Every architecture of six two-qubit gates is locally universal on three qubits

Abstract:We present the \emph{first} analytical determination of the exact accessible dimension for every fixed architecture of arbitrary two-qubit gates on three qubits. A support word represents which pair of qubits each two-qubit gate acts on, and its reduced length is obtained by merging consecutive gates on the same pair. If the reduced length is $r$, then the set of implementable three-qubit unitaries has accessible dimension $d(w)=\min\{63,9r+9\}$. Consequently, six arbitrary two-qubit gates are \emph{necessary and sufficient for local universality}: every fixed architecture reaches a nonempty open subset of $\mathrm{SU}(8)$ when its reduced length is at least six. The result is stronger than existence of one favorable architecture. Every reduced six-slot support word is locally universal, including the alternating nearest-neighbor line $AB,BC,AB,BC,AB,BC$. The upper bound follows from a standard parameter-counting argument. The matching lower bounds are proved by Jacobian certificates in the Pauli basis. Up to qubit relabeling and reversal, there are $22$ reduced architectures of lengths two through six. For each one, a product of rational Pauli rotations yields a nonzero maximal minor modulo the prime $1{,}000{,}003$. Thus, any obstruction to global universality with six two-qubit gates must go beyond parameter counting, connectivity, and differential rank.

arXiv:2609.22176 [quant-ph]

(or arXiv:2609.22176v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.22176 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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