Scrambling speed of random quantum circuits

TL;DR

This paper investigates how quickly random quantum circuits can scramble information, demonstrating that circuits with poly(log n) depth achieve strong scrambling and can encode significant quantum information with error correction, resolving a key conjecture in quantum information theory.

Contribution

It proves that random quantum circuits of poly(log n) depth achieve strong scrambling and can encode a linear number of qubits with error correction, resolving an open conjecture related to black hole information paradox.

Findings

Poly(log n) depth circuits satisfy strong scrambling properties.

Such circuits can encode alpha n qubits with up to beta n errors.

Resolved an outstanding conjecture about entanglement generation in quantum circuits.

Abstract

Random transformations are typically good at "scrambling" information. Specifically, in the quantum setting, scrambling usually refers to the process of mapping most initial pure product states under a unitary transformation to states which are macroscopically entangled, in the sense of being close to completely mixed on most subsystems containing a fraction fn of all n particles for some constant f. While the term scrambling is used in the context of the black hole information paradox, scrambling is related to problems involving decoupling in general, and to the question of how large isolated many-body systems reach local thermal equilibrium under their own unitary dynamics. Here, we study the speed at which various notions of scrambling/decoupling occur in a simplified but natural model of random two-particle interactions: random quantum circuits. For a circuit representing the dynamics generated by a local Hamiltonian, the depth of the circuit corresponds to time. Thus, we consider the depth of these circuits and we are typically interested in what can be done in a depth that is sublinear or even logarithmic in the size of the system. We resolve an outstanding conjecture raised in the context of the black hole information paradox with respect to the depth at which a typical quantum circuit generates an entanglement assisted encoding against the erasure channel. In addition, we prove that typical quantum circuits of poly(log n) depth satisfy a stronger notion of scrambling and can be used to encode alpha n qubits into n qubits so that up to beta n errors can be corrected, for some constants alpha, beta > 0.