Experimental Hybrid Shadow Tomography and Distillation

A hybrid shadow (HS) tomography [31] has been proposed aiming to enhance the efficiency for estimating nonlinear functions, which combines the advantages of shadow estimation and the generalized SWAP test [32, 33]. HS can effectively conduct randomized measurements on $\rho^{k}$ by introducing a coherent controlled-(generalized) SWAP operation on $k$-copy $\rho^{\otimes k}$, and therefore construct the estimator $\widehat{\rho^{k}}$. In this way, HS is capable of estimating higher-degree functions by patching lower-degree estimator $\widehat{\rho^{k}}$ via post-processing, which can significantly reduce the sampling and post-processing costs [31]. The theoretical and numerical findings there depend on the ideal multi-qubit coherent operation, and thus it remains an open question whether HS holds its effectiveness and superiority in real quantum platforms.

In this work, we give this question a positive answer by realizing HS in an optical system. We design and implement a novel quantum Fredkin gate with single photon, which is able to deterministically apply controlled-SWAP operation on two single-qubit states with high fidelity. Inspired by this novel Fredkin gate, we implement HS and show the efficiency of HS in estimation of higher-degree functions, i.e., the moment functions $P_{L}$ with $L=\{2,3,4\}$. We demonstrate that the measurement budget and sample complexity are significantly reduced in HS compared to OS. Moreover, we showcase the usefulness of HS in VD and error-mitigated quantum metrology.

For a (linear) observable $O$, OS provides an unbiased estimation of $\langle O\rangle=\text{Tr}(O\rho)$ via $\hat{o}=\text{Tr}(O\hat{\rho})$. Meanwhile, OS is capable of estimating nonlinear functions, such as the higher-order moment $P_{L}$, and the corresponding unbiased estimator shows

where $\mathbb{S}_{L}$ denotes the generalized SWAP operation, i.e., the SHIFT operation on $\mathcal{H}^{\otimes L}$. Here, $\hat{\rho}_{(j)}$ are distinct shadow snapshots. Meanwhile, one can construct the estimator for $\text{Tr}(O\rho^{L})$ as $\widehat{o_{L}}_{(\text{OS})}=\text{Tr}\left[O\prod_{j=1}^{L}\hat{\rho}_{(j)}\right]$. One would enhance the estimation accuracy by repeating the experiment for $M$ times to collect the shadow set $\{\hat{\rho}_{(i)}\}_{i=1}^{M}$. However, the exponentially increased number of measurements with the qubit number, along with the cost for classical post-processing retards efficiency of OS in estimation of nonlinear functions [5, 17].

HS [31] advances OS on estimating nonlinear functions. As shown in Fig. 1(b), a controlled-$\mathbb{S}_{k}$ gate is applied on $\rho^{\otimes k}$ with the control qubit initialized as $\ket{+}$, and in general $k<L$. The measurement outcome of the control qubit in the X-basis is denoted as ${b_{c}}$, and one implements OS on one copy of $\rho$ with measurement outcome $\bm{b}$. For each single-shot experimental run, one can construct the estimator of $\rho^{k}$ as

such that $\mathbb{E}_{\{U,\bm{b},b_{c}\}}\widehat{\rho^{k}}=\rho^{k}$, and similarly one can collect the set $\{\widehat{\rho^{k}}_{(i)}\}_{i=1}^{M}$. Interestingly, by ignoring the information of $b_{c}$ in the hybrid setting, one can also get the classical shadow of $\rho$ with $\bm{U}$ and $\bm{b}$ via Eq. (1). (See Supplementary Materials for a detailed illustration.)

In this way, one can build estimators for many nonlinear properties. For example, $\text{Tr}(O\rho^{k})$ can be directly estimated with $\widehat{o_{k}}=\text{Tr}(O\widehat{\rho^{k}})$. Furthermore, more complex nonlinear functions with a higher degree can be estimated by combining low-degree shadow snapshots. Specifically, the unbiased estimator of $P_{L}$ shows

where $r=L\ (\text{mod}\ k)$ with $L=qk+r$, $\widehat{\rho^{k}}_{(j)}$ and $\hat{\rho}_{(j^{\prime})}$ are distinct snapshots. Similarly, one can construct the corresponding estimator for $\text{Tr}(O\rho^{L})$ as $\widehat{o_{L}}_{(\text{HS})}=\text{Tr}\left[O\prod_{j=1}^{q}\widehat{\rho^{k}}_{(j)}\prod_{j^{\prime}=1}^{r}\hat{\rho}_{(j^{\prime})}\right]$. Compared to the one by OS in Eq. (2), HS requires $q+r$ snapshots while OS requires $L$ ones, thus the classical post-processing cost for HS is significantly reduced. Moreover, as the estimator of HS effectively transfers an $L$-degree nonlinear function to a $(q+r)$-degree one, the statistical variance can also be reduced, thus saving the sample number of $\rho$ in practise.

Apart from many applications [36, 37, 38], nonlinear functions are vital in purification-based quantum error mitigation, say VD [24, 25, 27]. VD enables to approach the noiseless value of the observable $O$ over noisy state $\rho$ by $\langle O\rangle^{(L)}_{\text{VD}}=\text{Tr}\left(O\rho^{L}\right)/\text{Tr}\left(\rho^{L}\right)$, which is more accurate than the noisy value $\text{Tr}(O\rho)$. Such enhancement increases exponentially with the degree $L$. VD benefits a few quantum information tasks, such as quantum metrology [27, 39]. In quantum metrology [40, 41], a probe state $\ket{\psi}$ evolves under the dynamics $V_{\theta}$, and the estimator $\hat{\theta}$ of the unknown parameter $\theta$ can be constructed via measuring the evolved state $\ket{\psi_{\theta}}$ using some observable $O$. A noisy preparation $\rho$ instead of $\ket{\psi}$ would introduce considerable systematic errors [27, 42, 43]. Here in the hybrid framework, we follow the spirit of VD to feed one of the output port of $\rho^{\otimes k}$ after the control gate to the evolution $V_{\theta}$ and finally make the measurement in the eigenbasis of some observable $O$, as shown in Fig. 1(c). After repeating the experiment, $\langle O\rangle^{(k)}_{\text{VD}}$ can be estimated by

where $o_{i}$ is the outcome of $O$ for the $i$-th shot. Such approach significantly enhances the estimating accuracy, as it effectively realizes quantum metrology with the purified state $\rho^{k}/\text{Tr}(\rho^{k})$, which is much closer to the ideal one $\ket{\psi}$ than $\rho$. We should remark that our scheme only needs to feed the purified port to the environment (other than all $k$ copies) and do not require the controlled-$O$ operation compared to Ref. [27]. (See Supplementary Materials for proofs and more details.)

We first characterize the Fredkin gate by measuring the truth table, according to which we observe the classical fidelity of $0.9898\pm 0.0003$ with respect to the ideal case. The Fredkin gate is also capable to generate three-qubit Greenberger-Horne-Zeilinger (GHZ) state $\ket{\text{GHZ}}=(\ket{010}+\ket{101})/\sqrt{2}$ by inputting $\ket{+10}$. The fidelity of generated GHZ state is $0.933\pm 0.002$. Furthermore, we estimate the process fidelity $0.935\pm 0.001$ of the implemented gate. (See Supplementary Materials for more experimental results regarding the characterization of Fredkin gate.)

We implement HS on two copies of single-qubit noisy state $\rho=0.8\ket{+}\bra{+}+0.2\mathbb{I}/2$, where the two-qubit target state $\rho^{\otimes 2}$ can be written as

and can be prepared by randomly setting angles $[\theta_{1},\theta_{2},\theta_{3}]$ to generate $\ket{t_{1}t_{2}}\in\{\ket{++}$, $\ket{+1}$, $\ket{+0}$, $\ket{1+}$, $\ket{11}$, $\ket{10}$, $\ket{0+}$, $\ket{01}$, $\ket{00}\}$, with probability of 0.64, 0.08, 0.08, 0.08, 0.01, 0.01, 0.08, 0.01, 0.01 (See supplementary Materials for the angle settings.) After the Fredkin gate, the control qubit is measured on Pauli-$X$ basis. The target qubit is randomly detected on Pauli-$X$, $Y$ and $Z$ bases. By running the experiment many times, the unbiased estimators of $P_{2}$, $P_{3}$ and $P_{4}$ are constructed by

according to Eq. (4), with $\{\hat{\rho}_{(i)}\}$ and $\{\widehat{\rho^{2}}_{(i)}\}$ constructed via Eq. (1) and (3), respectively. The experimental results of $\widehat{P_{2}}$, $\widehat{P_{3}}$ and $\widehat{P_{4}}$ are shown with red dots in Fig. 3(a)-Fig. 3(c), respectively. To give a comparison, we also implement OS to estimate $\widehat{P_{2}}$, $\widehat{P_{3}}$ and $\widehat{P_{4}}$ according to Eq. (2), and the results are shown with blue dots in Fig. 3(a)-Fig. 3(c) respectively. Note that the HS consumes two copies of $\rho$ in each experiment run while the OS consumes one, so that the comparison is carried out with the same number of consumed copies (denoted by $N$) instead of experimental runs $M$. As reflected in Fig. 3(a)-Fig. 3(c), the results of HS converge faster than that of OS, which clearly demonstrate the reduction of sample complexity using the hybrid framework.

The sample complexity is further investigated by the variance of $\widehat{P_{L}}$ [5, 48]

Note that Var$(\widehat{P_{L}})\to 0$ when $N\to\infty$. In the experiment, the converged value $P_{L}$ is estimated by using a large number ($3\times 10^{4}$) of copies. The results of Var$(\widehat{P}_{2})$, Var$(\widehat{P}_{3})$, Var$(\widehat{P}_{4})$ are shown in Fig. 3(d)-Fig. 3(f) respectively, and we observe Var$(\widehat{P_{L}})$ of HS converges to zero faster than that of OS.

Next, we show VD of the expectation value of some observable $O$ with the assistance of $\widehat{P_{L}}$ and $\widehat{O_{L}}$. That is, estimating $\langle O\rangle^{(L)}_{\text{VD}}$ with $\widehat{O_{L}}/\widehat{P_{L}}$, where $\widehat{O_{L}}$ is an estimator of $\text{Tr}(O\rho^{L})$. For example, $\widehat{O_{3}}$ for HS is constructed as

in a similar way as $\widehat{P_{3}}_{(\text{HS})}$ in Eq. (7). The result of estimating $\langle X\rangle^{(3)}_{\text{VD}}$ for observable $O=X$ is shown in Fig. 4(a). After VD, $\langle X\rangle^{(3)}_{\text{VD}}$ is closer to 1, which is the theoretical prediction of $\langle X\rangle$ on state $\ket{+}$ ( green dashed line in insert of Fig. 4(a)). Similarly, the variance of $\langle X\rangle^{3}_{\text{VD}}$ converges faster compared to OS as reflected in Fig. 4(b). See Supplementary Materials for the results of $\langle X\rangle^{(2)}_{\text{VD}}$ and $\langle X\rangle^{(4)}_{\text{VD}}$. Note that $\langle X\rangle^{(4)}_{\text{VD}}$ is more accurate than $\langle X\rangle^{(3)}_{\text{VD}}$.

Finally, we demonstrate quantum metrology with the assistance of VD in such a hybrid framework. We consider the probe qubit passing through a phase gate ${V_{\theta}}=e^{-i\frac{\theta}{2}Z}$ as shown in Fig. 2(e), and the value of $\theta$ is the parameter to be estimated. By measuring the probe qubit in the eigenbasis of Pauli $Y$, one can obtain the estimation $\hat{\theta}=\widehat{Y_{L}}/\widehat{P_{L}}$. The optimal accuracy is achieved with the perfect probe state of $\ket{+}$. In our experiment, we set $\theta=\pi/15$ and the initial probe state is a noisy state $\rho=0.8\ket{+}\bra{+}+0.2\mathbb{I}/2$. Without VD, $\delta\hat{\theta}$ is illustrated with blue dots in Fig. 4(c). The results of $\delta\hat{\theta}$ with VD are shown with red dots in Fig. 4(c), and we observe $\delta\hat{\theta}$ with VD is smaller than that without it when around $2^{10}$ copies of $\rho$ are consumed in the experiment. Also, we perform the metrology with perfect probe state of $\ket{+}$, and the results are shown with green dots in Fig. 4(c). The accuracy of $\hat{\theta}$ with noisy state $\rho$ achieves that with (almost) pure state $\ket{+}$ when the virtual distillation is performed with around $2^{12}$ copies.