A Butterfly Effect in Encoding-Decoding Quantum Circuits

TL;DR

This paper explores how tiny noise in quantum encoding-decoding circuits can cause large-scale information scrambling, revealing a butterfly effect in quantum systems, supported by analytic and numerical results.

Contribution

It introduces an analytic expression for the $ ext{A}$-OTOC in noisy quantum circuits and demonstrates a butterfly effect where minimal noise leads to macroscopic scrambling.

Findings

Infinitesimal noise causes macroscopic scrambling in the thermodynamic limit.

Derived an explicit formula for the $ ext{A}$-OTOC depending on system size and noise.

Numerical simulations suggest the butterfly effect may occur in broader circuit classes.

Abstract

The study of information scrambling has profoundly deepened our understanding of many-body quantum systems. Much recent research has been devote to understanding the interplay between scrambling and decoherence in open systems. Continuing in this vein, we investigate scrambling in a noisy encoding-decoding circuit model. Specifically, we consider an $L$-qubit circuit consisting of a Haar-random unitary, followed by noise acting on a subset of qubits, and then by the inverse unitary. Scrambling is measured using the bipartite algebraic out-of-time-order correlator ($\mathcal{A}$-OTOC), which allows us to track information spread between extensively sized subsystems. We derive an analytic expression for the $\mathcal{A}$-OTOC that depends on system size and noise strength. In the thermodynamic limit, this system displays a \textit{butterfly effect} in which infinitesimal noise induces macroscopic information scrambling. We also perform numerical simulations while relaxing the condition of Haar-randomness, which preliminarily suggest that this effect may manifest in a larger set of circuits.