Quantum computation promises to solve many hard or infeasible problems substantially faster than classical solutions. The involvement of big players like Google, IBM, Intel, Rigetti, or Microsoft furthermore led to a momentum which increases the demand for automated design methods for quantum computations. In this context, decision diagrams for quantum computation provide a major pillar as they allow to efficiently represent quantum states and quantum operations which, otherwise, have to be described in terms of exponentially large state vectors and unitary matrices. However, current decision diagrams for the quantum domain suffer from a tradeoff between accuracy and compactness, since: 1) small errors that are inevitably introduced by the limited precision of floating-point arithmetic can harm the compactness (i.e., the size of the decision diagram) significantly and 2) overcompensating these errors (to increase compactness) may lead to an information loss and introduces numerical instabilities. In this article, we describe and evaluate the effects of this tradeoff which clearly motivates the need for a solution that is perfectly accurate and compact at the same time. More precisely, we show that the tradeoff indeed weakens current design automation approaches for quantum computation (possibly leading to corrupted results or infeasible run-times). To overcome this, we propose an alternative approach that utilizes an algebraic representation of the occurring complex and irrational numbers and outline how this can be incorporated in a decision diagram which is suited for quantum computation. Evaluations show that—at the cost of an overhead which is moderate in many cases—the proposed algebraic solution indeed overcomes the tradeoff between accuracy and compactness that is present in current numerical solutions.

中文翻译：

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克服量子计算决策图中准确性和紧凑性之间的权衡

量子计算有望比经典解决方案更快地解决许多困难或不可行的问题。谷歌、IBM、英特尔、Rigetti 或微软等大公司的参与进一步推动了对量子计算自动化设计方法的需求。在这种情况下，量子计算的决策图提供了一个主要支柱，因为它们允许有效地表示量子状态和量子操作，否则，必须用指数级大状态向量和酉矩阵来描述。然而，量子域的当前决策图在准确性和紧凑性之间存在折衷，因为：1）由浮点运算的有限精度不可避免地引入的小错误会损害紧凑性（即，决策图的大小）显着和 2）过度补偿这些错误（以增加紧凑性）可能会导致信息丢失并引入数值不稳定性。在本文中，我们描述并评估了这种权衡的影响，这清楚地激发了对同时完全准确和紧凑的解决方案的需求。更准确地说，我们表明这种权衡确实削弱了当前用于量子计算的设计自动化方法（可能导致结果损坏或运行时间不可行）。为了克服这个问题，我们提出了一种替代方法，该方法利用出现的复数和无理数的代数表示，并概述了如何将其合并到适合量子计算的决策图中。