Hamiltonian quantum computation is a form ofquantum computing. Unlike methods of quantum computation such as theadiabatic,measurement-based andcircuit model where eternal control is used to apply operations on a register of qubits, Hamiltonian quantum computers operate without external control.^[1][2][3]

Hamiltonian quantum computation was the pioneering model of quantum computation, first proposed byPaul Benioff in 1980.Benioff's motivation for building a quantum mechanical model of a computer was to have a quantum mechanical description ofartificial intelligence and to create a computer that would dissipate theleast amount of energy allowable by the laws of physics.^[1] However, his model was not time-independent andlocal.^[4]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 acontinuous-time quantum walk that could perform universal quantum computation.^[2]Superconducting qubits,^[5]Ultracold atoms andnon-linear photonics^[6] have been proposed as potential experimental implementations of Hamiltonian quantum computers.

Given a list of quantum gates described as unitaries${\displaystyle U_1,U_2...U_k}$, define a hamiltonian

${\displaystyle H=\sum _{i=1}^{k-1}|i+1⟩⟨i|\otimes U_{i+1}+|i⟩⟨i+1|\otimes U_{i+1}^{†}}$

Evolving this Hamiltonian on a state${\displaystyle |{\varphi }_0⟩=|\mathrm{100..00}⟩\otimes |{\psi }_0⟩}$ composed of a clock register (${\displaystyle |\mathrm{100..00}⟩}$) that constaines${\displaystyle k+1}$ qubits and a data register (${\displaystyle |{\psi }_0⟩}$) will output${\displaystyle |{\varphi }_k⟩=e^{-iHt}|{\varphi }_0⟩}$. At a time${\displaystyle t}$, the state of the clock register can be${\displaystyle |\mathrm{000..01}⟩}$. When that happens, the state of the data register will be${\displaystyle U_1,U_2...U_k|{\psi }_0⟩}$. The computation is complete and${\displaystyle |{\varphi }_k⟩=|\mathrm{000..01}⟩\otimes U_1,U_2...U_k|{\psi }_0⟩}$.^[7]

• Adiabatic quantum computation
• One-way quantum computer

• Quantum computing
• Quantum information science

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