Hamiltonian quantum computation

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

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

Evolving this Hamiltonian on a state ${\displaystyle |\phi _{0}\rangle =|100..00\rangle \otimes |\psi _{0}\rangle }$ composed of a clock register ( ${\displaystyle |100..00\rangle }$) that constaines ${\displaystyle k+1}$ qubits and a data register (${\displaystyle |\psi _{0}\rangle }$) will output ${\displaystyle |\phi _{k}\rangle =e^{-iHt}|\phi _{0}\rangle }$. At a time ${\displaystyle t}$, the state of the clock register can be ${\displaystyle |000..01\rangle }$. When that happens, the state of the data register will be ${\displaystyle U_{1},U_{2}...U_{k}|\psi _{0}\rangle }$. The computation is complete and ${\displaystyle |\phi _{k}\rangle =|000..01\rangle \otimes U_{1},U_{2}...U_{k}|\psi _{0}\rangle }$.^[7]

• Adiabatic quantum computation
• One-way quantum computer