One way to study the physical plausibility of closed timelike curves (CTCs) is to examine their computational power. This has been done for Deutschian CTCs (D-CTCs) and post-selection CTCs (P-CTCs), with the result that they allow for the efficient solution of problems in PSPACE and PP, respectively. Since these are extremely powerful complexity classes, which are not expected to be solvable in reality, this can be taken as evidence that these models for CTCs are pathological. This problem is closely related to the nonlinearity of this models, which also allows for example cloning quantum states, in the case of D-CTCs, or distinguishing non-orthogonal quantum states, in the case of P-CTCs. In contrast, the process matrix formalism allows one to model indefinite causal structures in a linear way, getting rid of these effects, and raising the possibility that its computational power is rather tame. In this paper we show that process matrices correspond to a linear particular case of P-CTCs, and therefore that its computational power is upperbounded by that of PP. We show, furthermore, a family of processes that can violate causal inequalities but nevertheless can be simulated by a causally ordered quantum circuit with only a constant overhead, showing that indefinite causality is not necessarily hard to simulate.

翻译：研究闭合类时曲线物理可行性的一个途径是考察其计算能力。针对 Deutsch 型闭合类时曲线与后选择闭合类时曲线的研究已表明，它们分别能高效求解 PSPACE 与 PP 复杂度类中的问题。由于这两类复杂度异常强大，现实中预期不可解，该结果可视为这些闭合类时曲线模型存在病理特征的证据。此问题与模型的非线性特性密切相关——Deutsch 型曲线允许量子态克隆，后选择型曲线则可区分非正交量子态。相较而言，过程矩阵形式体系能以线性方式建模不定因果结构，规避上述效应，从而使其计算能力可能趋于温和。本文证明过程矩阵对应于后选择闭合类时曲线的线性特例，因此其计算能力以 PP 为上界。进一步，我们构建了一类可违反因果不等式却仅需恒定开销即可由因果有序量子电路模拟的过程，表明不定因果性未必难以模拟。