[论文解读] Detecting and quantifying entanglement on near-term quantum devices

Kun Wang, Zhixin Song+3 more|arXiv (Cornell University)|Dec 28, 2020

Quantum Information and Cryptography参考文献 17被引用 6

一句话总结

该论文提出变分纠缠检测（VED）与变分对数负性估计（VLNE）两种变分量子算法，利用准概率分解与NISQ友好型量子线路，在近期量子设备上实现纠缠的检测与量化。VED通过变分估计变换后密度矩阵的最小特征值，结合正映射判据来检测纠缠；VLNE则通过迹距离估计方法变分估计对数负性纠缠度量。两种方法在贝尔态、混旋态与Breuer态上均通过数值验证，精度较高。

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.

研究动机与目标

• 为解决在近期含噪声中等规模量子（NISQ）设备上检测与量化量子纠缠的挑战，传统方法如量子态层析由于资源呈指数级增长而不可行。
• 开发可在当前量子硬件上实现的实用化变分量子算法，无需完整态重建。
• 通过NISQ兼容线路实现基于正映射判据的可靠纠缠检测与基于对数负性度量的纠缠量化。
• 通过在贝尔态、混旋态与Breuer态等基准纠缠态上的数值模拟，展示所提方法的可行性与准确性。

提出的方法

• VED通过将选定的正映射分解为NISQ可实现的量子操作的线性组合，利用变分方法估计所得态的最小特征值，从而应用正映射判据检测纠缠。
• 提出两种输出态平均方法：一种基于准概率分布的确定性方法，另一种基于采样技术的随机方法，用于估计平均态。
• 若优化后的最小特征值为负，则判定存在纠缠，此结果由正映射判据保证。
• VLNE基于对转置映射的线性分解为泡利项，并利用近期开发的迹距离估计算法，变分估计对数负性纠缠度量。
• 算法采用参数化量子线路（PQCs）与经典优化，最小化对应于纠缠判据或度量的损失函数。
• 该框架应用于多种双粒子态，数值结果显示，纠缠检测与量化估计值与理论值高度一致。

Figure 1: The simplified quantum circuit that estimates the overlap $\langle\psi|{\cal O}(\rho_{AB})|\psi\rangle$ in ( 12 ) for a given implementable operation ${\cal O}$ , where $|\psi\rangle\mathrel{\mathop{\mathchar 58\relax}}=U_{\bm{\alpha}}|0\rangle^{\otimes 2n}$ is the parameterized input stat

实验结果

研究问题

• RQ1 能否在近期量子设备上无需完整量子态层析，可靠检测纠缠？
• RQ2 正映射判据如何适配为适用于NISQ硬件的变分量子算法？
• RQ3 能否通过变分量子线路与迹距离估计高效估计对数负性这一经典纠缠度量？
• RQ4 所提出的VED与VLNE框架在贝尔态、混旋态与Breuer态等基准纠缠态上的准确度与可扩展性如何？

主要发现

• 数值模拟表明，VED在贝尔态、混旋态及四量子比特Breuer态中成功检测到纠缠，最小化损失值与理论预测一致。
• 对于四量子比特Breuer态，增强的简化判据检测出所有纠缠态，优于标准简化判据与PPT判据，验证了方法的鲁棒性。
• VLNE准确估计了两量子比特混旋态的对数负性，估计值与精确计算的理论值高度吻合，尤其在p > 1/3（纠缠存在区域）表现更优。
• 该方法在纠缠检测与量化中实现高保真度，最小特征值与对数负性估计值在所有测试态中均收敛至理论边界。

Figure 3: Estimated $\lambda_{\text{min}}$ by VED using the reduction criterion on the Bell state $|\Phi\rangle$ . The red curve records the results from ibmq-santiago with shots = $8192$ for each circuit evaluation. The blue curve records the simulation results on Quantum Leaf platform using the Ba