Compare Papers

Paper 1

Quantum Universality in Composite Systems: A Trichotomy of Clifford Resources

Alejandro Borda, Julian Rincon, César Galindo

Year

2025

Journal

arXiv preprint

DOI

arXiv:2512.20787

arXiv

2512.20787

The Clifford group is efficiently classically simulable, and universality is obtained by supplementing it with non-Clifford resources. We determine which single-qudit gates suffice to achieve universality. We show that the structure of such resources is governed by the prime factorization of the qudit dimension $d$. Using the adjoint action on the space of complex trace-zero matrices, we relate density to irreducibility together with an infiniteness criterion, yielding a trichotomy based on the factorization of $d$. When $d$ is prime, any non-Clifford gate generates a dense subgroup of the determinant-one unitaries. If $d$ is a prime power, the adjoint action is reducible, and universality requires gates that couple the resulting invariant subspaces. For composite $d$ with pairwise coprime factors, generalized intra-qudit controlled-NOT gates connecting the factors already suffice. These findings suggest that ``composite architectures'' -- hybrid registers combining incommensurate dimensions -- offer a route to bypass the standard overhead associated with magic-state injection.

Open paper

Paper 2

Geometry and Entanglement of Super-Qubit Quantum States

Oktay K. Pashaev, Aygul Kocak

Year

2024

Journal

arXiv preprint

DOI

arXiv:2410.04361

arXiv

2410.04361

We introduce the super-qubit quantum state, determined by superposition of the zero and the one super-particle states, which can be represented by points on the super-Bloch sphere. In contrast to the one qubit case, the one super-particle state is characterized by points in extended complex plain, equivalent to another super-Bloch sphere. Then, geometrically, the super-qubit quantum state is represented by two unit spheres, or the direct product of two Bloch spheres. By using the displacement operator, acting on the super-qubit state as the reference state, we construct the super-coherent states, becoming eigenstates of the super-annihilation operator, and characterized by three complex numbers, the displacement parameter and stereographic projections of two super-Bloch spheres. The states are fermion-boson entangled, and the concurrence of states is the product of two concurrences, corresponding to two Bloch spheres. We show geometrical meaning of concurrence as distance from point-state on the sphere to vertical axes - the radius of circle at horizontal plane through the point-state. Then, probabilities of collapse to the north pole state and to the south pole state are equal to half-distances from vertical coordinate of the state to corresponding points at the poles. For complimentary fermion number operator, we get the flipped super-qubit state and corresponding super-coherent state, as eigenstate of transposed super-annihilation operator. The infinite set of Fibonacci oscillating circles in complex plain, and corresponding set of quantum states with uncertainty relations as the ratio of two Fibonacci numbers, and in the limit at infinity becoming the Golden Ratio uncertainty, is derived.

Open paper