线性代数中困难问题的量子近似优化
Quantum Approximate Optimization for Hard Problems in Linear Algebra
摘要：
量子近似优化算法（QAOA）是Farhi等人提出的一种用于解决量子或经典优化任务的量子计算框架。在这里，我们探索使用QAOA的二进制线性最小二乘（BLLS）;一个问题，可以作为一个积木的其他几个困难的问题在线性代数，如非负二进制矩阵分解（NBMF）和其他变种的非负矩阵分解（NMF）问题。大多数以前的努力在量子计算解决这些问题是使用量子退火范例。在这项工作的范围内，我们的实验是在无噪声量子模拟器、包括设备真实噪声模型的模拟器和两台IBM Q 5量子比特机器上完成的。我们强调使用QAOA和QAOA变分算法解决这些问题的可能性，其中可以直接作为样本获得试验解决方案，而不是在量子波函数中进行振幅编码。我们的数值计算表明，模拟退火可以优于QAOA的BLLS在QAOA深度为$p\leq3$的概率采样的基态。最后，我们指出了目前在基于云的量子计算机上实验实现这种技术所面临的一些挑战。
英文摘要：
The Quantum Approximate Optimization Algorithm (QAOA) by Farhi et al. is a quantum computational framework for solving quantum or classical optimization tasks. Here, we explore using QAOA for Binary Linear Least Squares (BLLS); a problem that can serve as a building block of several other hard problems in linear algebra, such as the Non-negative Binary Matrix Factorization (NBMF) and other variants of the Non-negative Matrix Factorization (NMF) problem. Most of the previous efforts in quantum computing for solving these problems were done using the quantum annealing paradigm. For the scope of this work, our experiments were done on noiseless quantum simulators, a simulator including a device-realistic noise-model, and two IBM Q 5-qubit machines. We highlight the possibilities of using QAOA and QAOA-like variational algorithms for solving such problems, where trial solutions can be obtained directly as samples, rather than being amplitude-encoded in the quantum wavefunction. Our numerics show that Simulated Annealing can outperform QAOA for BLLS at a QAOA depth of $p\leq3$ for the probability of sampling the ground state. Finally, we point out some of the challenges involved in current-day experimental implementations of this technique on cloud-based quantum computers.
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