Importance Sampling of Randomized Measurements for Probing Entanglement

We show that combining randomized measurement protocols with importance sampling allows for characterizing entanglement in significantly larger quantum systems and in a more efficient way than in previous work. A drastic reduction of statistical errors is obtained using classical techniques of machine learning and tensor networks using partial information on the quantum state. In current experimental settings of engineered many-body quantum systems this significantly increases the (sub-)system sizes for which entanglement can be measured. In particular, we show an exponential reduction of the required number of measurements to estimate the purity of product states and GHZ states.

Figure 1

Randomized measurement protocol with importance sampling. In the first phase, we construct a classical function $X_{\mathrm{IS}}(u)$. In the second phase, unitaries are sampled from the appropriate classical representation. In the last phase, measurements are performed in the quantum system, and are analyzed to construct different properties accessible by randomized measurements. The measurement data obtained during the experiment could also be considered as additional samples to obtain improved classical function for future experiments.

Figure 2

Statistical error scalings for product and GHZ states Average statistical error $\mathcal{E}$ of the estimated purity for (a) 10-qubit product state and (b) 5-qubit GHZ state in function of $N_u$ with $N_M=1000$ for a uniform sampling (Uniform) and importance sampling from a machine learning model (ML). (c)–(d) Scaling of the required total number of measurements $N_u{N}_M$ as a function of $N$ for uniform and importance sampling for a product state, to obtain a statistical error of $\mathcal{E}=0.1$ (c), and $\mathcal{E}=0.05$ (d), respectively. We represent with a cross the analytical prediction (cf. Supplemental Material [31]) and with circles the numerical simulations. Panels (e)–(f) show the numerical simulations for the GHZ states for $\mathcal{E}=0.1$ and $\mathcal{E}=0.05$ with corresponding exponential fits of the type $2^{b+aN}$.

Figure 3

Purity estimation of a highly entangled 10 qubit state with ML and MPS samplers. Panel (a) shows the average statistical error $\mathcal{E}$ of the estimated purity in function of $N_u$ with $N_M=7500$ for a uniform sampling and importance sampling done from a neural network and a MPS representation of the corresponding state, respectively. Panel (b) illustrates the scaling of the error $\mathcal{E}$ with respect to different bond dimensions $D$ used for the MPS representation of the state for $N_u=5$ and $N_M=7500$.