Controlled jump in the Clifford hierarchy

• URL: http://arxiv.org/abs/2602.22201v1
• Date: Wed, 25 Feb 2026 18:51:04 GMT
• Title: Controlled jump in the Clifford hierarchy
• Authors: Yichen Xu, Xiao Wang,
• Abstract summary: We develop a simple and systematic route to higher levels of the qubit Clifford hierarchy by coherently controlling Clifford operations.<n>Our approach is based on Pauli periodicity, defined for a Clifford unitary $U$ as the smallest integer $mge 1$ such that $U2m$ is a Pauli operator up to phase.
• Score: 8.338159347871155
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We develop a simple and systematic route to higher levels of the qubit Clifford hierarchy by coherently controlling Clifford operations. Our approach is based on Pauli periodicity, defined for a Clifford unitary $U$ as the smallest integer $m\ge 1$ such that $U^{2^{m}}$ is a Pauli operator up to phase. We prove a sharp controlled-jump rule showing that the controlled gate $CU$ lies strictly in level $m+2$ of the hierarchy, and equivalently that $CU$ lies in level $k$ if $U^{2^{k-2}}$ is Pauli while no smaller positive power of $U$ is Pauli. We further quantify the resources required to realize large level jumps in the Clifford hierarchy by proving an essentially tight upper bound on Pauli periodicity as a function of the number of qubits, which implies that accessing high hierarchy levels through controlled Cliffords requires a number of target qubits that grows exponentially with the desired level. We complement this limitation with explicit infinite families of Pauli-periodic Cliffords whose controlled versions achieve asymptotically optimal jumps. As an application, we propose a protocol for preparing logical catalyst states that enable logical $Z^{1/2^k}$ phase gates via phase kickback from a single jumped Clifford.

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