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arXiv 2103.04963quant-phcs.ET
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降低量子退火偏差求解图划分问题

Reducing quantum annealing biases for solving the graph partitioning problem

Elijah Pelofske, Georg Hahn, Hristo N. Djidjev

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摘要：

量子退火器提供了一种有效的方法来计算高质量的解决方案的NP难问题时，表示在一个QUBO（二次无约束二进制优化）或伊辛形式。这是通过将一个问题映射到量子芯片的物理量子位和耦合器上来完成的，在一个称为量子退火的过程之后，从量子芯片读取解决方案。然而，该过程受到多个偏置源的影响，包括校准不良、相邻量子位之间的泄漏、控制偏置等，这可能对退火结果的质量产生负面影响。在这项工作中，我们的目标是减轻这种偏见的影响，解决约束优化问题，通过提供一个两步的方法，并将其应用到图分割。在第一步中，我们测量并减少由于实现问题的约束而导致的任何偏差。在第二种情况下，我们将目标函数添加到约束的偏差校正实现中，并将问题发送到量子退火机。我们将这个概念应用于图划分，这是一个重要的NP-难问题，它要求找到一个平衡的（约束）和最小化切割大小（目标）的图的顶点的分区。我们首先量化量子退火器上的约束的实现的偏差，也就是说，我们要求，在无偏实现中，任何两个顶点被分配到分区的相同或不同部分的可能性相同。然后，我们提出了一个迭代的方法来纠正任何这样的偏见。我们证明，添加目标后，解决由此产生的偏置校正伊辛问题的量子退火机的结果在一个更高的解决方案的精度。

英文摘要：

Quantum annealers offer an efficient way to compute high quality solutions of NP-hard problems when expressed in a QUBO (quadratic unconstrained binary optimization) or an Ising form. This is done by mapping a problem onto the physical qubits and couplers of the quantum chip, from which a solution is read after a process called quantum annealing. However, this process is subject to multiple sources of biases, including poor calibration, leakage between adjacent qubits, control biases, etc., which might negatively influence the quality of the annealing results. In this work, we aim at mitigating the effect of such biases for solving constrained optimization problems, by offering a two-step method, and apply it to Graph Partitioning. In the first step, we measure and reduce any biases that result from implementing the constraints of the problem. In the second, we add the objective function to the resulting bias-corrected implementation of the constraints, and send the problem to the quantum annealer. We apply this concept to Graph Partitioning, an important NP-hard problem, which asks to find a partition of the vertices of a graph that is balanced (the constraint) and minimizes the cut size (the objective). We first quantify the bias of the implementation of the constraint on the quantum annealer, that is, we require, in an unbiased implementation, that any two vertices have the same likelihood of being assigned to the same or to different parts of the partition. We then propose an iterative method to correct any such biases. We demonstrate that, after adding the objective, solving the resulting bias-corrected Ising problem on the quantum annealer results in a higher solution accuracy.

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