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arXiv:2107.02301 · Physical Review Applied

Machine-Learning-Derived Entanglement Witnesses

Alexander C. B. Greenwood, Larry Wu, Eric Y. Zhu, Brian T. Kirby, Li Qian

Published Mar 17, 2023• 17 citations
arXivHTMLPDF

Abstract

In this work, we show a correspondence between linear support vector machines (SVMs) and entanglement witnesses, and use this correspondence to generate entanglement witnesses for bipartite and tripartite qubit (and qudit) target entangled states. An SVM allows for the construction of a hyperplane that clearly delineates between separable states and the target entangled state; this hyperplane is a weighted sum of observables (``features'') whose coefficients are optimized during the training of the SVM. We demonstrate with this method the ability to obtain witnesses that require only local measurements even when the target state is a nonstabilizer state. Furthermore, we show that SVMs are flexible enough to allow us to rank features, and to reduce the number of features systematically while bounding the inference error. This allows us to derive $W$-state witnesses capable of detecting entanglement with fewer measurement terms than the fidelity method dominant in today's literature. The utility of this approach is demonstrated on quantum hardware furnished through the IBM Quantum Experience.

DOI
10.1103/physrevapplied.19.034058

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Abstract We propose a support vector machine (SVM) based approach for generating an entanglement witness that requires exponentially less training data than previously proposed methods. SVMs generate hyperplanes represented by a weighted sum of expectation values of local observables whose coefficients are optimized to sum to a positive number for all separable states and a negative number for as many entangled states as possible near a specific target state. Previous SVM-based approaches for entanglement witness generation used large amounts of randomly generated separable states to perform training, a task with considerable computational overhead. Here, we propose a method for orienting the witness hyperplane using only the significantly smaller set of states consisting of the eigenstates of the generalized Pauli matrices and a set of entangled states near the target entangled states. With the orientation of the witness hyperplane set by the SVM, we tune the plane’s placement using a differential program that ensures perfect classification accuracy on a limited test set as well as maximal noise tolerance. For N qubits, the SVM portion of this approach requires only O ( 6 N ) training states, whereas an existing method needs O ( 2 4 N ) . We use this method to construct witnesses of 4 and 5 qubit GHZ states with coefficients agreeing with stabilizer formalism witnesses to within 3.7 percent and 1 percent, respectively. We also use the same training states to generate novel 4 and 5 qubit W state witnesses. Finally, we computationally verify these witnesses on small test sets and propose methods for further verification.
We are entering a time in which quantum channel capacity is becoming increasingly dependent on having high-dimensional entangled systems. However, existing methods of detecting high-dimensional entanglement are hindered by their need for an unwieldy amount of measurements and computational power. Indeed, the use of entanglement witnesses circumvents this problem by only considering a small subset of the entangled states in Hilbert Space. Although efficient witnesses for 2-qubit Bell States and 3-Qubit GHZ states are well-known, the predominant “fidelity method" witnesses used for detecting multipartite W states remain relatively complex in the amount of measurements that they require. The work outlined in this thesis aims to improve on existing entanglement witnesses by drawing their correspondence with supervised learning techniques, specifically Support Vector Machines. In doing so, one can derive entanglement witnesses whose features (observables obtained via local operations and classical communications) can be ranked according to their impact on the classification of entangled and separable states. This method allows us to derive W state witnesses that require fewer measurements than those derived via the fidelity method. The process of Recursive Feature Elimination is also presented, by which we can define witnesses that can be decomposed into fewer Pauli observables at the cost of inferior tolerance to noise. Finally, we present the practical utility of our derived witnesses through physical experiments performed on the superconducting qubit devices provided by IBM.
Entanglement witnesses are essential for certifying entanglement, yet constructing ones that are both noise-robust and economical in measurement settings remains challenging - particularly beyond qubits and for non-stabilizer ("magic") states. We present a machine-learning method that, given a target state and a user-specified number of measurement settings, generates an entanglement witness optimized for noise tolerance in the neighborhood of that state, requiring only local measurements. The approach is fully general, applying to multipartite qubit and qudit systems alike, including non-stabilizer states. For N qudits of dimension d, we train on the fully-separable eigenstates of each qudit's SU(d) generators to find a prototype witness, then tune the witness's bias term via gradient descent to maximize noise tolerance. Adversarial training further strengthens the witnesses, delivering greater noise tolerance with even fewer settings; critically, under this scheme the required training-set size becomes independent of system size. We package the entire pipeline as an automated script that, in every case we tested, produces witnesses surpassing all existing methods in noise tolerance and/or number of measurement settings. We demonstrate the method on Bell, GHZ, W, and hypergraph states, along with a range of qudit states, spanning 2-6 qubits, bipartite qudits up to d=10, and tripartite qutrits. Our witnesses achieve perfect accuracy across both physical experimental test states and large numerical sets of separable mixed states-including 30 million test states for a 3-qubit W-state witness and 10 million for a 4-qubit hypergraph-state witness-and we experimentally confirm the noise tolerance of Bell- and hypergraph state witnesses on both photonic and superconducting platforms, respectively.
Abstract An entanglement witness (EW) is a Hermitian operator that can distinguish an entangled state from all separable states. We developed and implemented a numerical method based on machine learning to create multipartite EW. Using a Support Vector Machine (SVM) algorithm, we construct EW's based on local orthogonal observables in the form of a hyperplane that separates the separable region from the entangled state for two, three, and four qubits Bell-diagonal mixed states which can be generalized to a multipartite mixed state. One of the important features of this method is that the EW s are optimal and are completely tangent to the separable region. Also, we generate non-decomposable EW's that can detect positive partial transpose entangled states (PPTES).
Abstract This work presents a machine learning approach based on support vector machines (SVMs) for quantum entanglement detection. Particularly, we focus on bipartite systems of dimensions 3×33\times 33×3 3 × 3 , 4×44\times 44×4 4 × 4 , and 5×55\times 55×5 5 × 5 , where the positive partial transpose criterion (PPT) provides only partial characterization. Using SVMs with quantum-inspired kernels, we develop a classification scheme that distinguishes between separable and entangled states, including PPT-detectable entangled states, and entangled states that evade PPT detection. Our method achieves increasing accuracy with system dimension, reaching 80%80\%80% 80 % , 90%90\%90% 90 % , and nearly 100%100\%100% 100 % for 3×33\times 33×3 3 × 3 , 4×44\times 44×4 4 × 4 , and 5×55\times 55×5 5 × 5 systems, respectively. Our results show that principal component analysis significantly enhances performance for small training sets. The study reveals important practical considerations regarding purity biases in the generation of data for this problem and examines the challenges of implementing these techniques on near-term quantum hardware. Our results demonstrate that machine learning can be an effective alternative to entanglement detection of higher-dimensional systems where conventional entanglement detection methods struggle. Our approach provides improved data generation protocols and can be readily implemented in hybrid classical-quantum architectures, overcoming current limitations.