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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

Asymptotic Replacement for Quantum Channel Products with Applications to Inhomogeneous Matrix Product States

Lubashan Pathirana

Year

2026

Journal

arXiv preprint

DOI

arXiv:2605.00157

arXiv

2605.00157

We develop a product-level trace-Dobrushin theory for finite-dimensional quantum channel products and apply it to deterministic and stationary random inhomogeneous matrix product states in left-canonical CPTP gauge. For a product of channels, the centered trace-Dobrushin coefficient quantifies the residual dependence on the input state, and its decay is the criterion for trace-norm forgetting. In the deterministic setting, this decay is equivalent to asymptotic replacement by a moving replacement channel. For two-sided products, pullback forgetting produces a unique boundary state, which determines the canonical replacement family. For stationary random CPTP cocycles, submultiplicativity of the product coefficient yields a trace-Dobrushin Lyapunov exponent. We prove that the almost sure negativity of this exponent is equivalent to quenched trace-norm memory loss and gives exponential forward and pullback convergence to a unique dynamically stationary random replacement channel. When the \(\varrho\)-mixing profile of the channel environment tends to zero, we obtain annealed super-polynomial estimates, while independence gives annealed exponential estimates. Finally, we transfer these estimates to inhomogeneous matrix product states whose auxiliary transfer maps are CPTP. These channel estimates transfer to deterministic and stationary random inhomogeneous MPS, giving infinite-volume limits of trace-closed finite-volume states, quantitative boundary stability, and correlation bounds governed by the same auxiliary product coefficients.

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