Unscrambling Quantum Information with Clifford decoders

TL;DR

This work demonstrates that even without previous knowledge of the internal dynamics, information can be efficiently decoded from an unknown scrambler by monitoring the outgoing information of a local subsystem.

Abstract

Quantum information scrambling is a unitary process that destroys local correlations and spreads information throughout the system, effectively hiding it in nonlocal degrees of freedom. In principle, unscrambling this information is possible with perfect knowledge of the unitary dynamics [B. Yoshida and A. Kitaev, arXiv:1710.03363.]. However, this Letter demonstrates that even without previous knowledge of the internal dynamics, information can be efficiently decoded from an unknown scrambler by monitoring the outgoing information of a local subsystem. Surprisingly, we show that scramblers with unknown internal dynamics, which are rapidly mixing but not fully chaotic, can be decoded using Clifford decoders. The essential properties of a scrambling unitary can be efficiently recovered, even if the process is exponentially complex. Specifically, we establish that a unitary operator composed of $t$ non-Clifford gates admits a Clifford decoder up to $t\le n$.

Figures (1)

• Figure 1: (a) diagrammatic representation of $\ket{\Psi_V}$, where the upward direction represents the progression of time. The steps of the decoding algorithm generating $\ket{\Psi}_V$ are: (1) parallel application of $U_t$ (obtaining so $\ket{\Psi_t}$, diagrammatically shown in the purple box) and of the decrypter $V^\prime$ on the initial state $\ket{RA}\ket{BB^\prime}\ket{A^\prime R^\prime}$ (2) application of $\mathcal{D}^{\dag}$ (obtaining $\mathcal{D}\ket{\Psi}_t$, as diagrammatically shown in the violet box) and $\mathcal{D}^{T}$. This process dumps the stabilizer entropy onto the $F, F^\prime$ subspaces. (3) The final steps involve the application of $\mathcal{R}^*$ and of a projective measurement on $DD^\prime$. Panel (b) provides an example of a doped random Clifford circuit $U_t$, whereas panel (c) displays the circuit $\mathcal{D}^{\dagger}U_t$, where $\mathcal{D}$ is a diagonalizer for the circuit $U_t$ given in panel (b). Panel (d) illustrates the adjoint action of both circuits on the generators of the group $\mathbb{P}_E$. The circuit $U_t$ shown in panel (b) only preserves the generator $IIZII$, transforming it into another Pauli operator, whereas the adjoint action of $\mathcal{D}^\dagger U_t$ preserves all generators, showing how the diagonalizer can move non-Cliffordness away from the subsystem of interest.