基于量子近似优化算法的最大似然检测
Quantum Approximate Optimization Algorithm Based Maximum Likelihood Detection
摘要：
量子技术的最新进展为有噪声的中间尺度量子（NISQ）器件铺平了道路，其中量子近似优化算法（QAOA）构成了基于NISQ器件展示有形量子优势的有希望的候选者。在本文中，我们考虑的最大似然（ML）检测问题的二进制符号传输的多输入多输出（MIMO）信道，其中找到最佳的解决方案是指数困难的使用经典的计算机。在这里，我们将QAOA用于ML检测，通过将感兴趣的问题编码到具有2p个变分参数的p级QAOA电路中，该电路可以通过经典优化器进行优化。这种p级QAOA电路是通过将准备好的哈密顿量应用于我们的问题和初始哈密顿量交替地应用于p个连续轮来构造的。更明确地说，我们首先将ML检测问题的最优解编码到问题哈密顿量的基态中。使用量子绝热演化技术，我们提供了分析和数值结果表征用于ML检测的量子系统的本征值的演变。然后，对于一级QAOA电路，我们推导了QAOA期望值的解析表达式，并讨论了基于QAOA的ML检测器的复杂度。解释，我们评估的计算复杂性的经典优化器和模拟QAOA的存储要求。最后，我们评估的比特出错率（BER）的QAOA基于ML检测器，并将其与经典的ML检测器和经典的MMSE检测器进行比较，表明基于QAOA的ML检测器是能够接近经典的ML检测器的性能。这为NISQ计算机解决大量的经典优化问题铺平了道路。
英文摘要：
Recent advances in quantum technologies pave the way for noisy intermediate-scale quantum (NISQ) devices, where quantum approximation optimization algorithms (QAOAs) constitute promising candidates for demonstrating tangible quantum advantages based on NISQ devices. In this paper, we consider the maximum likelihood (ML) detection problem of binary symbols transmitted over a multiple-input and multiple-output (MIMO) channel, where finding the optimal solution is exponentially hard using classical computers. Here, we apply the QAOA for the ML detection by encoding the problem of interest into a level-p QAOA circuit having 2p variational parameters, which can be optimized by classical optimizers. This level-p QAOA circuit is constructed by applying the prepared Hamiltonian to our problem and the initial Hamiltonian alternately in p consecutive rounds. More explicitly, we first encode the optimal solution of the ML detection problem into the ground state of a problem Hamiltonian. Using the quantum adiabatic evolution technique, we provide both analytical and numerical results for characterizing the evolution of the eigenvalues of the quantum system used for ML detection. Then, for level-1 QAOA circuits, we derive the analytical expressions of the expectation values of the QAOA and discuss the complexity of the QAOA based ML detector. Explicitly, we evaluate the computational complexity of the classical optimizer used and the storage requirement of simulating the QAOA. Finally, we evaluate the bit error rate (BER) of the QAOA based ML detector and compare it both to the classical ML detector and to the classical MMSE detector, demonstrating that the QAOA based ML detector is capable of approaching the performance of the classical ML detector. This paves the way for a host of large-scale classical optimization problems to be solved by NISQ computers.
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