![]() We can immediately derive the Page curve, using the Ryu-Takayanagi formula, and the Hayden-Preskill decoding criterion, using entanglement wedge reconstruction. The new RT surface lies slightly inside the event horizon, at an infalling time approximately the scrambling time β/ 2 π log S BH into the past. When absorbing boundary conditions are used to evaporate a black hole in AdS/CFT, we show that there is a phase transition in the location of the quantum Ryu-Takayanagi surface, at precisely the Page time. On the other hand, if the radiation is not pseudorandom, an exterior observer may be able to create a firewall by applying a polynomial-time quantum computation to the radiation. Thus, under our pseudorandomness assumption, the black hole interior is well protected from exterior observers as long as the remaining black hole is macroscopic. Specifically, efficient operations acting on the radiation, those with quantum computational complexity polynomial in the entropy of the remaining black hole, commute with a complete set of logical operators acting on the encoded interior, up to corrections which are exponentially small in the entropy. This encoded interior is entangled with the late outgoing Hawking quanta emitted by the old black hole, and is inaccessible to computationally bounded observers who are outside the black hole. We then infer the existence of a subspace of the radiation system which we interpret as an encoding of the black hole interior. We assume that the Hawking radiation emitted by an old black hole is pseudorandom, meaning that it cannot be distinguished from a perfectly thermal state by any efficient quantum computation acting on the radiation alone. We reconsider the black hole firewall puzzle, emphasizing that quantum error- correction, computational complexity, and pseudorandomness are crucial concepts for understanding the black hole interior. ![]() We then study the success probability of QEC under such coherent errors, and confirm that the exact success probability under coherent error is smaller than the results using Pauli twirling approximation at physical level. In addition, we also find that if the code with a fixed distance d is $\epsilon$-correctable, the value of $\epsilon$ describing the accuracy of the approximate QEC cannot be smaller than a lower bound. The extra term in the generalized K-L criterion corresponds to the coherent part of the error channel at logical level, and then show that the generalized K-L criterion approaches the normal K-L criterion when the code distance becomes large. We find that the surface code under coherent error satisfies generalized Knill-Laflamme (K-L) criterion and falls into the category of approximate QEC. In this paper, we study the independent coherent error due to the imperfect unitary rotation on each physical qubit of the toric code. Their impacts are believed to be very subtle, more detrimental and hard to analyze compared to those ideal stochastic errors. The realistic coherent errors could induce very different behaviors compared with their stochastic counterparts in the quantum error correction (QEC) and fault tolerant quantum computation. On the other hand, if the radiation is not pseudorandom, an exterior observer may be able to create a firewall by applying a polynomial-time quantum computation to the radiation.A preprint version of the article is available at ArXiv.
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