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

Abstract

We explore the relationship between renormalization group (RG) flow and error correction by constructing quantum algorithms that exactly recognize 1D 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 wavefunction 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.

10 pages + appendices v2: extended discussion on RG convergence; added refs

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Cited by in corpus (4)

• Quantum Convolutional Neural Network for Phase Recognition in Two Dimensions
• Quantum circuit complexity and unsupervised machine learning of topological order
• Non-onsite symmetries and quantum teleportation in split-index matrix product states
• Quantum phase classification via partial tomography-based quantum hypothesis testing