Universal computation with magic Hamiltonians
Abstract
In the conventional theory of quantum computation, universality is discussed in terms of properties of unitary gate sets. In many experimental setups, however, we instead have access to a parametrized set of Hamiltonians, which can be exponentiated for any desired time. Accordingly, we formulate a continuous analogue of universality, where adding one expensive Hamiltonian control to a cheap set of local control operators allows for universal computation. We find several concrete sets of 1- or 2-local Pauli generators S and n-qubit ``magic" Hamiltonians H that together generate the full Lie algebra su(2n), with a special focus on the Ising Hamiltonian J. We propose a method to directly implement any unitary by exponentiating these generator sets, and give bounds on the needed time of application of the magic Hamiltonian H. We find that the number of applications of H needed to approximate any unitary up to accuracy ε scales exponentially in the number of qubits as expected from the standard application of the Solovay-Kitaev theorem. We also find phase transitions in the size of the generated Lie algebras as the parameters of the Ising Hamiltonian are varied. We find an algorithmic application of the Chow/Rashevskii technique to implement unitaries corresponding to iterated commutators.
Related
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BibTeX
@misc{junge2026universal,
title = {{Universal computation with magic Hamiltonians}},
author = {Marius Junge and Jason Pollack and Luke Visser},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14757},
eprint = {2609.14757},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14757},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Universal computation with magic Hamiltonians
AU - Marius Junge
AU - Jason Pollack
AU - Luke Visser
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14757
UR - https://arxiv.org/abs/2609.14757
AB - In the conventional theory of quantum computation, universality is discussed in terms of properties of unitary gate sets. In many experimental setups, however, we instead have access to a parametrized set of Hamiltonians, which can be exponentiated for any desired time. Accordingly, we formulate a continuous analogue of universality, where adding one expensive Hamiltonian control to a cheap set of local control operators allows for universal computation. We find several concrete sets of 1- or 2-local Pauli generators S and n-qubit ``magic" Hamiltonians H that together generate the full Lie algebra su(2^n), with a special focus on the Ising Hamiltonian J. We propose a method to directly implement any unitary by exponentiating these generator sets, and give bounds on the needed time of application of the magic Hamiltonian H. We find that the number of applications of H needed to approximate any unitary up to accuracy ε scales exponentially in the number of qubits as expected from the standard application of the Solovay-Kitaev theorem. We also find phase transitions in the size of the generated Lie algebras as the parameters of the Ising Hamiltonian are varied. We find an algorithmic application of the Chow/Rashevskii technique to implement unitaries corresponding to iterated commutators.
ER -