Exact Quantum Algorithms for Quantum Phase Recognition: Renormalization Group and Error Correction

We explore the relationship between renormalization-group (RG) flow and error correction by constructing quantum algorithms that exactly recognize one-dimensional symmetry-protected topological (SPT) phases protected by finite internal Abelian symmetries. For each SPT phase, our algorithm runs a quantum circuit, which emulates RG flow: an arbitrary input ground-state wave function in the phase is mapped to a unique minimally entangled reference state, thereby allowing for efficient phase identification. This construction is enabled by viewing a generic input state in the phase as a collection of coherent “errors” applied to the reference state, and engineering a quantum circuit to efficiently detect and correct such errors. Importantly, the error-correction threshold is proven to coincide exactly with the phase boundary. We discuss the implications of our results in the context of condensed-matter physics, machine learning, and near-term quantum algorithms.