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.^{[1] [2] [3]}

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.^[1] However, his model was not time-independent and local.^[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 a continuous-time quantum walk that could perform universal quantum computation.^[2] Superconducting qubits,^[5] Ultracold atoms and non-linear photonics ^[6] have been proposed as potential experimental implementations of Hamiltonian quantum computers.

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

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

• Quantum computing
• Quantum information science

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