Enhanced Estimation of Quantum Properties with Common Randomized Measurements

We present a technique for enhancing the estimation of quantum state properties by incorporating approximate prior knowledge about the quantum state. This consists in performing randomized measurements on a quantum processor and comparing the results with those obtained from a classical computer that stores an approximation of the quantum state. We provide unbiased estimators for expectation values of multicopy observables and present performance guarantees in terms of variance bounds that depend on the prior knowledge accuracy. We demonstrate the effectiveness of our approach through experimental and numerical examples detecting mixed-state entanglement, and estimating polynomial approximations of the von Neumann entropy and state fidelities.

Popular Summary

Randomized measurements are now routinely implemented in quantum computers and quantum simulators. These protocols allow us to measure the physical properties of a quantum state on the basis of an experimentally friendly procedure. This is useful in particular to access expectation values of observables that represent the result of the computation but also to verify the presence of fundamental quantum properties such as entanglement. However, the physical quantities that are estimated with randomized measurements are subject to statistical errors. We propose significantly reducing the role of such statistical errors by introducing the framework of common randomized measurements.

Our method consists in performing randomized measurements on a quantum processor and comparing the results with those obtained from a classical computer that stores an approximation of the quantum state. Using the fact that these two measurements are statistically correlated, we can propose estimators for quantum state properties with small and controlled statistical errors. We experimentally demonstrate the advantage of our method for entanglement detection, reanalyzing the randomized measurement data from a previous experiment. We also present further illustrations regarding the estimation of the von Neumann entropy and of quantum state fidelities.

Our method can be readily used in existing experiments to enhance the estimation of quantum properties. We believe that it should also find broader applications, in particular for the estimation of quantum processes or the study of measurement-induced entanglement transitions.

Figure 1

Common randomized measurements. (a) Experimental data from RMs on a quantum state $\rho$ are combined with data from simulated RMs on classical approximations $\sigma$ to form CRM shadows ${\hat{\rho }}_{\sigma }^{(r)}$ for enhanced property prediction. (b) Enhanced entanglement detection via the $p_3$ positive partial transpose condition ($p_3$-PPT) condition [16] in a many-body quantum state using experimental data from Ref. [29] (with quench time $t=1$ ms). A value greater than unity indicates bipartite entanglement between two partitions $A=[1,2]$ and $B=[3,4]$ in a total system with ten qubits. This can be achieved with a statistical accuracy of 1 standard deviation (shaded areas) with only $N_U\approx 50$ random unitaries with use of CRM shadows compared with $N_U\approx 300$ with use of standard classical shadows (vertical dashed lines).

Figure 2

Estimation of the von Neumann entropy in the critical Ising chain. (a) Approximations $S_{n_{max}}$ as a function of subsystem size $N_A=N/2$ for $n_{\max }=3,5,7$ compared with $S$. The dashed line is a guide for the eye proportional to $c/6\log (N_A)$, with $c=1/2$, the central charge of the Ising universality class [52]. (b) Statistical relative error $\mathcal{E}({\hat{S}}_3)=\mathbb{E}[|{\hat{S}}_3-S_3|]/S_3$ of estimations of $S_3$, obtained with use of the standard batch shadow estimation [1, 25] (blue) and the CRM method (orange) with $\sigma$ obtained from MPS approximations $|{\psi }_{\chi }⟩$ of bond dimensions $\chi =1,2,3$. (c) Total relative error ${\mathcal{E}}_{\mathrm{tot}}({\hat{S}}_{n_{max}})=\mathbb{E}[|{\hat{S}}_{n_{\max }}-S|]/S$ for $n_{\max }=3,5,7$, and $\chi =2$. The horizontal line denotes $|S_3-S|/S$. In (b),(c), we use $N_A=N/2=8$, $N_M=1000$ is fixed, and we vary $N_U$. The dashed black lines are guides for the eye proportional $1/\sqrt{N_U{N}_M}$. The average errors are obtained by our averaging over $20$ simulations for (b) and $50$ simulations for (c).

Figure 3

Fidelity estimation in (noisy) random quantum circuits. (a) Estimated fidelities ${\hat{\mathcal{F}}}_{\varphi }$ of the prepared state $\rho$ and the theoretical prior states $\sigma =|\varphi ⟩⟨\varphi |$ as a function of their bond dimension $\chi$. Here $\rho$ is a $30$-qubit pure state generated from an ideal noiseless ($p=0$) random quantum circuit of depth $d=6$ and $|\varphi ⟩$ are obtained by our truncating $\rho$ to bond dimension $\chi$. (b) Each gate in the circuit is perturbed by local depolarization noise with strength $p$ resulting in a mixed state $\rho$. The prior state $\sigma$ is the same as in (a). For (a),(b), we compare CRM estimation (orange dots) with standard shadow estimation (blue dots). We fix $N_U=15$ and $N_M={10}^5$. The error bars are evaluated as standard errors of the mean over random unitaries. The solid black lines denote the exact fidelity ${\mathcal{F}}_{\varphi }$. The dashed black lines are guides for the eye for $0.5$ and $1$ respectively.

Figure 4

Estimation of the von Neumann entropy in the critical Ising chain, (as in Fig. 2), but using CRM shadows built from a companion experiment. Statistical relative error $\mathcal{E}({\hat{S}}_3)$, where the CRM shadows are formed from a companion experiment with use of $N_U^{\prime }$ additional unitaries ($N_M=1000$), as in Eq. (A1). Here we use $N_A=6$ ($N=12$).

Figure 5

Least-squares error $I_{n_{max}}$ as a function of $n_{max}$.