A Butterfly Effect in Encoding-Decoding Quantum Circuits

TL;DR

This work analyzes scrambling and decoherence in an encoding-decoding quantum circuit formed by a Haar-random unitary, identical noise on a subset of qubits, and the inverse unitary. By deriving an analytic, Haar-averaged expression for the bipartite A-OTOC, the authors reveal a striking butterfly effect: in the thermodynamic limit, infinitesimal noise can induce macroscopic scrambling across extensive subsystems. The main result, $\overline{G(\mathcal{C})} = \left\|\frac{X}{2}\right\|_2^{2k} - \left(\frac{\mathrm{Tr}(X)}{4}\right)^{2k}$, shows a clear distinction between unitary and non-unitary noise and signifies the competition between scrambling and decoherence. Numerical simulations with non-Haar random circuits and local Hamiltonian dynamics suggest the butterfly effect is robust beyond Haar randomness and extends to realistic many-body dynamics, highlighting potential experimental observability and implications for quantum information processing in noisy devices.

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.

Figures (8)

• Figure 1: Circuit diagram of $\mathcal{C}$.
• Figure 2: $\overline{G(\mathcal{C})}$ vs. $\theta$ for small values of $k$. As $k$ increases, $\overline{G(\mathcal{C})}$ approaches the step function $H(\theta)$.
• Figure 3: $\overline{G(\mathcal{C})}$ vs. $p$ for small values of $k$. As $k$ increases, $\overline{G(\mathcal{C})}$ flattens and eventually goes to $0$.
• Figure 4: Depolarizing noise strength and unitary rotation angle vs. $\overline{G(\mathcal{C})}$. The "EP" curves refer to repeated dual-unitary brickwork circuits with corresponding operator entangling power. The Haar-random circuits match the analytic results very closely. Increasing $E_p$ in dual-unitary brickwork circuits increases closeness to analytic results. The standard deviations for the EP curves under unitary noise are substantial enough for $\theta$ near $\pi/2$ that the non-monotonicity cannot be deemed significant. We omit the error bars for clarity.
• Figure 5: Plots of deviation of dual-unitary brickworks from analytic results at argmax noise parameter values, for depolarizing and unitary noise respectively. Both curves seem to reflect a power law scaling of this difference with respect to $E_p$.

Theorems & Definitions (13)

• Definition 1
• Proposition 1
• Proposition 2
• Definition 2: Natural representation
• Lemma 4.1: $\overline{G(\mathcal{C})}$ for finite qubit chain
• Theorem 4.2: $\overline{G(\mathcal{C})}$ for infinite qubit chains
• Corollary 4.2.1: Maximum value for extensively scaling noise
• proof
• Corollary 4.2.2: $\mathcal{A}$-OTOC for extensively scaling non-unitary noise
• proof