aiqbench analysis
This research establishes precise mathematical criteria for achieving quantum universality using Pauli strings and specific Hamiltonians, offering a rigorous framework for hardware designers to assess gate set sufficiency. By proving universality conditions for systems with limited control, such as the XYZ Heisenberg model with adjacent qubit coupling, the work provides actionable insights for optimizing control sequences in near-term quantum processors. The findings directly address the theoretical constraints of quantum compilation and error correction strategies where gate sets are restricted to Pauli-based operations.
Necessary and Sufficient Conditions for Universal Gates with Pauli Strings and Beyond
arXiv:2606.12096v1 Announce Type: new 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.
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.
8+10 pages, 0 figures. Comments are welcome!
arXiv:2606.12096 [quant-ph]
(or arXiv:2606.12096v1 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2606.12096 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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