Entanglement plays a crucial role in quantum physics and is the key resource in quantum information processing. In entanglement theory, Schmidt decomposition is a powerful tool to analyze the fundamental properties and structure of quantum entanglement. This work introduces a hybrid quantum-classical algorithm for Schmidt decomposition of bipartite pure states on near-term quantum devices. First, we show that the Schmidt decomposition task could be accomplished by maximizing a cost function utilizing bi-local quantum neural networks. Based on this, we propose a variational quantum algorithm for Schmidt decomposition (named VQASD) of which the cost function evaluation notably requires only one estimate of expectation with no extra copies of the input state. In this sense, VQASD outperforms existent approaches in resource cost and hardware efficiency. Second, by further exploring VQASD, we introduce a variational quantum algorithm to estimate the logarithm negativity, which can be applied to efficiently quantify entanglement of bipartite pure states. Third, we experimentally implement our algorithm on Quantum Leaf using the IoP CAS superconducting quantum processor. Both experimental implementations and numerical simulations exhibit the validity and practicality of our methods for analyzing and quantifying entanglement on near-term quantum devices.
Entanglement plays a crucial role in quantum physics and is the key resource in quantum information processing. In entanglement theory, Schmidt decomposition is a powerful tool to analyze the fundamental properties and structure of quantum entanglement. This work introduces a hybrid quantum-classical algorithm for Schmidt decomposition of bipartite pure states on near-term quantum devices. First, we show that the Schmidt decomposition task could be accomplished by maximizing a cost function utilizing bi-local quantum neural networks. Based on this, we propose a variational quantum algorithm for Schmidt decomposition (named VQASD) of which the cost function evaluation notably requires only one estimate of expectation with no extra copies of the input state. In this sense, VQASD outperforms existent approaches in resource cost and hardware efficiency. Second, by further exploring VQASD, we introduce a variational quantum algorithm to estimate the logarithm negativity, which can be applied to efficiently quantify entanglement of bipartite pure states. Third, we experimentally implement our algorithm on Quantum Leaf using the IoP CAS superconducting quantum processor. Both experimental implementations and numerical simulations exhibit the validity and practicality of our methods for analyzing and quantifying entanglement on near-term quantum devices.
Entanglement plays a crucial role in quantum physics and is the key resource in quantum information processing. However, entanglement detection and quantification are believed to be hard due to the operational impracticality of existing methods. This work proposes three near-term efficient algorithms that exploit the hybrid quantum-classical technique to address this difficulty. The first algorithm finds the Schmidt decomposition---a powerful tool to analyze the properties and structure of entanglement---for bipartite pure states. While the logarithm negativity can be calculated from the Schmidt decomposition, we propose the second algorithm to estimate the logarithm negativity for bipartite pure states, where the width of the parameterized quantum circuits is further reduced. Finally, we generalize our framework for mixed states, leading to our third algorithm that detects entanglement on specific families of states, and determines distillability in general. All three algorithms share a similar framework where the optimizations are accomplished by maximizing a cost function utilizing local parameterized quantum circuits, with better hardware efficiency and practicality compared to existing methods. The experimental implementation on Quantum Leaf using the Institute of Physics, Chinese Academy of Sciences superconducting quantum processor exhibits the validity and practicality of our methods for analyzing and quantifying entanglement on near-term quantum devices.
Abstract Quantum entanglement is a key resource in quantum technology, and its quantification is a vital task in the current noisy intermediate-scale quantum (NISQ) era. This paper combines hybrid quantum-classical computation and quasi-probability decomposition to propose two variational quantum algorithms, called variational entanglement detection (VED) and variational logarithmic negativity estimation (VLNE), for detecting and quantifying entanglement on near-term quantum devices, respectively. VED makes use of the positive map criterion and works as follows. Firstly, it decomposes a positive map into a combination of quantum operations implementable on near-term quantum devices. It then variationally estimates the minimal eigenvalue of the final state, obtained by executing these implementable operations on the target state and averaging the output states. Deterministic and probabilistic methods are proposed to compute the average. At last, it asserts that the target state is entangled if the optimized minimal eigenvalue is negative. VLNE builds upon a linear decomposition of the transpose map into Pauli terms and the recently proposed trace distance estimation algorithm. It variationally estimates the well-known logarithmic negativity entanglement measure and could be applied to quantify entanglement on near-term quantum devices. Experimental and numerical results on the Bell state, isotropic states, and Breuer states show the validity of the proposed entanglement detection and quantification methods.
Entanglement forging based variational algorithms leverage the bipartition of quantum systems for addressing ground-state problems. The primary limitation of these approaches lies in the exponential summation required over the numerous potential basis states, or bitstrings, when performing the Schmidt decomposition of the whole system. To overcome this challenge, we propose a method for entanglement forging employing generative neural networks to identify the most pertinent bitstrings, eliminating the need for the exponential sum. Through empirical demonstrations on systems of increasing complexity, we show that the proposed algorithm achieves comparable or superior performance compared to the existing standard implementation of entanglement forging. Moreover, by controlling the amount of required resources, this scheme can be applied to larger as well as non-permutation-invariant systems, where the latter constraint is associated with the Heisenberg forging procedure. We substantiate our findings through numerical simulations conducted on spin models exhibiting one-dimensional rings, two-dimensional triangular lattice topologies, and nuclear shell model configurations. Published by the American Physical Society 2024
Entanglement is a unique feature of quantum mechanics. In coupled systems of light and matter, entanglement manifests itself in the linear superposition of multipartite quantum states (e.g., parametrized by multiple spatial, spectral, or temporal degrees of freedom of a light field). In bipartite systems, the Schmidt decomposition provides a modal decomposition of the entanglement structure over independent, separable states. Although ubiquitous as a mathematical tool to describe and measure entanglement, there exists no general efficient experimental method to decompose a bipartite quantum state onto its Schmidt modes. Here, we propose a method that relies on bipartite self-configuring optics that automatically “learns” the Schmidt decomposition of an arbitrary pure quantum state. Our method is designed for entanglement distributed over spatial or spectral degrees of freedom of a light field and can reconstruct the Schmidt modes and values by variational optimization of the network’s output powers or coincidences. We illustrate our method with numerical examples of spectral entanglement analysis for biphotons generated via spontaneous parametric down conversion and provide experimental guidelines for its realization and generalization to other degrees of freedom of light, including the influence of losses and impurities. Our method provides a versatile and scalable way of analyzing entanglement in bipartite integrated quantum photonic systems.
In quantum image processing, a fundamental step is encoding classical image data into quantum states. This can be achieved using methods such as Flexible Representation of Quantum Images (FRQI), Quantum Probability Image Encoding (QPIE), and Novel Enhanced Quantum Representation (NEQR). However, on real quantum hardware, these encodings can quickly lead to circuits with many gates, large circuit depth, and high qubit usage, which is a problem for Noisy Intermediate-Scale Quantum (NISQ) devices. In this work, we investigate whether low-rank state approximation, formulated via Schmidt decomposition, can help reduce this complexity. The method keeps only the most significant parts of a quantum state's entanglement structure, making state preparation more efficient while preserving most of the image information. We compare the three encoding techniques in their original form and with low-rank approximation, evaluating metrics such as circuit depth, CNOT count, MSE, and visual quality of reconstructed images. The results reveal meaningful trade-offs between accuracy and resource efficiency, with the FRQI model achieving a 97 percent reduction in circuit depth while maintaining a near-perfect reconstruction (MSE of about 0.27). This demonstrates the potential of low-rank techniques for advancing practical quantum image processing on near-term hardware.
We study entanglement in a system of three coupled quantum harmonic oscillators. Specifically, we use the Schmidt decomposition to analyze how the entanglement is distributed among the three subsystems. The Schmidt decomposition is a powerful mathematical tool for characterizing bipartite entanglement in composite quantum systems. It allows to write a multipartite quantum state as a sum of product states between the subsystems, with coefficients known as Schmidt coefficients. We apply this decomposition to the general quantum state of three coupled oscillators and study how the Schmidt coefficients evolve as the interaction strengths between the oscillators are varied. This provides insight into how entanglement is shared between the different bipartitions of the overall three-particle system. Our results advance the fundamental understanding of multipartite entanglement in networked quantum systems. They also have implications for quantum information processing using multiple entangled nodes.