography on General Measurement Frames

We provide a new perspective on shadow tomography by demonstrating its deep connections with the general theory of measurement frames. By showing that the formalism of measurement frames offers a natural framework for shadow tomography—in which “classical shadows” correspond to unbiased estimators derived from a suitable dual frame associated with the given measurement—we highlight the intrinsic connection between standard state tomography and shadow tomography. Such a perspective allows us to examine the interplay between measurements, reconstructed observables, and the estimators used to process measurement outcomes, while paving the way to assessing the influence of the input state and the dimension of the underlying space on estimation errors. Our approach generalizes the method described by Huang et al. [H.-Y. Huang et al., Nat. Phys. 16, 1050 (2020)], whose results are recovered in the special case of covariant measurement frames. As an application, we demonstrate that a sought-after target of shadow tomography can be achieved for the entire class of tight rank-1 measurement frames—namely, that it is possible to accurately estimate a finite set of generic rank-1 bounded observables while avoiding the growth of the number of the required samples with the state dimension.

Popular Summary

A new perspective on shadow tomography, a technique to efficiently estimate properties of unknown quantum states, is proposed. Specifically, we unveil a tight link between shadow tomography and the broader theory of measurement frames, which allows the exploitation of the advantages held by shadow tomography in a previously unforeseen large class of experimental settings.

Such a link unravels the complex interaction between measurements, reconstructed observables, and estimators and provides a new tool to scrutinize how input states and space dimensions affect estimation errors. The proposed approach also demonstrates that the conceptual foundation of shadow tomography naturally emerges from broader metrological considerations and provides accurate estimations with a lower sample requirement as the state dimension grows.

Our work holds the promise of providing useful experimental routes for the inference of the properties of quantum states, a key task for the validation of quantum systems and processes that will increase in relevance as the size of noisy intermediate-scale quantum computing systems grow.

Figure 1

The probability distributions of the sample means. Histograms of the probability distribution of the sample mean ${\overline{o}}_N$ with $N={10}^3$, obtained taking the average of $\hat{o}(b)\equiv ⟨\hat{f}(b),\mathcal{O}⟩$ over $N$ randomly sampled outcomes $b$, for different choices of estimator $\hat{f}$. The histograms are computed using ${10}^4$ realizations of the sample mean. The input state is $\rho \equiv {\mathbb{P}}_0$ in all cases and the measurements are random rank-1 POVMs built as ${\mu }_b=V{\mathbb{P}}_b{V}^{†}$ with $V$ random isometries. In each case, we show the distribution of the sample mean for the nonrescaled estimator ${\mu }^{\star }$ [cf. Eq. (3)]; the estimators ${\tilde{\mu }}^{(\rho )}$ and ${\tilde{\mu }}^{(\sigma )}$ [cf. Eq. (9)] with $\sigma \equiv {\mathbb{P}}_1$; and the canonical estimator ${\tilde{\mu }}^{\mathrm{can}}$ [cf. Eq. (15)]. We show the data for (a) two-dimensional states with ten-outcome measurements and (b) five-dimensional states and 100-outcome measurements.

Figure 2

The sample variance for different estimators. Examples of the behavior of the sample variance ${\hat{S}}_N$ of the estimator ${\overline{o}}_N\equiv 1/N\sum _{k=1}^N\hat{o}(b_k)$ as a function of $N$, computed with respect to the estimators ${\mathit{\mu }}^{\star }$, ${\tilde{\mathit{\mu }}}^{(\rho )}$, and ${\tilde{\mathit{\mu }}}^{\mathrm{can}}$. The sample variance is defined as ${\hat{S}}_N\equiv 1/(N-1)\sum _{k=1}^N(\hat{o}(b_k)-{\overline{o}}_N{)}^2$. The dashed lines give the values of the variance $\mathrm{Var}[\hat{o}|\rho ]$ in each case, as computed via Eq. (20). The data are obtained using $d=2$-dimensional systems, with fixed input state $\rho ={\mathbb{P}}_0$, random rank-1 POVMs with ten outcomes, and random target observables with $\mathrm{tr}(\mathcal{O})=0$ and $\mathrm{tr}({\mathcal{O}}^2)=1$.

Figure 3

The average, minimum, and maximum variance for mutually unbiased basis (MUB) POVMs. We plot the values of ${\lambda }_{\min }(A)$, $\Vert A{\Vert }_{\mathrm{op}}$, and $\mathrm{tr}(A)/d$, as a function of the state dimension $d$, for the case of canonical estimators, with a random target observable for each $d$. The data are shown for prime $d$ because these are the values corresponding to which explicit constructions for MUBs are known [47]. These results give the range of possible values of $⟨A,\rho ⟩$ varying over the input states $\rho$, for the case of MUB measurements. These values are then tightly connected with the estimation variance via Eq. (28). The data shown correspond to a random target observable with $\mathrm{tr}(\mathcal{O})=0$ and $\mathrm{tr}({\mathcal{O}}^2)=1$.

Figure 4

The distributions of the estimators and their sample mean, corresponding to MUBs and Haar-random unitary POVMs. (a) Histograms of the probability distributions for the estimator $⟨\mathcal{O},\tilde{\mu }⟩$ for a random (fixed) observable $\mathcal{O}$ with $\mathrm{tr}(\mathcal{O})=0,\mathrm{tr}({\mathcal{O}}^2)=1$, and fixed qutrit state $\rho ={\mathbb{P}}_0$. The reported results correspond to MUBs, ${\mathit{\mu }}_{\mathrm{MUB}}$ (red), and random measurements ${\mathit{\mu }}_{\mathrm{Haar}}$, which have elements ${\mu }_{U,b}=U{\mathbb{P}}_b{U}^{†}$ with Haar-random unitaries $U$ (blue). For ${\mathit{\mu }}_{\mathrm{MUB}}$, there is a finite number of outcomes and we plot the probability associated with each outcome directly. For ${\mathit{\mu }}_{\mathrm{Haar}}$, owing to the infinitely many outcomes, we uniformly draw a number of random unitaries $U$ and plot a histogram of the observed estimator values $⟨{\tilde{\mathit{\mu }}}_{\mathrm{Haar}},\mathcal{O}⟩$. We show two different scales on the vertical axis: in the presence of a continuum of possible outcomes, as we have for $⟨\mathcal{O},{\tilde{\mathit{\mu }}}_{\mathrm{Haar}}⟩$, we plot the probability density function (PDF); while for finitely many outcomes, we show the probability mass function. (b) The histogram of possible outcomes of the sample mean ${\overline{o}}_N\equiv \frac{1}{N}\sum _{k=1}^N\hat{o}(b_k)$ of $\hat{o}(b)\equiv ⟨\mathcal{O},{\tilde{\mu }}_b⟩$, estimated with statistics of $N={10}^3$ samples. The histogram is drawn sampling ${10}^4$ realizations of this sample mean, in the same condition as the other histogram. The black solid line is a Gaussian with the same mean and variance as both estimators, ${\tilde{\mathit{\mu }}}_{\mathrm{MUB}}$ and ${\tilde{\mathit{\mu }}}_{\mathrm{Haar}}$—which have the same variance, both being tight measurement frames. Both histograms approach this Gaussian for $N\to \mathrm{\infty }$, due to the central-limit theorem.