Symmetry breaking and error correction in open quantum systems

TL;DR

This paper explores symmetry-breaking transitions in open quantum systems, revealing how different symmetry conditions affect steady states and demonstrating a method for error correction in photonic cat qubits.

Contribution

It characterizes $ ext{Z}_n$ symmetry breaking in open systems and links symmetry-breaking phases to quantum error correction capabilities.

Findings

Weak symmetry-breaking leads to classical steady states.

Strong symmetry-breaking can protect a qubit in the steady state.

Error correction in photonic cat qubits can be exponentially improved.

Abstract

Symmetry-breaking transitions are a well-understood phenomenon of closed quantum systems in quantum optics, condensed matter, and high energy physics. However, symmetry breaking in open systems is less thoroughly understood, in part due to the richer steady-state and symmetry structure that such systems possess. For the prototypical open system---a Lindbladian---a unitary symmetry can be imposed in a "weak" or a "strong" way. We characterize the possible $\mathbb{Z}_n$ symmetry breaking transitions for both cases. In the case of $\mathbb{Z}_2$, a weak-symmetry-broken phase guarantees at most a classical bit steady-state structure, while a strong-symmetry-broken phase admits a partially-protected steady-state qubit. Viewing photonic cat qubits through the lens of strong-symmetry breaking, we show how to dynamically recover the logical information after any gap-preserving strong-symmetric error; such recovery becomes perfect exponentially quickly in the number of photons. Our study forges a connection between driven-dissipative phase transitions and error correction.