Hamiltonian quantum computation

Hamiltonian quantum computation is a form of quantum computing. Unlike methods of quantum computation such as the adiabatic, measurement-based and circuit model where eternal control is used to apply operations on a register of qubits, Hamiltonian quantum computers operate without external control.

Background

Hamiltonian quantum computation was the pioneering model of quantum computation, first proposed by Paul Benioff in 1980. Benioff's motivation for building a quantum mechanical model of a computer was to have a quantum mechanical description of artificial intelligence and to create a computer that would dissipate the least amount of energy allowable by the laws of physics. However, his model was not time-independent and local. Richard Feynman, independent of Benioff, also wanted to provide a description of a computer based on the laws of quantum physics. He solved the problem of a time-independent and local Hamiltonian by proposing a continuous-time quantum walk that could perform universal quantum computation. Superconducting qubits, Ultracold atoms and non-linear photonics have been proposed as potential experimental implementations of Hamiltonian quantum computers.

Definition

Given a list of quantum gates described as unitaries $U_{1},U_{2}...U_{k}$ , define a hamiltonian

$H=\sum _{i=1}^{k-1}|i 1\rangle \langle i|\otimes U_{i 1} |i\rangle \langle i 1|\otimes U_{i 1}^{\dagger }$

Evolving this Hamiltonian on a state $|\phi _{0}\rangle =|100..00\rangle \otimes |\psi _{0}\rangle$ composed of a clock register ( $|100..00\rangle$ ) that constaines $k 1$ qubits and a data register ( $|\psi _{0}\rangle$ ) will output $|\phi _{k}\rangle =e^{-iHt}|\phi _{0}\rangle$ . At a time $t$ , the state of the clock register can be $|000..01\rangle$ . When that happens, the state of the data register will be $U_{1},U_{2}...U_{k}|\psi _{0}\rangle$ . The computation is complete and $|\phi _{k}\rangle =|000..01\rangle \otimes U_{1},U_{2}...U_{k}|\psi _{0}\rangle$ .