Wang David Z, Gauthier Aidan Q, Siegmund Ashley E, Hunt Katharine L C
Department of Chemistry, Michigan State University, East Lansing, MI 48824, USA.
Phys Chem Chem Phys. 2021 Mar 21;23(11):6370-6387. doi: 10.1039/d0cp05444e. Epub 2021 Feb 4.
This work provides quantitative tests of the extent of violation of two inequalities applicable to qubits coupled into Bell states, using IBM's publicly accessible quantum computers. Violations of the inequalities are well established. Our purpose is not to test the inequalities, but rather to determine how well quantum mechanical predictions can be reproduced on quantum computers, given their current fault rates. We present results for the spin projections of two entangled qubits, along three axes A, B, and C, with a fixed angle θ between A and B and a range of angles θ' between B and C. For any classical object that can be characterized by three observables with two possible values, inequalities govern relationships among the probabilities of outcomes for the observables, taken pairwise. From set theory, these inequalities must be satisfied by all such classical objects; but quantum systems may violate the inequalities. We have detected clear-cut violations of one inequality in runs on IBM's publicly accessible quantum computers. The Clauser-Horne-Shimony-Holt (CHSH) inequality governs a linear combination S of expectation values of products of spin projections, taken pairwise. Finding S > 2 rules out local, hidden variable theories for entangled quantum systems. We obtained values of S greater than 2 in our runs prior to error mitigation. To reduce the quantitative errors, we used a modification of the error-mitigation procedure in the IBM documentation. We prepared a pair of qubits in the state |00〉, found the probabilities to observe the states |00〉, |01〉, |10〉, and |11〉 in multiple runs, and used that information to construct the first column of an error matrix M. We repeated this procedure for states prepared as |01〉, |10〉, and |11〉 to construct the full matrix M, whose inverse is the filtering matrix. After applying filtering matrices to our averaged outcomes, we have found good quantitative agreement between the quantum computer output and the quantum mechanical predictions for the extent of violation of both inequalities as functions of θ'.
这项工作利用IBM的公开可用量子计算机，对适用于耦合到贝尔态的量子比特的两个不等式的违背程度进行了定量测试。不等式的违背情况已得到充分证实。我们的目的不是测试这些不等式，而是在考虑量子计算机当前错误率的情况下，确定在量子计算机上能多好地重现量子力学预测。我们给出了两个纠缠量子比特沿三个轴A、B和C的自旋投影结果，A和B之间有固定角度θ，B和C之间有一系列角度θ'。对于任何可以用具有两个可能值的三个可观测量来表征的经典物体，不等式支配着成对取出的可观测量结果概率之间的关系。从集合论的角度来看，所有这样的经典物体都必须满足这些不等式；但量子系统可能会违背这些不等式。我们在IBM的公开可用量子计算机上的运行中检测到了对一个不等式的明确违背。Clauser-Horne-Shimony-Holt（CHSH）不等式支配着成对取出的自旋投影乘积期望值的线性组合S。发现S>2排除了纠缠量子系统的局域隐变量理论。在进行错误缓解之前，我们在运行中得到了大于2的S值。为了减少定量误差，我们对IBM文档中的错误缓解程序进行了修改。我们将一对量子比特制备在|00〉态，在多次运行中找到观察|00〉、|01〉、|10〉和|11〉态的概率，并利用该信息构建误差矩阵M的第一列。我们对制备为|01〉、|10〉和|11〉态的情况重复此过程以构建完整矩阵M，其逆矩阵就是滤波矩阵。在将滤波矩阵应用于我们的平均结果后，我们发现量子计算机输出与量子力学预测在两个不等式违背程度作为θ'的函数方面有良好的定量一致性。
