Resource-efficient quantum correlation measurements via multicopy neural network methods

TL;DR

This approach simplifies and improves the error robustness of multicopy measurements, eliminating the need for complex nonlinear equation analysis, and represents a significant advancement in AI-assisted quantum measurements, making the practical implementation on current quantum hardware more feasible.

Abstract

Measuring complex properties in quantum systems, such as measures of quantum entanglement and Bell nonlocality, is inherently challenging. Traditional methods, like quantum state tomography (QST), necessitate a full reconstruction of the density matrix for a given system and demand resources that scale exponentially with system size. We propose an alternative strategy that reduces the required information by combining multicopy measurements with artificial neural networks (ANNs), resulting in a 67\% reduction in measurement requirements compared to QST. We have successfully measured two-qubit quantum correlations of Bell states subjected to a depolarizing channel (resulting in Werner states) and an amplitude damping channel (leading to Horodecki states) using the multicopy approach on real quantum hardware. Our experiments, conducted with transmon qubits on IBMQ processors, quantified the violation of Bell's inequality and the negativity of two-qubit entangled states. We compared these results with those obtained from the standard QST approach and applied a maximum likelihood method to mitigate noise. We trained ANNs to estimate entanglement and nonlocality measures using optimized sets of projections identified through Shapley's (SHAP) analysis for the Werner and Horodecki states. The ANN output, based on this reduced set of projections, aligns well with expected values and enhances noise robustness. This approach simplifies and improves the error-robustness of multicopy measurements, eliminating the need for complex nonlinear equation analysis. It represents a significant advancement in AI-assisted quantum measurements, making practical implementation on current quantum hardware more feasible.

Figures (10)

• Figure 1: Schematic representation of the approaches based on (a) quantum state tomography (QST) and (b) multicopy estimation (MCE). The symbols $\Delta \tau$ and $\Delta \tau'$ denote the delay time between the generation of each entangled pair or a set of copies of entangled pairs, respectively. In (a), detectors $D_1$ and $D_2$ measure products of eigenstates of Pauli's matrices. The multicopy state generation uses a quantum memory $Q$ to store and release the collected pairs. The detection rates are then processed using maximum likelihood estimation (MLE) to obtain the most probable physical combination of the measured experimental settings.
• Figure 2: Examples of graphs representing joint multicopy measurements: (a) the paired single--subsystem singlet projections $l_1$, (b) the paired cross--subsystem singlet projections $\bar{l}_2$, (c) the chained single--subsystem singlet projections $c_3$, and (d) the chained cross--subsystem singlet projections $\bar{c}_2$. Black lines combine subsystems (red and white circles) of the same copy of $\hat{\rho}$, while dotted lines correspond to projections of the multicopy system onto the singlet state. These graphical illustrations help in describing the concept of various projection settings used in our multicopy measurement approach.
• Figure 3: A schematic diagram illustrating the process of measuring and analyzing quantum correlations. The procedure consists of three key stages. The first stage involves preparing multiple copies of the quantum state with different qubit mappings and verifying them through fidelity measurements. The second stage focuses on selecting and executing measurements, followed by maximum likelihood estimation to ensure the physical validity of the results while mitigating noise effects. The third, optional stage optimizes projection selection through SHAP analysis and employs neural network processing to analyze measurement results and estimate quantum correlations. A feedback loop enables adaptive optimization based on the measurement outcomes.
• Figure 4: SHAP value analysis for quantum correlation measurements. Results shown for (a) negativity $N$ and (b) nonlocality measure $B$, computed from $5 \times 10^5$ random input states. The impact strength indicates each projection's contribution to the final measurement outcome.
• Figure 5: Experimental quantification of the entanglement (left column) and nonlocality (right column) for Werner and Horodecki states, determined by measuring the negativity $N$ and the Bell nonlocality measure $B$ as a function of the mixing parameter $p$. Solid curves show theoretical predictions for ideal states. The results were obtained using: (a, b) quantum state tomography and (c, d) multicopy estimation. Assumptions include shot noise (● for Werner states, ● for Horodecki states) using ibmq_qasm_simulator, and experimental data (✛ for Werner states, ✕ for Horodecki states) collected with the quantum processor ibm_hanoi. Standard deviations $\sigma$ were estimated by simulating $10^5$ experiments for ideal and noisy circuits, with noise models based on calibration data. The $y=0$ line separates separable and entangled states in (a), as well as between states that violate and satisfy the CHSH inequality in (b).