We incorporate active and passive quantum error-correcting techniques to protect a set of optical information modes of a continuous-variable quantum information system. Our method uses ancilla modes, entangled modes, and gauge modes (modes in a mixed state) to help correct errors on a set of information modes. A linear-optical encoding circuit consisting of offline squeezers, passive optical devices, feedforward control, conditional modulation, and homodyne measurements performs the encoding. The result is that we extend the entanglement-assisted operator stabilizer formalism for discrete variables to continuous-variable quantum information processing. © 2009 Springer Science+Business Media, LLC.
Noisy monitored quantum circuits have emerged as a versatile and unifying framework connecting quantum many-body physics, quantum information, and quantum computation. In this review, we provide a comprehensive overview of recent advances in understanding the dynamics of such circuits, with an emphasis on their entanglement structure, information-protection capabilities, and noise-induced phase transitions. A central theme is the mapping to classical statistical models, which reveals how quantum noise reshapes dominant spin configurations. This framework elucidates universal scaling behaviors, including the characteristic
q−1/3 entanglement scaling with noise probability
q and distinct timescales for information protection. We further highlight a broad range of constructions and applications inspired by noisy monitored circuits, spanning variational quantum algorithms, classical simulation methods, mixed-state phases of matter, and emerging approaches to quantum error mitigation and quantum error correction. These developments collectively establish noisy monitored circuits as a powerful platform for probing and controlling quantum dynamics in realistic, decohering environments.