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Comparing Deutsch and Postselected Closed Timelike Curves with Circuit Simulations

Naol Demisse

Published Jun 14, 2026• 0 citations
Publisher

Abstract

Closed timelike curves (CTCs) are a theoretical testbed for how quantum mechanics behaves when causal order breaks down. Two major quantum models exist: Deutsch CTCs, which impose a fixed-point self-consistency condition on the state inside the loop, and postselected CTCs, which model time travel via teleportation and postselection. Although both avoid logical paradoxes, they generally produce different quantum evolutions for the same circuit. This paper compares the two models directly at the circuit level using a small density-matrix and statevector simulator. The Deutsch model is implemented by solving for self-consistent fixed points; the postselected model by applying the corresponding nonlinear postselection map. Identical test circuits, including grandfather-paradox circuits, CNOT, SWAP, controlled-unitary families, and Haar-random unitaries, are run through both models, and the outputs compared using trace distance. The two models are found to disagree generically, not just numerically. The disagreement stems from a structural difference: Deutsch CTCs enforce consistency with mixed fixed-point states, while postselected CTCs renormalize only successful histories. For example, the X-gate "grandfather" circuit yields a well-defined mixed state under the Deutsch rule but an undefined (null-projection) output under postselection. Across 200 random unitaries, the two prescriptions never agree. All simulation code is included for full reproducibility.

DOI
10.5281/zenodo.20693268

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Closed timelike curves (CTCs) are a theoretical testbed for how quantum mechanics behaves when causal order breaks down. Two major quantum models exist: Deutsch CTCs, which impose a fixed-point self-consistency condition on the state inside the loop, and postselected CTCs, which model time travel via teleportation and postselection. Although both avoid logical paradoxes, they generally produce different quantum evolutions for the same circuit. This paper compares the two models directly at the circuit level using a small density-matrix and statevector simulator. The Deutsch model is implemented by solving for self-consistent fixed points; the postselected model by applying the corresponding nonlinear postselection map. Identical test circuits, including grandfather-paradox circuits, CNOT, SWAP, controlled-unitary families, and Haar-random unitaries, are run through both models, and the outputs compared using trace distance. The two models are found to disagree generically, not just numerically. The disagreement stems from a structural difference: Deutsch CTCs enforce consistency with mixed fixed-point states, while postselected CTCs renormalize only successful histories. For example, the X-gate "grandfather" circuit yields a well-defined mixed state under the Deutsch rule but an undefined (null-projection) output under postselection. Across 200 random unitaries, the two prescriptions never agree. All simulation code is included for full reproducibility.
Closed timelike curves (CTCs) are trajectories in spacetime that effectively travel backwards in time: a test particle following a CTC can interact with its former self in the past. A widely accepted quantum theory of CTCs was proposed by Deutsch. Here we analyze an alternative quantum formulation of CTCs based on teleportation and postselection, and show that it is inequivalent to Deutsch's. The predictions or retrodictions of our theory can be simulated experimentally: we report the results of an experiment illustrating how in our particular theory the "grandfather paradox" is resolved.
Although general relativity allows the existence of closed timelike curves (CTCs), self-consistency problems arise (the ‘grandfather paradox’ among others). It is known that quantum mechanical consideration of the matter formally removes all the paradoxes, but the questions about causal structure remain. On the other hand, the idea of postselected CTCs (P-CTC) in quantum teleportation has know been put forward and experimentally implemented. We consider these problems with the aid of quantum causal analysis, where causality is defined without invoking the time relation. It implements the Cramer principle of weak causality, which admits time reversal in entangled states. We analyze Deutsch CTCs (D-CTC) with different kinds of interactions between the chronology-violating and chronology-respecting particles, with refined inferences about mixedness, quantum/classical correlations, entanglement and thermodynamics in the D-CTC. The main result is that time reversal causality can really exist, however, the final quantum state does not place retrospective constraints on the initial state, instead the final state can influence the state inside the D-CTC. This is effectively the implementation of Novikov self-consistency principle. The P-CTC has radically different properties; in particular, if the initial state was pure, the final state is always pure too. Self-consistency is controlled by the initial state-dependent traversability of the P-CTC.
Closed timelike curves (CTCs) are non-intuitive theoretical solutions of general relativity field equations. The main paradox associated with the physical existence of CTCs, the so-called grandfather paradox, can be satisfactorily solved by a quantum model named Deutsch-CTC. An outstanding theoretical result that has been demonstrated in the Deutsch-CTC model is the computational equivalence of a classical and a quantum computer in the presence of a CTC. In this article, in order to explore the possible implications for the foundations of quantum mechanics of that equivalence, a fundamental particle is modelled as a classical-like system supplemented with an information space in which a randomizer and a classical Turing machine are stored. The particle could then generate quantum behavior in real time in case it was controlled by a classical algorithm coding the rules of quantum mechanics and, in addition, a logical circuit simulating a CTC was present on its information space. The conditions that, through the action of evolution under natural selection, might produce a population of such particles with both elements on their information spaces from initial sheer random behavior are analyzed.
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.
One out of many emerging implications from solutions of Einstein's general relativity equations are closed timelike curves (CTCs), which are trajectories through space-time that allow for time travel to the past without exceeding the speed of light. Two main quantum models of computation with the use of CTCs were introduced by Deutsch (D-CTC) and by Bennett and Schumacher (P-CTC). Unlike the classical theory in which CTCs lead to logical paradoxes, the quantum D-CTC model provides a solution that is logically consistent due to the self-consistency condition imposed on the evolving system, whereas the quantum P-CTC model chooses such a solution through postselection. Both models are nonequivalent and imply nonstandard phenomena in the field of quantum computation and quantum mechanics. In this paper, we study the implications of these two models on the second law of thermodynamics---the fundamental principle which states that in an isolated system the entropy never decreases. In particular, we construct CTC-based quantum circuits which lead to a decrease in entropy.
Deutsch closed timelike curves (D-CTCs) are described by a fixed-point condition for a chronology-violating register. We study a finite-dimensional circuit model that places a Hayden--Preskill/Yoshida--Kitaev recovery map inside such a consistency loop. A register-routing construction makes the Deutsch map explicit: an initial SWAP moves the incoming CTC state to an idle dump register, the scrambler and decoder act on the remaining active registers, and a final SWAP writes the recovered message back to the CTC register. When the active branch recovers the message, the induced map on the CTC register is the replacement channel σ↦ρMσ\mapsto ρ_Mσ↦ρM​, with the unique fixed point ρMρ_MρM​. We implement the associated Lloyd-type post-selected decoder circuits on quantum hardware and formulate a classical-feedback iteration for the experimentally estimated map. Qiskit simulations and IBM-hardware data for single-qubit instances quantify decoder fidelity, post-selection overhead, routing-dependent noise, and quantum-geometric susceptibility.