Hamiltonian quantum computation

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.{{cite journal| title= Review of quantum computation| journal= Vistas in Astronomy| volume=37| pages=291–295|author1=LLoyd, Seth|doi=10.1016/0083-6656(93)90051-K| year=1993}} 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,{{cite journal| title=Hamiltonian quantum computing with superconducting qubits| journal=IOP Publishing| volume=4| issue=3| pages=035002| arxiv=1310.5100|author1=Ciani, A. |author2= Terhal, B. M. |author3=DiVincenzo, D. P. |doi=10.1088/2058-9565/ab18dd| year=2019}} Ultracold atoms and non-linear photonics{{cite journal| title=Quantum logic using correlated one-dimensional quantum walks| journal=npj Quantum Information| volume=4| issue=1| pages=2| arxiv=1501.04349|author1=Lahini, Yoav|author2=Steinbrecher, Gregory R. |author3= Bookatz, Adam D.| author4=Englund, Dirk |doi=10.1038/s41534-017-0050-2| year=2018}} have been proposed as potential experimental implementations of Hamiltonian quantum computers.

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

$H\; =\; \backslash sum\_\{i=1\}^\{k-1\}\; |i+1\backslash rangle\backslash langle\; i|\; \backslash otimes\; U\_\{i+1\}\; +\; |i\backslash rangle\backslash langle\; i\; +\; 1|\; \backslash otimes\; U\_\{i+1\}^\{\backslash dagger\}$

Evolving this Hamiltonian on a state $|\backslash phi\_\{0\}\backslash rangle\; =\; |100..00\backslash rangle\; \backslash otimes\; |\backslash psi\_\{0\}\backslash rangle$ composed of a clock register ( $|100..00\backslash rangle$) that constaines $k+1$ qubits and a data register ($|\backslash psi\_\{0\}\backslash rangle$) will output $|\backslash phi\_\{k\}\backslash rangle\; =\; e^\{-\; iHt\}|\backslash phi\_\{0\}\backslash rangle$. At a time $t$, the state of the clock register can be $|000..01\backslash rangle$. When that happens, the state of the data register will be $U\_\{1\},\; U\_\{2\}...\; U\_\{k\}\; |\backslash psi\_\{0\}\backslash rangle$. The computation is complete and $|\backslash phi\_\{k\}\backslash rangle\; =\; |000..01\backslash rangle\; \backslash otimes\; U\_\{1\},\; U\_\{2\}...\; U\_\{k\}\; |\backslash psi\_\{0\}\backslash rangle$.{{cite journal| title=Efficiency of Feynman's quantum computer| journal=Physical Review A| volume=111| issue=2| pages=022615| arxiv=2309.09331|author1=Costales, R. J.|author2= Gunning, A.|author3=Dorlas, T.|doi=10.1103/PhysRevA.111.022615| year=2025}}

• Adiabatic quantum computation
• One-way quantum computer

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Category:Quantum computing

Category:Quantum information science