Geometry of Quantum Logic Gates

TL;DR

This paper explores the geometric structure of quantum logic gates using holomorphic representations, revealing their actions as canonical transformations on a toroidal space and characterizing entanglement geometrically.

Contribution

It derives explicit differential operator representations for universal quantum gates within a holomorphic framework and analyzes their geometric and topological properties.

Findings

Quantum gates act as canonical transformations on a toroidal space.

Entanglement is characterized via Segre embedding into complex projective space.

Topological protection arises from the $U(1)^N$ fiber bundle structure.

Abstract

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.