[Paper Review] Efficient decoding for the Hayden-Preskill protocol

Beni Yoshida, Alexei Kitaev|arXiv (Cornell University)|Oct 10, 2017

Cellular Automata and ApplicationsComputer Science13 references76 citations

TL;DR

The paper presents two decoding procedures to reconstruct a quantum state from Hawking radiation in the Hayden-Preskill setup: a probabilistic postselected teleportation method and a deterministic Grover-search-based decoder, both relying on scrambling and OTOCs.

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.

Motivation & Objective

• Demonstrate information-theoretically possible Hayden-Preskill decoding under a simplified model.
• Provide two concrete decoding procedures (probabilistic and deterministic) and analyze their fidelity and complexity.
• Relate decoding performance to scrambling behavior and out-of-time-order correlators (OTOCs).
• Discuss assumptions, limitations, and potential extensions to more realistic thermal states.

Proposed method

• Use a simplified model where the black hole and partner are n EPR pairs and the environment is described by a Haar-random unitary U.
• Define decoding fidelity parameter delta and relate it to Delta and OTOCs.
• Propose a probabilistic decoder that uses postselection on an EPR projection to teleport the diary to R′, achieving high fidelity when scrambling is near-perfect.
• Propose a deterministic decoder based on Grover’s search that applies a sequence of reflections (W and W̃A) to amplify the target state, achieving fidelity 1−O(delta).
• Express decoding success bounds in terms of Delta, delta, and dA, dR, dD, linking to the Rényi-2 mutual information I^(2)(R, DB′).
• Explain how scrambling (via almost-perfect OTOCs) guarantees faithful state recovery and bound delta by dA dR / dD^2.

Experimental results

Research questions

• RQ1 How can a quantum state be reconstructed from Hayden-Preskill Hawking radiation under scrambling assumptions?
• RQ2 What are the concrete decoding procedures (probabilistic vs. deterministic) and their fidelity/complexity trade-offs?
• RQ3 How do OTOCs and scrambling quantify decoding efficiency and fidelity bounds?
• RQ4 What are the limitations when moving from idealized Haar-random evolution to more realistic thermal states?
• RQ5 How does the decoding complexity scale with input/output dimensions and the black hole evolution circuit size?

Key findings

• Two decoding procedures are constructed: a probabilistic postselected teleportation method with success probability ~ 1/(dA dR), and a deterministic Grover-search-based decoder with complexity ∼ O(√(dA dR) C).
• Fidelity of the probabilistic decoder approaches 1−O(delta) when the evolution is almost perfectly scrambling and dD ≫ √(dAdR).
• Deterministic decoder achieves high fidelity by iterating Grover-style reflections; its analysis uses a Grover rotation with subspaces labeled by eigencomponents of a key projector (Pi).
• Delta is bounded by Δ ≤ 1/(dA dR) + 1/dD^2, and delta = dA dR Δ − 1 satisfies delta ≤ dA dR / dD^2, leading to fidelity bounds.
• Fidelity bounds extend to factorizable inputs/outputs and depend on the embedding Xi and the state rhoA, with generalized delta-like quantities (e.g., δ̂) under realistic extensions.