## Abstract

The circuit-to-Hamiltonian construction translates dynamics (a quantum circuit and its output) into statics (the groundstate of a circuit Hamiltonian) by explicitly defining a quantum register for a clock. The standard Feynman-Kitaev construction uses one global clock for all qubits while we consider a different construction in which a clock is assigned to each interacting qubit. This makes it possible to capture the spatio-temporal structure of the original quantum circuit into features of the circuit Hamiltonian. The construction is inspired by the original two-dimensional interacting fermion model in Mizel etal (2001 Phys. Rev. A 63 040302). We prove that for one-dimensional quantum circuits the gap of the circuit Hamiltonian is appropriately lowerbounded so that the applications of this construction for quantum Merlin-Arthur (and partially for quantum adiabatic computation) go through. For one-dimensional quantum circuits, the dynamics generated by the circuit Hamiltonian corresponds to the diffusion of a string around the torus.

Original language | English |
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Article number | 195304 |

Number of pages | 24 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Volume | 47 |

Issue number | 19 |

DOIs | |

Publication status | Published - 2014 |

Externally published | Yes |

## Keywords

- Heisenberg model
- Markov chains
- quantum complexity
- quantum information