Efficient decoding for the Hayden-Preskill protocol

TL;DR

This paper introduces two efficient decoding algorithms for the Hayden-Preskill protocol, enabling quantum state recovery from Hawking radiation with probabilistic and deterministic methods, leveraging scrambling and OTOC decay.

Contribution

The paper presents novel decoding procedures combining teleportation and Grover's search for the Hayden-Preskill protocol, improving efficiency and providing explicit algorithms.

Findings

Probabilistic decoding success probability scales as 1/d_A^2.

Deterministic decoding complexity is O(d_A * C).

Algorithms utilize scrambling and decay of out-of-time-order correlators.

Abstract

We present two particular decoding procedures for reconstructing a quantum state from the Hawking radiation in the Hayden-Preskill thought experiment. We work in an idealized setting and represent the black hole and its entangled partner by $n$ EPR pairs. The first procedure teleports the state thrown into the black hole to an outside observer by post-selecting on the condition that a sufficient number of EPR pairs remain undisturbed. The probability of this favorable event scales as $1/d_{A}^2$, where $d_A$ is the Hilbert space dimension for the input state. The second procedure is deterministic and combines the previous idea with Grover's search. The decoding complexity is $\mathcal{O}(d_{A}\mathcal{C})$ where $\mathcal{C}$ is the size of the quantum circuit implementing the unitary evolution operator $U$ of the black hole. As with the original (non-constructive) decoding scheme, our algorithms utilize scrambling, where the decay of out-of-time-order correlators (OTOCs) guarantees faithful state recovery.