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]}

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. ^[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.

Definition

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]

See also

• Adiabatic quantum computation
• One-way quantum computer

References

1. 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. doi:10.1007/BF01011339.
2. Feynman, Richard P. (1986). "Quantum mechanical computers". Foundations of Physics. 16 (6): 507–531. doi:10.1007/BF01886518.
3. 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.
4. LLoyd, Seth (1993). "Review of quantum computation". Vistas in Astronomy. 37: 291–295. doi:10.1016/0083-6656(93)90051-K.
5. 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.
6. 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.
7. 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.