Homotopy classification of loops of Clifford unitaries (Q6585688)

The paper under review explores the classification of topological phases in many-body quantum systems, specifically in multi-dimensional lattices of qudits. Topological phases are a distinct category of phases in condensed matter physics characterised by their robustness to local perturbations. Here, topological phases are understood as the path-connected components of the space of gapped local Hamiltonians, or, more precisely, as the lift of these components to the group of unitary transformations acting on this space. The authors restrict their analysis to unitary dynamics in a framework pertinent to quantum computing, namely, Clifford quantum cellular automata (QCA), i.e., unitary automorphisms that preserve local Pauli operators and remain invariant under lattice translations.\N\NPrevious work [\textit{J. Haah}, ``Topological phases of unitary dynamics: classification in Clifford category'', Preprint, \url{arXiv:2205.09141}] fully classified the topological phases of Clifford unitary dynamics on lattices of prime-dimensional qudits, where two Clifford QCA are considered topologically equivalent if they are equal up to lattice translations and layers of Clifford gates. A key ingredient in establishing this classification is the abelianisation of the Pauli group over the lattice of qudits, which gives rise to the so-called Pauli module -- a free module over the Laurent polynomials, equipped with an antihermitian form. This approach enables the study of Clifford unitary dynamics through Hermitian K-theory (or L-theory). Leveraging dimensional descent [\textit{A. A. Ranicki}, Proc. Lond. Math. Soc. (3) 27, 101--125, 126--158 (1973; Zbl 0269.18009); \textit{S. P. Novikov}, Math. USSR, Izv. 4, 257--292 (1971; Zbl 0216.45003)], the topological phases can ultimately be expressed in terms of the L-groups of prime fields.\N\NThis paper provides a new homotopical interpretation of topological phases of Clifford unitary dynamics. By recalling that path components of stabiliser Hamiltonians correspond to isotropic submodules of the Pauli module, the authors first identify Clifford QCA, up to lattice translations, with Lagrangian submodules of the Pauli module on which the Clifford QCA act transitively. By employing Sturm sequences [\textit{J. Barge} and \textit{J. Lannes}, Suites de Sturm, indice de Maslov et périodicité de Bott. Basel: Birkhäuser (2008; Zbl 1151.55001)], they then demonstrate that, under this identification, the topological phases of Clifford unitary dynamics correspond to the path-connected components of the Lagrangian Grassmannian.\N\NLastly, the authors extend their analysis to topological phases of periodic one-parameter families of Clifford QCA, referred to as Clifford Floquet circuits, thus broadening the scope to time-dependent quantum systems. Relying on the fundamental theorem of Hermitian K-theory [\textit{M. Karoubi}, Ann. Math. (2) 112, 259--282 (1980; Zbl 0483.18008)] and its interpretation by Barge and Lannes through the Maslov index, they calculate the algebraic fundamental group of the infinite Lagrangian Grassmannian associated to a \(d\)-dimensional lattice of \(p\)-dimensional qudits, for \(d\) equal to 0, 1, 2, 3, and 4, and with \(p\) an odd prime. The computations confirm, for these specific values of \(d\), a general correspondence between \(d\)-dimensional QCA and \((d + 1)\)-dimensional Floquet circuits.