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.4 However, his model was not time-independent and local.5 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.6 Superconducting qubits,7 Ultracold atoms and non-linear photonics8 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 {\displaystyle U_{1},U_{2}...U_{k}} , define a hamiltonian
H = ∑ i = 1 k − 1 | i + 1 ⟩ ⟨ i | ⊗ U i + 1 + | i ⟩ ⟨ i + 1 | ⊗ U i + 1 † {\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 | ϕ 0 ⟩ = | 100..00 ⟩ ⊗ | ψ 0 ⟩ {\displaystyle |\phi _{0}\rangle =|100..00\rangle \otimes |\psi _{0}\rangle } composed of a clock register ( | 100..00 ⟩ {\displaystyle |100..00\rangle } ) that constaines k + 1 {\displaystyle k+1} qubits and a data register ( | ψ 0 ⟩ {\displaystyle |\psi _{0}\rangle } ) will output | ϕ k ⟩ = e − i H t | ϕ 0 ⟩ {\displaystyle |\phi _{k}\rangle =e^{-iHt}|\phi _{0}\rangle } . At a time t {\displaystyle t} , the state of the clock register can be | 000..01 ⟩ {\displaystyle |000..01\rangle } . When that happens, the state of the data register will be U 1 , U 2 . . . U k | ψ 0 ⟩ {\displaystyle U_{1},U_{2}...U_{k}|\psi _{0}\rangle } . The computation is complete and | ϕ k ⟩ = | 000..01 ⟩ ⊗ U 1 , U 2 . . . U k | ψ 0 ⟩ {\displaystyle |\phi _{k}\rangle =|000..01\rangle \otimes U_{1},U_{2}...U_{k}|\psi _{0}\rangle } .9
See also
References
Benioff Paul (1980). "The computer as a physical system: A microscopic quantum mechanical Hamiltonian model of computers as represented by Turing machines". Journal of Statistical Physics. 22 (5): 563–591. Bibcode:1980JSP....22..563B. doi:10.1007/BF01011339. /wiki/Bibcode_(identifier) ↩
Feynman, Richard P. (1986). "Quantum mechanical computers". Foundations of Physics. 16 (6): 507–531. Bibcode:1986FoPh...16..507F. doi:10.1007/BF01886518. /wiki/Bibcode_(identifier) ↩
Janzing, Dominik (2007). "Spin-1∕2 particles moving on a two-dimensional lattice with nearest-neighbor interactions can realize an autonomous quantum computer". Physical Review A. 75 (1): 012307. arXiv:quant-ph/0506270. doi:10.1103/PhysRevA.75.012307. /wiki/ArXiv_(identifier) ↩
Benioff Paul (1980). "The computer as a physical system: A microscopic quantum mechanical Hamiltonian model of computers as represented by Turing machines". Journal of Statistical Physics. 22 (5): 563–591. Bibcode:1980JSP....22..563B. doi:10.1007/BF01011339. /wiki/Bibcode_(identifier) ↩
LLoyd, Seth (1993). "Review of quantum computation". Vistas in Astronomy. 37: 291–295. doi:10.1016/0083-6656(93)90051-K. /wiki/Doi_(identifier) ↩
Feynman, Richard P. (1986). "Quantum mechanical computers". Foundations of Physics. 16 (6): 507–531. Bibcode:1986FoPh...16..507F. doi:10.1007/BF01886518. /wiki/Bibcode_(identifier) ↩
Ciani, A.; Terhal, B. M.; DiVincenzo, D. P. (2019). "Hamiltonian quantum computing with superconducting qubits". IOP Publishing. 4 (3): 035002. arXiv:1310.5100. doi:10.1088/2058-9565/ab18dd. /wiki/ArXiv_(identifier) ↩
Lahini, Yoav; Steinbrecher, Gregory R.; Bookatz, Adam D.; Englund, Dirk (2018). "Quantum logic using correlated one-dimensional quantum walks". npj Quantum Information. 4 (1): 2. arXiv:1501.04349. doi:10.1038/s41534-017-0050-2. /wiki/ArXiv_(identifier) ↩
Costales, R. J.; Gunning, A.; Dorlas, T. (2025). "Efficiency of Feynman's quantum computer". Physical Review A. 111 (2): 022615. arXiv:2309.09331. doi:10.1103/PhysRevA.111.022615. /wiki/ArXiv_(identifier) ↩