The Clifford+T quantum computing gate library for single qubit gates can create all unitary matrices that are generated by the group 〈H, T〉. The matrix T can be considered the fourth root of Pauli Z, since T4= Z or also the eighth root of the identity I. The Hadamard matrix H can be used to translate between the Pauli matrices, since (HTH)4gives Pauli X. We are generalizing both these roots of the Pauli matrices (or roots of the identity) and translation matrices to investigate the groups they generate: the so-called Pauli root groups. In this work we introduce a formalization of such groups, study finiteness and infiniteness properties, and precisely determine equality and subgroup relations.
As problems in computer science become increasingly more computationally intense, researchers have begun to examine other methods of computation. One of the largest new models of computation is the quantum computer. Based on the concepts of quantum mechanics, quantum computers process bits of quantum information, or qubits, using quantum gates. So far some impressive results have been obtained using quantum computers on current problems (e.g. search problems). Many quantum algorithms make use of the Hadamard gate, and a few use a more general form of that gate. In this paper we present a generalized quantum gate that can be shown to encapsulate the standard gates used in quantum algorithms and quantum information processing: Pauli X, Y, and Z, Hadamard, T phase shift, S phase shift, and identity. This generalized gate may also lead to new, interesting, and useful gates for quantum computation.
Many well-known and well-studied four by four universal quantum logic gates in the literature are of a specific form, the so called eight-vertex form \eqref{8vertexform} \cite{kaufman etal 05-1,kaufman etal 05-2}, or {\it similar} to it. We present a formalism for universal quantum logic gates of such a form. First, we provide explicit formulas in terms of matrix entries, which are the necessary and sufficient conditions for such a matrix to be a solution to the Yang-Baxter equation \eqref{byb}. Then, combining this with the conditions needed for being unitary \eqref{unitarycond} and being entangling \eqref{entanglingcond}, we give a full description of entangling unitary solutions to the Yang-Baxter equation (hence, universal quantum logic gates) of such a specific form. We investigate in detail all the possible cases where some of the eight main entries might or might not be zero.
A unitary matrix is shown to be a product of certain {\em basic} unitary
matrices and the product is unique up to order. A basic unitary matrix itself
is a product of {\em minimal basic} matrices. A unitary
n×n matrix can
be expressed as a product of at most
n basic matrices. A basic matrix is
defined in terms of an idempotent matrix; the idempotent matrix used in the
definition of a minimal basic unitary matrix is a {\em density} matrix. This gives builders for unitary matrices. Quantum logic gates are represented
by unitary matrices thus giving unique building material for quantum logic
gates.
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
(zaj,zbj). 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
T2N, 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 Kähler 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.