Skip to content
DraconiticResearch
Search Sources API
206,690 studies31 sourcesUpdated 22 min ago
PreprintCC BY-NC-SA 4.0arXiv

How many switches does two-Hamiltonian universality require?

  • Matas Garbaravičius
  • Michael M. Wolf
Open at sourcePDFDOICite

Abstract

Any pair of sufficiently generic Hamiltonians on ℂd generates the full unitary group, i.e., any W∈ U(d) can be synthesized to arbitrary precision by alternating between the two evolutions for suitably chosen durations. For any given pair of Hamiltonians, the minimal number of switches required to cover the entire group is at least d and unbounded in general. Despite this, we show that it scales linearly in d in expectation. Assuming that the two spectra are sufficiently generic, we derive constructive upper bounds on the minimal switching number as a function of the change-of-basis matrix between the two eigenbases. In particular, if this matrix has a column whose entries all have equal moduli, as for a complex Hadamard matrix, then 9d switches suffice to reach every unitary. To obtain these results, we construct a short exact representation of any unitary as a product of unitaries that are alternately diagonal in two given bases. All bounds are extended to non-generic spectra and bounded evolution times.

Related

  1. Exact Fusion of Softmax Attention Heads Can Require Exponentially Many Atoms
    Hiroyasu Usami, Keisuke Hara, Ayato Tsuboi +1
    arXivPreprintarXiv licenceFull textStatisticsarXiv
  2. Beamfocusing Versus Airy Beams: Does Antenna Blockage Require Beam Bending?
    Emil Björnson, Vitaly Petrov, Erik G. Larsson
    arXivPreprintarXiv licenceFull textGeneral mathematicsarXiv
  3. Sustained growth of quantum circuit complexity in many-body Hamiltonian dynamics
    Wonjun Lee, Saúl Pilatowsky-Cameo, Soonwon Choi
    arXivPreprintarXiv licenceFull textMathematical physicsQuantum physics & quantum informationarXiv
  4. How many points does an affine algebraic set have in residue classes modulo n ?
    Mihai Prunescu
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryLogic & foundationsarXiv
  5. How Many Sides Does a Tire Have?
    Vanja Stojanović
    arXivPreprintCC BY 4.0Full textGeometry & topologyarXiv

Cite

BibTeXRIS
BibTeX
@misc{garbaravicius2026many,
  title = {{How many switches does two-Hamiltonian universality require?}},
  author = {Matas Garbaravičius and Michael M. Wolf},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.17826},
  eprint = {2610.17826},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.17826},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - UNPB
TI  - How many switches does two-Hamiltonian universality require?
AU  - Matas Garbaravičius
AU  - Michael M. Wolf
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2610.17826
UR  - https://arxiv.org/abs/2610.17826
AB  - Any pair of sufficiently generic Hamiltonians on ℂ^d generates the full unitary group, i.e., any W∈ U(d) can be synthesized to arbitrary precision by alternating between the two evolutions for suitably chosen durations. For any given pair of Hamiltonians, the minimal number of switches required to cover the entire group is at least d and unbounded in general. Despite this, we show that it scales linearly in d in expectation. Assuming that the two spectra are sufficiently generic, we derive constructive upper bounds on the minimal switching number as a function of the change-of-basis matrix between the two eigenbases. In particular, if this matrix has a column whose entries all have equal moduli, as for a complex Hadamard matrix, then 9d switches suffice to reach every unitary. To obtain these results, we construct a short exact representation of any unitary as a product of unitaries that are alternately diagonal in two given bases. All bounds are extended to non-generic spectra and bounded evolution times.
ER  -