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.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.

quant-pharXiv:2410.04630Oct 8, 20240 citations

Quantum computational complexity in the presence of closed timelike curves

Dave Bacon

Quantum computation with quantum data that can traverse closed timelike curves represents a new physical model of computation. We argue that a model of quantum computation in the presence of closed timelike curves can be formulated which represents a valid quantification of resources given the ability to construct compact regions of closed timelike curves. The notion of self-consistent evolution for quantum computers whose components follow closed timelike curves, as pointed out by Deutsch [Phys. Rev. D 44, 3197 (1991)], implies that the evolution of the chronology respecting components which interact with the closed timelike curve components is nonlinear. We demonstrate that this nonlinearity can be used to efficiently solve computational problems which are generally thought to be intractable. In particular we demonstrate that a quantum computer which has access to closed timelike curve qubits can solve $\mathit{NP}$ -complete problems with only a polynomial number of quantum gates.Quantum computation with quantum data that can traverse closed timelike curves represents a new physical model of computation. We argue that a model of quantum computation in the presence of closed timelike curves can be formulated which represents a valid quantification of resources given the ability to construct compact regions of closed timelike curves. The notion of self-consistent evolution for quantum computers whose components follow closed timelike curves, as pointed out by Deutsch [Phys. Rev. D 44, 3197 (1991)], implies that the evolution of the chronology respecting components which interact with the closed timelike curve components is nonlinear. We demonstrate that this nonlinearity can be used to efficiently solve computational problems which are generally thought to be intractable. In particular we demonstrate that a quantum computer which has access to closed timelike curve qubits can solve $\mathit{NP}$-complete problems with only a polynomial number of quantum gates.

arXiv:quant-ph/0309189Physical Review ASep 13, 200473 citations

\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.

\mathit{NP}

-complete problems with only a polynomial number of quantum gates.