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 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.
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BibTeX
@misc{cha2026every,
title = {{Every architecture of six two-qubit gates is locally universal on three qubits}},
author = {Hyunho Cha and Jungwoo Lee},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22176},
eprint = {2609.22176},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22176},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Every architecture of six two-qubit gates is locally universal on three qubits
AU - Hyunho Cha
AU - Jungwoo Lee
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.22176
UR - https://arxiv.org/abs/2609.22176
AB - 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 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.
ER -