[Paper Review] Experimental pairwise entanglement estimation for an N-qubit system :A machine learning approach for programming quantum hardware

Nathan L. Thompson, Nam H. Nguyen+2 more|arXiv (Cornell University)|Feb 20, 2019

Quantum Computing Algorithms and Architecture20 references4 citations

TL;DR

This paper proposes a machine learning approach to estimate pairwise entanglement in N-qubit systems by training a quantum system via time-dependent Hamiltonians, enabling efficient entanglement witness computation. Using bootstrapping and Q# simulation, the method achieves high-fidelity entanglement estimation with a 0.0015 confidence interval after 15,000 shots, generalizing effectively to systems up to seven qubits and suggesting scalability for larger N.

ABSTRACT

Designing and implementing algorithms for medium and large scale quantum computers is not easy. In previous work we have suggested, and developed, the idea of using machine learning techniques to train a quantum system such that the desired process is "learned," thus obviating the algorithm design difficulty. This works quite well for small systems. But the goal is macroscopic physical computation. Here, we implement our learned pairwise entanglement witness on Microsoft's Q#, one of the commercially available gate model quantum computer simulators; we perform statistical analysis to determine reliability and reproduceability; and we show that using the machine learning technique called "bootstrapping", we can infer the pattern for mesoscopic N from simulation results for three-, four-, five-, six-, and seven-qubit systems. Our results suggest a fruitful pathway for general quantum computer algorithm design and computation.

Motivation & Objective

• To address the lack of general algorithms for estimating entanglement in many-body quantum systems, which is NP-hard.
• To develop a scalable method for estimating pairwise entanglement in N-qubit systems without requiring explicit algorithm design.
• To leverage machine learning techniques, particularly bootstrapping, to infer patterns in entanglement from smaller system simulations.
• To implement the learned entanglement witness using gate-model quantum simulators like Microsoft's Q# for reproducibility and hardware compatibility.
• To assess the reliability, reproducibility, and scalability of the method across increasing qubit counts and discretized time parameters.

Proposed method

• The method uses a quantum neural network framework where the initial state is the input and a measurement outcome at the final time is the output, trained via a quantum version of backpropagation.
• The time evolution is governed by a time-dependent Hamiltonian with tunable parameters: tunneling amplitudes {K}, qubit biases {ε}, and qubit-qubit coupling ζ, treated as trainable weights.
• A training set of only four pure states enables generalization to mixed and entangled states across multi-qubit systems.
• The learned witness is decomposed into a sequence of universal quantum gates (single-qubit and CNOT) for implementation on gate-model simulators like Q#.
• Bootstrapping is applied to infer behavior in larger N-qubit systems using simulation data from 3- to 7-qubit systems.
• Statistical analysis, including confidence intervals and reproducibility testing, is performed over 15,000 measurement shots to validate reliability.

Experimental results

Research questions

• RQ1 Can a machine learning approach trained on small N-qubit systems reliably estimate pairwise entanglement in larger systems?
• RQ2 To what extent can bootstrapping techniques generalize entanglement witness patterns from small simulations to mesoscopic N?
• RQ3 How accurately can a learned time-dependent Hamiltonian approximate a pairwise entanglement witness in a gate-model quantum simulator?
• RQ4 What is the statistical reliability and reproducibility of the entanglement estimation under realistic hardware constraints?
• RQ5 Can the method be extended to systems with higher N while maintaining fidelity and reducing required training effort?

Key findings

• The entanglement witness was successfully approximated with a confidence interval of approximately 0.0015 after 15,000 measurement shots, indicating high statistical reliability.
• The method generalized well to systems with up to seven qubits, demonstrating strong scalability of the learned patterns.
• Two of the learned parameter functions (e.g., tunneling amplitudes) showed signs of asymptotic convergence, suggesting minimal further training is needed for larger N.
• The qubit-qubit coupling function showed strong convergence trends, indicating that a large fraction of training effort had already been completed, reducing future training burden.
• Despite favorable gate count (28 single-qubit and 8 two-qubit gates), reproducibility on IBM hardware was poor due to decoherence and gate fidelity limitations.
• The approach remains robust to noise and decoherence, and its value increases with improved hardware coherence times and finer discretization of training parameters.