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 postselection 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 nonorthogonal 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.
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 postselection 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 nonorthogonal 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.
Born in the intersection between quantum mechanics and general relativity, indefinite causal structure is the idea that in the continuum of time, some sets of events do not have an inherent causal order between them. Process matrices, introduced by Oreshkov, Costa and Brukner (Nature Communications, 2012), define quantum information processing with indefinite causal structure -- a generalization of the operations allowed in standard quantum information processing, and to date, are the most studied such generalization.
Araujo et al. (Physical Review A, 2017) defined the computational complexity of process matrices, and showed that polynomial-time process matrix computation is equivalent to standard polynomial-time quantum computation with access to a weakening of post-selection Closed Timelike Curves (CTCs), that are restricted to be linear. Araujo et al. accordingly defined the complexity class for efficient process matrix computation as BQPℓCTC (which trivially contains BQP), and posed the open question of whether BQPℓCTC contains computational tasks that are outside BQP.
In this work we solve this open question under a widely believed hardness assumption, by showing that NP⊆BQPℓCTC. Our solution is captured by an even more restricted subset of process matrices that are purifiable (Araujo et al., Quantum, 2017), which (1) is conjectured more likely to be physical than arbitrary process matrices, and (2) is equivalent to polynomial-time quantum computation with access to unitary (which are in particular linear) post-selection CTCs. Conceptually, our work shows that the previously held belief, that non-linearity is what enables CTCs to solve NP, is false, and raises the importance of understanding whether purifiable process matrices are physical or not.
Can quantum computation become more powerful when operations do not need to follow a definite causal order? We study this question in the process matrix formalism of Oreshkov et al. (Nature Communications, 2012), which allows quantum operations to be connected without a fixed global order. In the computational model, a quantum circuit succinctly describes a process matrix, and the algorithm is given access to its generated channel, with computational cost measured by the size of the generating circuit.
We give polynomial-time algorithms in two settings. First, general process matrix computation suffices to solve every problem in NP. The valid processes used by our algorithm satisfy the purification postulate of Araújo et al. (Quantum, 2017), and their generated channels are unitary, although their generating circuits are not. Second, requiring the process matrices themselves to be pure, we give a polynomial-time algorithm for unique-sink orientation, a general problem capturing several longstanding computational problems in optimization. Our unitary generators are constructed explicitly and efficiently. Assuming that unique-sink orientation has no quantum polynomial-time algorithm, our result shows that indefinite causal order can provide computational power beyond BQP within the process matrix formalism, even while preserving unitarity.
Closed Timelike Curves (CTCs) are intriguing relativistic objects that allow for time travel to the past and can be used as computational resources. In Deutschian Closed Timelike Curves (D-CTCs), due to the monogamy of entanglement, non-local correlations between entangled states are destroyed. In contrast, for Postselected Closed Timelike Curves (P-CTCs), a second variant of CTCs, the non-local correlations are preserved. P-CTCs can be harnessed for the signaling of non-orthogonal states to the past without a disruption of causality. In this paper, we take up signaling to the past and show a method of sending four non-orthogonal states to the past using P-CTCs. After constructing our signaling protocol, we study the causality violations that our protocol results in and put forward two consistency relations to prevent them.
Closed Timelike Curves (CTCs) are intriguing relativistic objects that allow for time travel to the past and can be used as computational resources. In Deutschian Closed Timelike Curves (D-CTCs), due to the monogamy of entanglement, non-local correlations between entangled states are destroyed. In contrast, for Postselected Closed Timelike Curves (P-CTCs), a second variant of CTCs, the non-local correlations are preserved. P-CTCs can be harnessed for the signaling of non-orthogonal states to the past without a disruption of causality. In this paper, we take up signaling to the past and show a method of sending four non-orthogonal states to the past using P-CTCs. After constructing our signaling protocol, we study the causality violations that our protocol results in and put forward two consistency relations to prevent them.
Abstract This study examines closed timelike curves (CTCs) in quantum mechanics, focusing on the Deutsch model, one of the most widely studied frameworks for describing quantum systems in the presence of time loops. In this model, a quantum state ∣ψ⟩|\psi\rangle∣ψ⟩ interacts unitarily with its past version, effectively forming a two-qubit system where the chronologically consistent qubit interacts with a qubit ρCTC\rho_{\text{CTC}}ρCTC trapped in a closed timelike curve. The state ρCTC\rho_{\text{CTC}}ρCTC is determined by the Deutsch consistency condition, ensuring self-consistent evolution along the CTC. This approach allows for rigorous modeling of quantum systems under non-classical temporal structures, providing insights into temporal correlations, causality violations, and the potential computational advantages offered by CTC-assisted quantum operations. While alternative CTC models based on postselection have been proposed, the Deutsch model remains a foundational reference for studying quantum evolution in non-linear temporal frameworks. Its analysis informs research in quantum information, retrocausality, and the theoretical limits of computation in exotic spacetime geometries.
The fundamental nature of causality has been challenged by quantum theory. In particular, recent developments have shown that by relaxing the standard assumption that quantum events are embedded in a fixed background causal structure, a new kind of causal relations, indefinite, can emerge. These indefinite causal structures have been interpreted as causal loops allowing for the violation of a form of causality without paradoxes. On the other hand, quantum contextuality shows that any attempt to extend quantum theory such that classical logic holds in the extended domain fails due to the rise of inconsistent logical loops. There has been increasing evidence that it is a key feature in the understanding quantum paradoxes. This thesis aims at improving our understanding of indefinite causal orders and quantum paradoxes in order to address the question: “Could the non-fundamental nature of causality be related to a form of contextuality ?”The process matrix formalism is a mathematical framework which, in analogy withentanglement and the violation of Bell inequalities (nonlocality), offers new tools - witnesses of causal nonseparability and the violation of causal inequalities with noncausalcorrelations - to study and identify indefinite causal orders. In a bottom-up approach, wehave identified a large class of physically implementable processes, the quantum circuitswith quantum control of causal order (QC-QCs), that includes causally nonseparable processes such as the celebrated “quantum switch”. However these circuits cannot be used to violate causal inequalities. Nevertheless, in another work, I have proposed a significant relaxation of assumptions to certify the causal nonseparability of many processes, showing that it can be done in a semi-device-independent scenario with untrusted local operations and trusted quantum inputs. In particular, I have shown that the quantum switch can generate noncausal correlations in this kind of scenario.In order to refine my understanding of quantum contextuality, I have investigated theorigins of the Kochen-Specker theorem and I have analysed how the Hardy paradox entails a form of logical contextuality. I have identified that the two possible intermediatemeasurements involved in a pre-post-selection version of the paradox differ causally fromeach other. This analysis might be useful to study of the Frauchiger-Renner paradox, a“meta-version” of the Hardy paradox. On a more metaphysical aspect, I have argued fora “Gödelian hunch” from quantum theory, the idea that quantum paradoxes emerge froma lack of distinction between theoretical and metatheoretical objects.Finally, in order to establish a link between indefinite causal orders and contextuality,I have proposed a new causal game, in which the grandfather paradox is formalized interms of a logical inequality. Some process matrices can violate the causal inequalitywithout violating the logical one. I suggest that the emergence of logical inconsistenciesfrom the violation of a logical inequality might be interpreted as a form of contextuality.