Hamiltonian quantum computation

<h2 id="background">Background</h2>
<p>Hamiltonian quantum computation was the pioneering model of quantum computation, first proposed by <a href="/facts/Paul_Benioff/G53CYo26">Paul Benioff</a> in 1980.
Benioff's motivation for building a quantum mechanical model of a computer was to have a quantum mechanical description of <a href="/facts/Artificial_intelligence/lJGauQwX">artificial intelligence</a> and to create a computer that would dissipate the <a href="/facts/Reversible_computing/yEum13Vs">least amount of energy allowable by the laws of physics</a>.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> However, his model was not time-independent and <a href="/facts/Principle_of_locality/umaVy92L">local</a>.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> <a href="/facts/Richard_Feynman/32vGFrtB">Richard Feynman</a>, 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 <a href="/facts/Continuous-time_quantum_walk/wVIHq3Aa">continuous-time quantum walk</a> that could perform universal quantum computation.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> <a href="/facts/Superconducting_quantum_computing/3Dj2BN2v">Superconducting qubits</a>,<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> <a href="/facts/Ultracold_atom/LdM0IeD7">Ultracold atoms</a> and <a href="/facts/Nonlinear_optics/maerngXd">non-linear photonics</a><a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> have been proposed as potential experimental implementations of Hamiltonian quantum computers.
</p>
<h2 id="definition">Definition</h2>
<p>Given a list of quantum gates described as unitaries 
  
    
      
        
          U
          
            1
          
        
        ,
        
          U
          
            2
          
        
        .
        .
        .
        
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    {\displaystyle U_{1},U_{2}...U_{k}}
  
, define a hamiltonian
</p><p>
  
    
      
        H
        =
        
          ∑
          
            i
            =
            1
          
          
            k
            −
            1
          
        
        
          |
        
        i
        +
        1
        ⟩
        ⟨
        i
        
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        ⊗
        
          U
          
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            +
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        +
        
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        ⟨
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            †
          
        
      
    
    {\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 }}

</p><p>Evolving this Hamiltonian on a state 
  
    
      
        
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          ϕ
          
            0
          
        
        ⟩
        =
        
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        100..00
        ⟩
        ⊗
        
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          ψ
          
            0
          
        
        ⟩
      
    
    {\displaystyle |\phi _{0}\rangle =|100..00\rangle \otimes |\psi _{0}\rangle }
  
 composed of a clock register ( 
  
    
      
        
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        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 
  
    
      
        
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            −
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            H
            t
          
        
        
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          ϕ
          
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    {\displaystyle |\phi _{k}\rangle =e^{-iHt}|\phi _{0}\rangle }
  
. At a time 
  
    
      
        t
      
    
    {\displaystyle t}
  
, the state of the clock register can be 
  
    
      
        
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        000..01
        ⟩
      
    
    {\displaystyle |000..01\rangle }
  
. When that happens, the state of the data register will be 
  
    
      
        
          U
          
            1
          
        
        ,
        
          U
          
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        .
        .
        .
        
          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 }
  
.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a>
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Adiabatic_quantum_computation/USqCIzRf">Adiabatic quantum computation</a></li>
<li><a href="/facts/One-way_quantum_computer/w3vd4DFA">One-way quantum computer</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>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. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Feynman, Richard P. (1986). "Quantum mechanical computers". Foundations of Physics. 16 (6): 507–531. Bibcode:1986FoPh...16..507F. doi:10.1007/BF01886518. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>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. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>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. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>LLoyd, Seth (1993). "Review of quantum computation". Vistas in Astronomy. 37: 291–295. doi:10.1016/0083-6656(93)90051-K. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Feynman, Richard P. (1986). "Quantum mechanical computers". Foundations of Physics. 16 (6): 507–531. Bibcode:1986FoPh...16..507F. doi:10.1007/BF01886518. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>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. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>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. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
<li id="fn:9"><p>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. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li>
</ol>

Hamiltonian quantum computation open-in-new

Hamiltonian quantum computation