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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.

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Paper 2

Quantum Calculus of Fibonacci Divisors and Fermion-Boson Entanglement for Infinite Hierarchy of N = 2 Supersymmetric Golden Oscillators

Oktay K. Pashaev

Year

2024

Journal

arXiv preprint

DOI

arXiv:2410.04169

arXiv

2410.04169

The quantum calculus with two bases, as powers of the Golden and the Silver ratio, relates Fibonacci divisor derivative with Binet formula of Fibonacci divisor number operator, acting in Fock space of quantum states.It provides a tool to study the hierarchy of Golden oscillators with energy spectrum in form of Fibonacci divisor numbers. We generalize this model to supersymmetric number operator and corresponding Binet formula for supersymmetric Fibonacci divisor number operator. The operator determines the Hamiltonian of hierarchy of supersymmetric Golden oscillators, acting in fermion-boson Hilbert space and belonging to N=2 supersymmetric algebra. The eigenstates of the super Fibonacci divisor number operator are double degenerate and can be characterized by a point on the super-Bloch sphere. By the supersymmetric Fibonacci divisor annihilation operator, we construct the hierarchy of supersymmetric coherent states as eigenstates of this operator. Entanglement of fermions with bosons in these states is calculated by the concurrence, represented by the Gram determinant and hierarchy of Golden exponential functions. We show that the reference states and corresponding von Neumann entropy, measuring fermion-boson entanglement, are characterized completely by the powers of the Golden ratio. The simple geometrical classification of entangled states by the Frobenius ball and meaning of the concurrence as double area of parallelogram in Hilbert space are given.

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