The Quantum Dance of Rings on a Doughnut express
Each qubit has its own little ring, freely sliding on the doughnut. All quantum logic boils down to smooth movements: shift, flip, connect. Entanglement, for example, is when two doughnuts stick together, and the rings on them begin to spin synchronously, like figure skaters in pairs skating.
This approach builds on the works of Schwinger and Jordan. But the most astonishing surprise: even after a full loop around the doughnut, the ring doesn't exactly return to its initial state — it acquires a barely perceptible shift, a memory of the journey. This subtle geometric phase is already used in experiments, and in the future it will become the basis for quantum sensors capable of detecting gravitational waves.
Original abstract
as written by the authors arXiv:2602.15080
In this work, we investigate the geometry of quantum logic gates within the holomorphic representation of quantum mechanics. We begin by embedding the physical qubit subspace into the space of holomorphic functions that are homogeneous of degree one in each Schwinger boson pair $$(z_{a_j}, z_{b_j})$$. Within this framework, we derive explicit closed-form differential operator representations for a universal set of quantum gates, including the Pauli operators, Hadamard, CNOT, CZ, and SWAP, and demonstrate that they preserve the physical subspace exactly. Restricting to unit-magnitude variables ($$|z| = 1$$) reveals a toroidal space $$\mathbb{T}^{2N}$$, on which quantum gates act as canonical transformations: Pauli operators generate Hamiltonian flows, the Hadamard gate induces a nonlinear automorphism, and entangling gates produce correlated diffeomorphisms that couple distinct toroidal factors. Beyond the torus, the full Segal--Bargmann space carries a natural K\"ahler geometry that governs amplitude dynamics. Entanglement is geometrically characterized via the Segre embedding into complex projective space, while topological protection arises from the $$U(1)^N$$ fiber bundle structure associated with the Jordan--Schwinger constraint.
The geometric shift after a full loop — an analogue of the Berry phase — is already used in prototypes of quantum gyroscopes and gravitational antennas.