Decoupling with random quantum circuits

TL;DR

This paper demonstrates that random quantum circuits with nearly optimal gate complexity can achieve decoupling efficiently, significantly improving the understanding of quantum dynamics leading to decoupling.

Contribution

It proves that random quantum circuits with O(n log^2 n) gates can effectively achieve decoupling, reducing the required circuit depth to polylogarithmic scale.

Findings

Decoupling can be achieved with O(n log^2 n) gates.

Decoupling circuits can be implemented in depth O(log^3 n).

Decoupling occurs in polylogarithmic time for all interacting particles.

Abstract

Decoupling has become a central concept in quantum information theory with applications including proving coding theorems, randomness extraction and the study of conditions for reaching thermal equilibrium. However, our understanding of the dynamics that lead to decoupling is limited. In fact, the only families of transformations that are known to lead to decoupling are (approximate) unitary two-designs, i.e., measures over the unitary group which behave like the Haar measure as far as the first two moments are concerned. Such families include for example random quantum circuits with O(n^2) gates, where n is the number of qubits in the system under consideration. In fact, all known constructions of decoupling circuits use \Omega(n^2) gates. Here, we prove that random quantum circuits with O(n log^2 n) gates satisfy an essentially optimal decoupling theorem. In addition, these circuits can be implemented in depth O(log^3 n). This proves that decoupling can happen in a time that scales polylogarithmically in the number of particles in the system, provided all the particles are allowed to interact. Our proof does not proceed by showing that such circuits are approximate two-designs in the usual sense, but rather we directly analyze the decoupling property.