Dynamic syndrome decoder in volume-law phases of hybrid quantum circuits

Phases of matter with volume-law entanglement are frequently observed in quantum circuits and have numerous applications, ranging from deepening our understanding of quantum mechanics to advancements in quantum computing and cryptography. Their capacity to host entangled, complex quantum information is complemented by their ability to efficiently obscure it from quantum measurements through scrambling, reminiscent of quantum error-correction. However, the issue of initial-state decodability has primarily been studied in measurement-only models with area-law phases, which limit the entanglement of the encoded state. In this work, we introduce a class of Clifford circuits in one and two dimensions that feature a decodable volume law phase, allowing for information retrieval in logarithmic circuit depths. We present the Sign-Color Decoder that tracks stabilizers revealing the initial state, akin to monitoring a dynamically-changing syndrome for error-correcting codes. We demonstrate this approach in scenarios where error locations are either known or unknown to the decoder, and we provide new insights about the relationship between the decodability transition and measurement-induced phase transition. We propose that this decodability transition is universal across various settings, including different circuit geometries. Our findings pave the way for using volume law states as encoders with mid-circuit measurements, opening potential applications in quantum error correction and quantum cryptography.