Necessary and Sufficient Conditions for Universal Gates with Pauli Strings and Beyond

• URL: http://arxiv.org/abs/2606.12096v1
• Date: Wed, 10 Jun 2026 13:54:37 GMT
• Title: Necessary and Sufficient Conditions for Universal Gates with Pauli Strings and Beyond
• Abstract summary: We show a sufficient condition for universality based on the Pauli-basis expansion of the Hamiltonian.<n>We prove two corollaries: (i) a necessary and sufficient condition for the universality of a general Hamiltonian given arbitrary single-qubit control on all qubits, and (ii) the universality of an XYZ Heisenberg Hamiltonian with local control of just two adjacent qubits.
• Score: 0.41510732880787476
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Any quantum computation consists of a sequence of unitary evolutions described by a finite set of Hamiltonians. For the case where this set consists of only products of Pauli operators, known as Pauli strings, we provide a necessary and sufficient condition for it to generate $\mathfrak{su}(2^n)$, i.e., to be universal for quantum computation on $n$ qubits. When combining Pauli strings with a general Hamiltonian, we show a sufficient (and in certain circumstances even necessary) condition for universality based on the Pauli-basis expansion of the Hamiltonian. As an application of these results, we prove two corollaries: (i) a necessary and sufficient condition for the universality of a general Hamiltonian given arbitrary single-qubit control on all qubits, and (ii) the universality of an XYZ Heisenberg Hamiltonian with local control of just two adjacent qubits.

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