Quantum Barcodes: Persistent Homology for Quantum Phase Transitions
Khyathi Komalan
TL;DR
This work proposes quantum barcodes, a framework that applies persistent homology to classify quantum topological phases by mapping quantum states to a classical data cloud via measurements of a fixed observable set. Barcode features persistently characterize phases, and phase transitions appear as discontinuities in the spectrum of the persistent Dirac operator $B^{\varepsilon,\varepsilon'}_k$ and as changes in persistent Betti numbers $\beta_k^{\varepsilon \to \varepsilon'}$, demonstrated concretely in the 1-D SSH model. The framework provides a formal link between persistent homology and quantum topology, offering a complementary topological classification tool with potential implications for quantum computation, phase diagram analysis, and robust experimental signatures. By establishing stability, unitary-invariance, and information completeness of quantum barcodes, the approach opens avenues for efficient phase diagrams and new insights into quantum phases beyond traditional invariant-based methods.
Abstract
We introduce "quantum barcodes," a theoretical framework that applies persistent homology to classify topological phases in quantum many-body systems. By mapping quantum states to classical data points through strategic observable measurements, we create a "quantum state cloud" analyzable via persistent homology techniques. Our framework establishes that quantum systems in the same topological phase exhibit consistent barcode representations with shared persistent homology groups over characteristic intervals. We prove that quantum phase transitions manifest as significant changes in these persistent homology features, detectable through discontinuities in the persistent Dirac operator spectrum. Using the SSH model as a demonstrative example, we show how our approach successfully identifies the topological phase transition and distinguishes between trivial and topological phases. While primarily developed for symmetry-protected topological phases, our framework provides a mathematical connection between persistent homology and quantum topology, offering new methods for phase classification that complement traditional invariant-based approaches.