### Abstract

Universal quantum computation on decoherence-free subspaces and subsystems (DFSs) is examined with particular emphasis on using only physically relevant interactions. A necessary and sufficient condition for the existence of decoherence-free (noiseless) subsystems in the Markovian regime is derived here for the first time. A stabilizer formalism for DFSs is then developed which allows for the explicit understanding of these in their dual role as quantum error correcting codes. Conditions for the existence of Hamiltonians whose induced evolution always preserves a DFS are derived within this stabilizer formalism. Two possible collective decoherence mechanisms arising from permutation symmetries of the system-bath coupling are examined within this framework. It is shown that in both cases universal quantum computation which always preserves the DFS (natural fault-tolerant computation) can be performed using only two-body interactions. This is in marked contrast to standard error correcting codes, where all known constructions using one- or two-body interactions must leave the code space during the on-time of the fault-tolerant gates. A further consequence of our universality construction is that a single exchange Hamiltonian can be used to perform universal quantum computation on an encoded space whose asymptotic coding efficiency is unity. The exchange Hamiltonian, which is naturally present in many quantum systems, is thus asymptotically universal.

Original language | English (US) |
---|---|

Article number | 042307 |

Pages (from-to) | 1-29 |

Number of pages | 29 |

Journal | Physical Review A - Atomic, Molecular, and Optical Physics |

Volume | 63 |

Issue number | 4 |

DOIs | |

State | Published - Jan 1 2001 |

### Fingerprint

### ASJC Scopus subject areas

- Atomic and Molecular Physics, and Optics

### Cite this

*Physical Review A - Atomic, Molecular, and Optical Physics*,

*63*(4), 1-29. [042307]. https://doi.org/10.1103/PhysRevA.63.042307

**Theory of decoherence-free fault-tolerant universal quantum computation.** / Kempe, Julia; Bacon, D.; Lidar, D. A.; Whaley, K. B.

Research output: Contribution to journal › Article

*Physical Review A - Atomic, Molecular, and Optical Physics*, vol. 63, no. 4, 042307, pp. 1-29. https://doi.org/10.1103/PhysRevA.63.042307

}

TY - JOUR

T1 - Theory of decoherence-free fault-tolerant universal quantum computation

AU - Kempe, Julia

AU - Bacon, D.

AU - Lidar, D. A.

AU - Whaley, K. B.

PY - 2001/1/1

Y1 - 2001/1/1

N2 - Universal quantum computation on decoherence-free subspaces and subsystems (DFSs) is examined with particular emphasis on using only physically relevant interactions. A necessary and sufficient condition for the existence of decoherence-free (noiseless) subsystems in the Markovian regime is derived here for the first time. A stabilizer formalism for DFSs is then developed which allows for the explicit understanding of these in their dual role as quantum error correcting codes. Conditions for the existence of Hamiltonians whose induced evolution always preserves a DFS are derived within this stabilizer formalism. Two possible collective decoherence mechanisms arising from permutation symmetries of the system-bath coupling are examined within this framework. It is shown that in both cases universal quantum computation which always preserves the DFS (natural fault-tolerant computation) can be performed using only two-body interactions. This is in marked contrast to standard error correcting codes, where all known constructions using one- or two-body interactions must leave the code space during the on-time of the fault-tolerant gates. A further consequence of our universality construction is that a single exchange Hamiltonian can be used to perform universal quantum computation on an encoded space whose asymptotic coding efficiency is unity. The exchange Hamiltonian, which is naturally present in many quantum systems, is thus asymptotically universal.

AB - Universal quantum computation on decoherence-free subspaces and subsystems (DFSs) is examined with particular emphasis on using only physically relevant interactions. A necessary and sufficient condition for the existence of decoherence-free (noiseless) subsystems in the Markovian regime is derived here for the first time. A stabilizer formalism for DFSs is then developed which allows for the explicit understanding of these in their dual role as quantum error correcting codes. Conditions for the existence of Hamiltonians whose induced evolution always preserves a DFS are derived within this stabilizer formalism. Two possible collective decoherence mechanisms arising from permutation symmetries of the system-bath coupling are examined within this framework. It is shown that in both cases universal quantum computation which always preserves the DFS (natural fault-tolerant computation) can be performed using only two-body interactions. This is in marked contrast to standard error correcting codes, where all known constructions using one- or two-body interactions must leave the code space during the on-time of the fault-tolerant gates. A further consequence of our universality construction is that a single exchange Hamiltonian can be used to perform universal quantum computation on an encoded space whose asymptotic coding efficiency is unity. The exchange Hamiltonian, which is naturally present in many quantum systems, is thus asymptotically universal.

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U2 - 10.1103/PhysRevA.63.042307

DO - 10.1103/PhysRevA.63.042307

M3 - Article

VL - 63

SP - 1

EP - 29

JO - Physical Review A

JF - Physical Review A

SN - 2469-9926

IS - 4

M1 - 042307

ER -