Quantum computation with indefinite causal structures
Mateus Araújo, Philippe Allard Guérin, Ämin Baumeler
TL;DR
The paper investigates quantum computation with indefinite causal structures by comparing nonlinear CTC models (D-CTCs, P-CTCs) to the linear process-matrix framework. It proves that process matrices are exactly linear P-CTCs and defines the complexity class $\mathrm{BQP}_{\ell\mathrm{CTC}}$, showing $\mathrm{BQP}_{\ell\mathrm{CTC}} \subseteq \mathrm{PP}$ and linking it to PostBQP. It analyzes concrete examples, including the quantum switch and a deterministic acausal process, showing that indefiniteness in causality does not necessarily imply computational hardness or a large power advantage; some acausal processes can be simulated with modest overhead. Overall, the work argues that linear, indefinite causal-order formalisms like process matrices may be physically plausible and computationally tame, while still capturing nontrivial causal structures.
Abstract
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.