### Abstract

We present a theoretical analysis of the paradigm of encoded universality, using a Lie algebraic analysis to derive specific conditions under which physical interactions can provide universality. We discuss the significance of the tensor product structure in the quantum circuit model and use this to define the conjoining of encoded qudits. The construction of encoded gates between conjoined qudits is discussed in detail. We illustrate the general procedures with several examples from exchange-only quantum computation. In particular, we extend our earlier results showing universality with the isotropic exchange interaction to the derivation of encoded universality with the anisotropic exchange interaction, i.e., to the XY model. In this case the minimal encoding for universality is into qutrits rather than into qubits as was the case for isotropic (Heisenberg) exchange. We also address issues of fault-tolerance, leakage and correction of encoded qudits.

Original language | English (US) |
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Pages (from-to) | 33-55 |

Number of pages | 23 |

Journal | Quantum Information and Computation |

Volume | 1 |

Issue number | SUPPL. 1 |

State | Published - Dec 2001 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Statistical and Nonlinear Physics
- Nuclear and High Energy Physics
- Mathematical Physics
- Physics and Astronomy(all)
- Computational Theory and Mathematics

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## Cite this

*Quantum Information and Computation*,

*1*(SUPPL. 1), 33-55.