Combinatorial quantum method in 3-dimensional topology

Mauro Carfora B, Annalisa Marzuoli

Adam Wesołowski, Dimitrios Thanos, Daniel Leykam +1 more

Many datasets encountered across a wide range of domains possess rich geometric and topological structure that is difficult to capture using conventional vector-based representations. Quantum machine learning offers the possibility of processing high-dimensional data in Hilbert spaces, but its practical success depends critically on how classical data is encoded into quantum states. We introduce \emph{quantum topological data encoding} (QTDE), a general framework for encoding topological information into quantum states via topology-driven quantum evolution. Our method generalises an existing topology-driven quantum encoding framework to higher-dimensional data. We test the proposed method on clique-complexes classification tasks, and provide preliminary evidence that topology-driven quantum representations can capture discriminative information beyond that available through direct comparisons of classical topological descriptors. The proposed quantum representations consistently outperform a baseline based on direct comparisons of the combinatorial Laplacians describing the underlying topological structure. We indicate several areas of application where the framework can be used to provide a more efficient and reliable data representation.

Mauro Carfora B, Annalisa Marzuoli

Rio Honda, Katsuhiro Endo, Yudai Suzuki +3 more

ABSTRACT Quantum annealing is emerging as a practical tool for large‐scale combinatorial optimization. We map continuum displacements and element densities to binary variables, turning both deformation analysis and topology optimization into quadratic unconstrained binary optimization (QUBO) problems that run on today's annealers. Two formulations are compared: one derived from the stiffness equation and another from the minimum total potential energy. Benchmarks show the energy‐based QUBO matches finite‐element displacements and stresses more closely and avoids the rapid term‐count growth that slows the stiffness‐based form. The numerical demonstrations reported here include deformation analysis of a 100‐element plate model and topology optimization of cantilever beams discretized with up to 60 triangular elements or 32 rectangular elements. These examples are intended as proof‐of‐concept demonstrations under current hardware and encoding limitations rather than as large‐scale industrial benchmarks. Leveraging this model, a single QUBO embedding density variables–implemented with combinatorial random‐number sums and a quadratic volume penalty–yields two‐dimensional beam layouts that are quantitatively compared with those obtained using the classical Optimality Criteria (OC) method.

Claudio Chamon, Dmitry Green, Andrew J. Kerman

We propose a superconducting quantum circuit based on a general symmetry principle-combinatorial gauge symmetry-designed to emulate topologically ordered quantum liquids and serve as a foundation for the construction of topological qubits.The proposed circuit exhibits rich features: in the classical limit of large capacitances its ground state consists of two superimposed loop structures; one is a crystal of small loops containing disordered U(1) degrees of freedom, and the other is a gas of loops of all sizes associated with Z 2 topological order.We show that these classical results carry over to the quantum case, where phase fluctuations arise from the presence of finite capacitances, yielding Z 2 quantum topological order.A key feature of the exact gauge symmetry is that amplitudes connecting different Z 2 loop states arise from paths having zero classical energy cost.As a result, these amplitudes are controlled by dimensional confinement rather than tunneling through energy barriers.We argue that this effect may lead to larger energy gaps than previous proposals that are limited by such barriers, potentially making it more likely for a topological phase to be experimentally observable.Finally, we discuss how our superconducting circuit realization of combinatorial gauge symmetry can be implemented in practice.

Ravi Shankar