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arXiv:2410.04630

Unitary Closed Timelike Curves can Solve all of NP

Omri Shmueli

quant-phSubmitted Oct 6, 2024• Announced Oct 8, 2024• 0 citations• Revised Oct 29, 2026 (v2)
v1 · Oct 6, 2024v2 · Oct 29, 2026
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Abstract

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 $\textit{linear}$. Araujo et al. accordingly defined the complexity class for efficient process matrix computation as $\mathbf{BQP}_{\ell CTC}$ (which trivially contains $\mathbf{BQP}$), and posed the open question of whether $\mathbf{BQP}_{\ell CTC}$ contains computational tasks that are outside $\mathbf{BQP}$. In this work we solve this open question under a widely believed hardness assumption, by showing that $\mathbf{NP} \subseteq \mathbf{BQP}_{\ell CTC}$. 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 $\textit{unitary}$ (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 $\mathbf{NP}$, is false, and raises the importance of understanding whether purifiable process matrices are physical or not.

Comments
The paper is withdrawn due to an error in the definition of the computational model, which invalidates the proof of the main result. The model imposes consistency only for identity local operations but does not impose the stronger consistency requirement under arbitrary local operations, which is needed for valid process matrices

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To study which are the most general causal structures which are compatible with local quantum mechanics, Oreshkov et al. [1] introduced the notion of a process: a resource shared between some parties that allows for quantum communication between them without a predetermined causal order. These processes can be used to perform several tasks that are impossible in standard quantum mechanics: they allow for the violation of causal inequalities, and provide an advantage for computational and communication complexity. Nonetheless, no process that can be used to violate a causal inequality is known to be physically implementable. There is therefore considerable interest in determining which processes are physical and which are just mathematical artefacts of the framework. Here we make key progress in this direction by proposing a purification postulate: processes are physical only if they are purifiable. We derive necessary conditions for a process to be purifiable, and show that several known processes do not satisfy them.

•Jan 1, 2017•52 citations
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\textit{linear}linear. Araujo et al. accordingly defined the complexity class for efficient process matrix computation as BQPℓCTC\mathbf{BQP}_{\ell CTC}BQPℓCTC​ (which trivially contains BQP\mathbf{BQP}BQP), and posed the open question of whether BQPℓCTC\mathbf{BQP}_{\ell CTC}BQPℓCTC​ contains computational tasks that are outside BQP\mathbf{BQP}BQP.
In this work we solve this open question under a widely believed hardness assumption, by showing that NP⊆BQPℓCTC\mathbf{NP} \subseteq \mathbf{BQP}_{\ell CTC}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\textit{unitary}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\mathbf{NP}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\textbf{NP}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\textbf{BQP}BQP within the process matrix formalism, even while preserving unitarity.
In quantum causality and quantum information, there is a vast landscape of abstract quantum protocols permitting cyclic or non-acyclic causal structures between operations, including frameworks for indefinite causal order and higher-order quantum processes such as process matrices. A longstanding open question is what is the largest class of abstract processes that admit physical realisations without post-selection. In this work, we provide a rigorous answer using a top-down approach grounded in relativistic causality principles. Building on the framework of causal boxes, which characterise the most general quantum information-processing protocols compatible with fixed background spacetimes, we formalise additional constraints (Acting Once + Local Order) capturing the closed-laboratory assumptions of the process matrix framework at a fine-grained spacetime level. We prove that any protocol in a classical acyclic spacetime satisfying these conditions is behaviourally equivalent to a quantum circuit with quantum control of causal order (QC-QC), providing a top-down derivation of QC-QCs from physical principles. Our results show that QC-QCs constitute precisely the class of higher-order quantum processes, including those with indefinite order, that can be physically realised within classical spacetime, ruling out more general non-causal processes under the closed-labs assumption. This clarifies the relationship between abstract higher-order process matrix frameworks and experimentally accessible quantum protocols, as well as the interplay between coarse-grained cyclic and fine-grained acyclic operational causal structures. We also develop characterisation techniques for process box protocols that lead to new causality-based open questions concerning spacetime quantum protocols and relativistic quantum experiments.
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.
Recently, the possible existence of quantum processes with indefinite causal order has been extensively discussed, in particular using the formalism of process matrices. Here we give a new perspective on this question, by establishing a direct connection to the theory of multi-time quantum states. Specifically, we show that process matrices are equivalent to a particular class of pre- and post-selected quantum states. This offers a new conceptual point of view to the nature of process matrices. Our results also provide an explicit recipe to experimentally implement any process matrix in a probabilistic way, and allow us to generalize some of the previously known properties of process matrices. Finally we raise the issue of the difference between the notions of indefinite temporal order and indefinite causal order, and show that one can have indefinite causal order even with definite temporal order.
Process matrices are a framework to model causal relations in the absence of a well-defined acyclic causal order. The framework is very general and does not even assume the existence of a background spacetime. As a result, it is an open question how the framework should be interpreted physically and how and even if composition can be defined. On the other hand, so-called causal boxes define a framework that allows for arbitrary composition. In this work, we treat quantum circuits with quantum control of causal order (QC-QC), a subset of process matrices, which can be interpreted as generalized quantum circuits, and process box, a subset of causal boxes, which can be interpreted as processes. We analyze their state spaces and define a notion of operational equivalence between QC-QCs and process boxes based on this analysis. We then explicitly construct for each QC-QC an operationally equivalent process box. This allows us to define composition of QC-QCs in terms of composition of causal boxes which is well-defined. We further show that process boxes admit a unitary extension and conjecture that the background spacetime can be assumed to have a specific simple form. Based on this conjecture, we construct an operationally equivalent QC-QC for each process box. Our results indicate that the only class of processes that can be physically implemented in a fixed background spacetime are those that can be interpreted as quantum circuits with quantum controlled superpositions of orders. Further, they also reveal that the composability issue can be resolved by embedding processes in a spacetime structure. This in turn sheds light on the connection between physical realizability in a spacetime and composability.
To study which are the most general causal structures which are compatible with local quantum mechanics, Oreshkov et al. [1] introduced the notion of a process: a resource shared between some parties that allows for quantum communication between them without a predetermined causal order. These processes can be used to perform several tasks that are impossible in standard quantum mechanics: they allow for the violation of causal inequalities, and provide an advantage for computational and communication complexity. Nonetheless, no process that can be used to violate a causal inequality is known to be physically implementable. There is therefore considerable interest in determining which processes are physical and which are just mathematical artefacts of the framework. Here we make key progress in this direction by proposing a purification postulate: processes are physical only if they are purifiable. We derive necessary conditions for a process to be purifiable, and show that several known processes do not satisfy them.