Experimental Quantum State Measurement with Classical Shadows

A crucial subroutine for various quantum computing and communication algorithms is to efficiently extract different classical properties of quantum states. In a notable recent theoretical work by Huang, Kueng, and Preskill [Nat. Phys. 16, 1050 (2020)], a thrifty scheme showed how to project the quantum state into classical shadows and simultaneously predict $M$ different functions of a state with only $\mathcal{O}({\log }_{2}M)$ measurements, independent of the system size and saturating the information-theoretical limit. Here, we experimentally explore the feasibility of the scheme in the realistic scenario with a finite number of measurements and noisy operations. We prepare a four-qubit GHZ state and show how to estimate expectation values of multiple observables and Hamiltonians. We compare the measurement strategies with uniform, biased, and derandomized classical shadows to conventional ones that sequentially measure each state function exploiting either importance sampling or observable grouping. We next demonstrate the estimation of nonlinear functions using classical shadows and analyze the entanglement of the prepared quantum state. Our experiment verifies the efficacy of exploiting (derandomized) classical shadows and sheds light on efficient quantum computing with noisy intermediate-scale quantum hardware.

Figure 1

Schematic illustration of the experimental setup. (a) The setup to generate maximally polarization-entangled photon pair. (b) Two photons are sent into BD to generate a four-qubit hyperentangled state. (c) Experimental setup to implement the Pauli measurements. (d) The single-qubit Clifford operations ($U_i$) are realized with different settings of wave plates. Narrow-band filter (NBF); Dichroic mirror (DM).

Figure 2

The error of observable estimations with five different measurement schemes. (a) The maximum absolute error of $⟨{\mathbf{O}}_l{⟩}^{\mathrm{ES}}$ over 50 local observables ${\mathbf{O}}_l$ that are randomly selected from the Pauli set with different number of $N_s$. (b) The maximum absolute error of $⟨{\mathbf{O}}_l{⟩}^{\mathrm{ES}}$ with different number of local observables, each of which we fix $N_s=2000$. (c) and (d) Errors of estimated energy $⟨H{⟩}^{\mathrm{ES}}$ and that of estimated Hamiltonian moment $⟨H^2{⟩}^{\mathrm{ES}}$ with different $N_s$. In each measurement basis we collect five coincidences for data processing, and we run 20 independent repetitions of the entire setup.

Figure 3

Experimental results for estimating nonlinear functions with classical shadows. (a) The estimation of subsystem purity $⟨{\mathcal{P}}_A{⟩}^{\mathrm{ES}}$ with the different subsystems $A$. The colored bars represent the results from the CS method, while the red sticks represent the results from QST for comparison. (b) The estimation of $p_2^2-p_3$ for different subsystem partitioning of the prepared GHZ state, which clearly shows the violation of the $p_3$-PPT condition. The number of measurements $N_s$ in (a) and (b) is fixed as 1000 for the CS method, and the standard deviation is obtained by repeating the experiment 10 times. Here, we collect one coincidence at each measurement. The dots in (c) and (d) represent the errors of $⟨{\mathcal{P}}_A{⟩}^{\mathrm{ES}}$ and that of $⟨p_2{⟩}^{\mathrm{ES}}$ with different $N_s$, respectively. The dashed line represents the scaling of $\propto 1/N_s$. The $|{\mathrm{GHZ}}_4⟩$ is a specific graph state, corresponding to a star graph, which is exhibited in the insets.