SPICE Workshop on Quantum materials and quantum information science May 19th - 21st, 2026
Eric Akkermans
A unified topological perspective on quantum entanglement is presented, connecting topological defects in two-dimensional materials with the classification of two-qubit Hamiltonians and the geometric phases of quantum gates. The central insight is that the tenfold classification of condensed matter operates equally well on finite-dimensional Hilbert spaces: for two-qubit systems, topology becomes a necessary condition for entanglement, and the space of time-reversal symmetric Hamiltonians decomposes into five disconnected topological phases (ℤ₅), with Bell states appearing as zero-dimensional analogs of topological edge states at phase boundaries. A topological sum rule equates the winding number of any two-qubit gate to the sum of Berry phases over a complete basis, providing a robust, tomography-free signature of entanglement capability. These ideas are illustrated using the framework with graphene vacancies protected by an index theorem, and with experimentally distinguishable geometric phases for two implementations of the SWAP gate relevant to transmon and spin-qubit platforms.