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

TL;DR

This paper presents quantum algorithms that leverage renormalization group flow and error correction principles to precisely identify 1D symmetry-protected topological phases, linking phase boundaries with error correction thresholds.

Contribution

The authors develop exact quantum algorithms that emulate RG flow for phase recognition, connecting error correction with phase boundaries in 1D SPT phases.

Findings

Quantum algorithms can recognize SPT phases via RG flow emulation.

Error correction thresholds match phase boundaries.

Efficient phase identification is achieved through coherent error detection.

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.