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 $\textbf{NP}$. 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 $\textbf{BQP}$ within the process matrix formalism, even while preserving unitarity.
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The paper asks whether indefinite causal structure increases computational power in the process-matrix formalism, including under purity and purification requirements. Computation is modeled by polynomial-size ordinary circuits with designated feedback connections; access to the resulting channel is granted at cost proportional to the generating circuit’s size, rather than its implementation cost.
For general processes, it proves $\mathbf{NP}\subseteq\mathbf{BQP}_{\mathrm{lCTC}}$. The valid processes used are purifiable and generate unitary channels, although their generating circuits are nonunitary. The main construction converts an exact circuit preparing an isometry’s Choi state into a process generator implementing that isometry. Teleportation measurements and feedback enforce the desired outcome while preserving normalization under arbitrary inserted channels. An efficiently prepared Choi state for a unique-witness swap, combined with Valiant–Vazirani isolation and classical verification, yields the NP algorithm. The construction also enables coherent permutation inversion.
For pure processes, it explicitly constructs polynomial-size unitary generators that coherently invert unique-sink-orientation outmaps. One generated-channel call finds the sink with probability $1$. Validity follows from the pairwise USO characterization, which guarantees operator-valued row orthogonality even with arbitrary local auxiliary systems. The argument extends to certain product-basis transformations.
A separation from $\mathbf{BQP}$ for pure processes remains conditional on USO lacking a quantum polynomial-time algorithm. Physical realizability and efficient implementation of these channels are unknown. Purifiability establishes existence of a pure extension, not an efficient way to construct it; existing switch-based proposals cover only particular cases.
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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.
The paper asks whether indefinite causal structure increases computational power in the process-matrix formalism, including under purity and purification requirements. Computation is modeled by polynomial-size ordinary circuits with designated feedback connections; access to the resulting channel is granted at cost proportional to the generating circuit’s size, rather than its implementation cost.
For general processes, it proves NP⊆BQPlCTC. The valid processes used are purifiable and generate unitary channels, although their generating circuits are nonunitary. The main construction converts an exact circuit preparing an isometry’s Choi state into a process generator implementing that isometry. Teleportation measurements and feedback enforce the desired outcome while preserving normalization under arbitrary inserted channels. An efficiently prepared Choi state for a unique-witness swap, combined with Valiant–Vazirani isolation and classical verification, yields the NP algorithm. The construction also enables coherent permutation inversion.
For pure processes, it explicitly constructs polynomial-size unitary generators that coherently invert unique-sink-orientation outmaps. One generated-channel call finds the sink with probability 1. Validity follows from the pairwise USO characterization, which guarantees operator-valued row orthogonality even with arbitrary local auxiliary systems. The argument extends to certain product-basis transformations.
A separation from BQP for pure processes remains conditional on USO lacking a quantum polynomial-time algorithm. Physical realizability and efficient implementation of these channels are unknown. Purifiability establishes existence of a pure extension, not an efficient way to construct it; existing switch-based proposals cover only particular cases.
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.
In recent years, the investigation of causal relations in quantum theory has attracted a lot of interest. In particular, it has been found that it is possible to conceive of scenarios where some parties perform operations that are compatible with quantum theory locally, but that cannot be embedded into a global background causal structure. Such indefinite causal structures are of interest from a fundamental point of view, but also from the perspective of quantum information processing, since they do not fit into the usual paradigm of quantum circuits, which assumes a definite causal order. The main aim of this thesis is to study indefinite quantum causal relations involving more than two parties. Compared to the bipartite case, there are many new aspects and complications that arise in multipartite situations, which need to be clarified in order to fundamentally understand quantum causal structures, and to shed light on their implications and potential usefulness for quantum information processing.A suitable mathematical framework for the investigation of quantum causal relations is the process matrix formalism. In this framework, the notion of causal nonseparability was introduced in order to qualify scenarios that are incompatible with a definite causal order. In Chapter 2, we study how to generalise this concept to the multipartite case, how to characterise multipartite causally (non)separable quantum processes, and how to certify multipartite causal nonseparability. Another important topic is to determine which quantum scenarios with indefinite causal order are physically implementable, and how they can be realised concretely. In Chapter 3, we introduce two new classes of physically realisable multipartite quantum processes, and characterise them in terms of their process matrix descriptions. In particular, we define the class of quantum circuits with quantum control of causal order. The simplest example of such a circuit is the quantum switch, a protocol in which the order between two operations is controlled by a qubit in a superposition state, and which defines a causally nonseparable process. The class we introduce also contains more general examples of causally nonseparable processes with new features. We then show how the process matrix characterisation of this class allows us to search for new quantum information processing applications of causal nonseparability. In Chapter 4, we investigate a particular quantum communication effect in a scenario involving coherent control between two quantum channels. This leads us to a more general analysis of the notion of a coherently controlled channel, which involves certain subtilities. In Chapter 5, we turn to another problem that arises in multipartite scenarios, namely whether a given phenomenon is genuinely multipartite or not. More particularly, we study genuinely multipartite (non)causal correlations. In Chapter 6, we show that anomalous weak values are possible without post-selection. In Chapter 7, we show that certain tripartite process matrices, namely those that are unitarily extensible, have a realisation on so-called time-delocalised subsystems, i.e., quantum subsystems that are not associated with a definite time. The class of unitarily extensible tripartite process matrices is larger than the class of quantum circuits with quantum control, and in particular contains processes that violate so-called causal inequalities.
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.
Indefinite causal order extends the conventional circuit paradigm by allowing local quantum operations to be connected without a predetermined global order. Within the process-matrix framework, it is characterized by causal nonseparability, whereas extensible causal separability (ECS) requires that, under arbitrary input-ancilla extensions, a process admit a recursive decomposition into components, each compatible with a particular operation acting first. Quantum circuits with classical control of causal order (QC-CC), in which previous outcomes dynamically determine which operation acts next, are known to generate ECS processes, but whether every ECS process admits such a realization has remained a longstanding open problem. We settle it through a generalized teleportation construction that realizes every multipartite ECS process as a QC-CC. We further establish consistency between ECS definitions for trivial and nontrivial global past and future systems. These results provide ECS with an exact operational interpretation and identify QC-CC as the complete circuit structure underlying this class of processes.
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.
In the last few years, there has been increasing interest in quantum processes with indefinite causal order. Process matrices are a convenient framework to study such processes. Ref. [1] defines higher order transformations from process matrices to process matrices and shows that no continuous and reversible transformation can change the causal order of a process matrix. Ref. [2] argues, based on a set of examples, that there are situations where a process can change its causal order over time. Ref. [2] claims that the formalism of higher order transformations is not general enough to capture its examples. Here we show that this claim is incorrect. Moreover, a crucial example of Ref. [2] has already been explicitly considered in Ref. [1] and shown to be compatible with its results.